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835c2398f7e8c8249c224d48e7cfa05006ad2d48 | abstract | 0 | 29 | Abstract | We propose a variant of the classical augmented Lagrangian method for
constrained optimization problems in Banach spaces. Our theoretical framework
does not require any convexity or second-order assumptions and allows the
treatment of inequality constraints with infinite-dimensional image space.
Moreover, we discuss th... | {
"cite_spans": []
} | 10.1137/16M1107103 | 1807.04467 | An Augmented Lagrangian Method for Optimization Problems in Banach
Spaces | [
"Christian Kanzow",
"Daniel Steck",
"Daniel Wachsmuth"
] | [
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24f17002cb59f835a0ffe18e5c9a310152e57553 | subsection | 1 | 29 | Introduction | Let X, Y be (real) Banach spaces and let f:X\rightarrow \mathbb {R}, g:X\rightarrow Y be
given mappings. The aim of this paper is to describe an augmented Lagrangian
method for the solution of the constrained optimization problem\min \ f(x) \quad \text{subject to (s.t.)}\quad g(x)\le 0.We assume that Y\hookrightarrow L... | {
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"raw": "D. Bertsekas. Nonlinear Programming. Athena Scientific, 1995.",
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... | 10.1137/16M1107103 | 1807.04467 | An Augmented Lagrangian Method for Optimization Problems in Banach
Spaces | [
"Christian Kanzow",
"Daniel Steck",
"Daniel Wachsmuth"
] | [
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] | 2,018 | en | Mathematics | [
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c6774eeaa31db2b0bba9c7a4ebbb16cde32a113d | subsection | 2 | 29 | Introduction | The norms on X, Y, etc. are
denoted by \Vert \cdot \Vert , where an index (as in \Vert \cdot \Vert _X) is appended if necessary.
Furthermore, we write \rightarrow , \rightharpoonup , and \rightharpoonup ^* for strong, weak, and weak-^*
convergence, respectively. Finally, we use the abbreviation lsc for a lower
semicont... | {
"cite_spans": []
} | 10.1137/16M1107103 | 1807.04467 | An Augmented Lagrangian Method for Optimization Problems in Banach
Spaces | [
"Christian Kanzow",
"Daniel Steck",
"Daniel Wachsmuth"
] | [
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9442428e98990dfcb9863210809d1c637a7cbad6 | subsection | 3 | 29 | Preliminaries and Assumptions | We denote by e:Y\rightarrow Z the (linear and continuous) dense embedding of Y into
Z:=L^2(\Omega ), and by K_Y, K_Z the respective nonnegative cones in
Y and Z, i.e.K_Z:=\lbrace z\in Z\mid z(t)\ge 0~\text{a.e.}\rbrace \quad \text{and}\quad K_Y:= \lbrace y\in Y \mid e(y) \in K_Z\rbrace .Note that the adjoint mapping e^... | {
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"raw": "C. Baiocchi and A. Capelo. Variational and Quasivariational Inequalities. John Wiley & Sons, Inc., New York, 1984.",
"source_ref_id": "b8dc03b226a5c6f4e5aceb781... | 10.1137/16M1107103 | 1807.04467 | An Augmented Lagrangian Method for Optimization Problems in Banach
Spaces | [
"Christian Kanzow",
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f1ab019fe369649a5766d80f63daae0213b96fdc | subsection | 4 | 29 | Preliminaries and Assumptions | Hence, if \Vert g_+\Vert is convex (which is true if g is convex with respect to the order in Y), then
the (strong) lower semicontinuity of g already implies the weak lower
semicontinuity. We conclude that (A1) holds, in particular, for every
lsc. convex function f and any mapping g\in \mathcal {L}(X,Y).On a further no... | {
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Spaces | [
"Christian Kanzow",
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60da3cf59d9c7494d114e3685aee14fef517830e | subsection | 5 | 29 | Preliminaries and Assumptions | For instance, consider the case where \Omega =\lbrace 1\rbrace and
Z=L^2(\Omega ), which can be identified with \mathbb {R}. Then the sequences
a^k=k and b^k=1/k provide a simple counterexample. | {
"cite_spans": []
} | 10.1137/16M1107103 | 1807.04467 | An Augmented Lagrangian Method for Optimization Problems in Banach
Spaces | [
"Christian Kanzow",
"Daniel Steck",
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eb0f67bdbb7a5023f11763a778802f595b66bc66 | subsection | 6 | 29 | An Augmented Lagrangian Method | This section gives a detailed statement of our augmented Lagrangian
method for the solution of the optimization problem (REF ).
It is motivated by the finite-dimensional discussion in, e.g.,
and differs from the traditional augmented
Lagrangian method as applied, e.g., in , to a class
of infinite-dimensional problems,... | {
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"raw": "E. G. Birgin and J. M. Martínez. Practical Augmented Lagrangian Methods for Constrained Optimization. Society for Industrial and Applied Mathema... | 10.1137/16M1107103 | 1807.04467 | An Augmented Lagrangian Method for Optimization Problems in Banach
Spaces | [
"Christian Kanzow",
"Daniel Steck",
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047382410681af3ef9dc5009574dd699071d0e5c | subsection | 7 | 29 | An Augmented Lagrangian Method | Going a
little further, our method also includes the Moreau-Yosida regularization scheme
(see , and Section ) as a special
case, which arises if (w^k) is chosen as a constant sequence. However, the
most natural choice, which also brings the method closer to traditional augmented
Lagrangian schemes, is w^k:=\min \lbrace... | {
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Spaces | [
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7654f86924960682501f2cfe947ba575ad46a01c | subsection | 8 | 29 | Global Minimization | We begin by considering Algorithm REF from a global optimization
perspective. Note that most of the analysis in this section can be carried
out in the more general case where f is an extended real-valued function,
i.e. f maps to \mathbb {R}\cup \lbrace +\infty \rbrace .The global optimization perspective is particularl... | {
"cite_spans": []
} | 10.1137/16M1107103 | 1807.04467 | An Augmented Lagrangian Method for Optimization Problems in Banach
Spaces | [
"Christian Kanzow",
"Daniel Steck",
"Daniel Wachsmuth"
] | [
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9cfbf62dd0ef0291621a5f640f345a0db0d9bc07 | subsection | 9 | 29 | Global Minimization | Let \mathcal {K}\subset \mathbb {N}
be such that x^{k+1}\rightharpoonup _{\mathcal {K}}\bar{x} and assume that there is an
x\in X with \Vert g_+(x)\Vert _Z^2<\Vert g_+(\bar{x})\Vert _Z^2.
By (REF ), the boundedness of (w^k), and the fact
that \rho _k\rightarrow \infty , there is a constant c>0 such that\left\Vert \left... | {
"cite_spans": []
} | 10.1137/16M1107103 | 1807.04467 | An Augmented Lagrangian Method for Optimization Problems in Banach
Spaces | [
"Christian Kanzow",
"Daniel Steck",
"Daniel Wachsmuth"
] | [
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a3b6a38612babdc3acd23dcc58a29ac6e8b6c2d7 | subsection | 10 | 29 | Global Minimization | Using the feasibility
of x and a similar inequality to above, it follows thatf(x^{k+1})+\frac{\rho _k}{2} \left\Vert \left( g(x^{k+1})+
\frac{w^k}{\rho _k} \right)_+ \right\Vert _Z^2 \le f(x)+\frac{\rho _k}{2}\left\Vert \frac{w^k}{\rho _k}\right\Vert _Z^2+\varepsilon _k.But\left( g(x^{k+1})+\frac{w^k}{\rho _k} \right)_... | {
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Spaces | [
"Christian Kanzow",
"Daniel Steck",
"Daniel Wachsmuth"
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961a48f847780f0ec23d735d01cee6ad1b9a4562 | subsection | 11 | 29 | Global Minimization | Therefore, existence and uniqueness of the solution \bar{x} follow from
standard arguments.Now, denoting by c>0 the modulus of convexity of f, it follows that\frac{c}{8} \Vert x^{k+1}-\bar{x}\Vert _X^2 \le \frac{f(x^{k+1})+f(\bar{x})}{2}-
f( \frac{x^{k+1}+\bar{x}}{2} )for all k. By the proof of Theorem REF (b), it is e... | {
"cite_spans": []
} | 10.1137/16M1107103 | 1807.04467 | An Augmented Lagrangian Method for Optimization Problems in Banach
Spaces | [
"Christian Kanzow",
"Daniel Steck",
"Daniel Wachsmuth"
] | [
"math.OC"
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f64ec1d39d0355868b31bd34b96d58ca0438f9a4 | subsection | 12 | 29 | Sequential KKT conditions | Throughout this section, we assume that f and g are continuously
Fréchet-differentiable on X, and discuss the KKT conditions of the
optimization problem (REF ). Recalling that K_Y is the nonnegative
cone in Y, we denote byK_Y^+ := \lbrace f\in Y^* \mid \left\langle f,y \right\rangle \ge 0~
\forall y\in K_Y \rbraceits d... | {
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Spaces | [
"Christian Kanzow",
"Daniel Steck",
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fcaa62f7335da7b0af90efd5e4820c57df9e8722 | subsection | 13 | 29 | Sequential KKT conditions | Due to (x^k)\subset B_r(\bar{x}),
there is a \mathcal {K}\subset \mathbb {N} such that x^k\rightharpoonup _{\mathcal {K}}\bar{y}
for some \bar{y}\in B_r(\bar{x}). Since x^k is a solution of
(REF ), we havef(x^k)+k\Vert g_+(x^k)\Vert _Z^2+\Vert x^k-\bar{x}\Vert _X^2 \le f(\bar{x})for every k. Dividing by k and taking th... | {
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Spaces | [
"Christian Kanzow",
"Daniel Steck",
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6a1748eac7260116154f1fbdec4d1004befb1205 | subsection | 14 | 29 | Sequential KKT conditions | However, in the infinite-dimensional
setting, our choice of constraint qualification is much more restricted. For
instance, we are not aware of any infinite-dimensional analogues of the (very
amenable) CPLD condition. Hence, we have decided to employ the Zowe-Kurcyusz
regularity condition , which is known to be equival... | {
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8801bd7a1b8f4262037c4a105bf087996fac3a1e | subsection | 15 | 29 | Sequential KKT conditions | By the AKKT conditions and REF , there is a
k_0\in \mathbb {N} such that\Vert g(x^k)-g(x)\Vert _Y\le \frac{r}{4}
\quad \text{and}\quad \Vert g^{\prime }(x^k)-g^{\prime }(x)\Vert _{\mathcal {L}(X,Y)}\le \frac{r}{4}for every k\ge k_0. Now, let u\in B_r^Y and k\ge k_0. It follows that
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"cite_spans": []
} | 10.1137/16M1107103 | 1807.04467 | An Augmented Lagrangian Method for Optimization Problems in Banach
Spaces | [
"Christian Kanzow",
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f4522f2b5faeebf6f239aced890383513dd77e2a | subsection | 16 | 29 | Sequential KKT conditions | We conclude that\Vert \lambda ^k\Vert _{Y^*}=\sup _{\Vert u\Vert \le r}\left\langle \lambda ^k,\frac{1}{r}u \right\rangle \le \frac{1}{r}
\left(C+\frac{r}{2}\Vert \lambda ^k\Vert _{Y^*}\right)and, hence, \Vert \lambda ^k\Vert _{Y^*}\le 2C/r.(b): Since (\lambda ^k) is bounded in Y^* and the unit ball in Y^* is
weak-^* s... | {
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0.051... |
bba8e39c0fc1133726f55bc5a6c01f70f88ecb76 | subsection | 17 | 29 | Convergence to KKT Points | We now discuss the convergence properties of Algorithm REF
from the perspective of KKT points. To this end, we make the following
assumption.Assumption 6.1
In Step 2 of Algorithm REF , we obtain x^{k+1} such that
L_{\rho _k}^{\prime }(x^{k+1},w^k)\rightarrow 0 as k\rightarrow \infty .The above is a very natural assum... | {
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"doi": "10.1007/978-3-319-48311-5",
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"raw": "H. H. Bauschke and P. L. Combettes. Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York, 2011.",
"source... | 10.1137/16M1107103 | 1807.04467 | An Augmented Lagrangian Method for Optimization Problems in Banach
Spaces | [
"Christian Kanzow",
"Daniel Steck",
"Daniel Wachsmuth"
] | [
"math.OC"
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b2aeb732a4ee750ade0d528cfdc5e7e470e8b827 | subsection | 18 | 29 | Convergence to KKT Points | To this end, recall that Assumption REF implies thatf^{\prime }(x^{k+1})+g^{\prime }(x^{k+1})^*\lambda ^{k+1}\rightarrow 0,which already suggests that the sequence of tuples (x^k,\lambda ^k) satisfies
AKKT for the optimization problem (REF ). In fact, the only missing
ingredient is the asymptotic complementarity of g a... | {
"cite_spans": []
} | 10.1137/16M1107103 | 1807.04467 | An Augmented Lagrangian Method for Optimization Problems in Banach
Spaces | [
"Christian Kanzow",
"Daniel Steck",
"Daniel Wachsmuth"
] | [
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7bb8f4896b334fda7301d111792f09f6435a11b2 | subsection | 19 | 29 | Convergence to KKT Points | Now, the claim essentially follows from Theorem REF (b),
the only difference here is that we are working in the Hilbert space
Z instead of Y or Y^* , hence the two conditions REF and
REF formally required in Theorem REF (b) are
automatically satisfied in the current Hilbert space situation.Some further remarks about t... | {
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"source_r... | 10.1137/16M1107103 | 1807.04467 | An Augmented Lagrangian Method for Optimization Problems in Banach
Spaces | [
"Christian Kanzow",
"Daniel Steck",
"Daniel Wachsmuth"
] | [
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] | 2,018 | en | Mathematics | [
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033b1f4fb37c0c9edcc52d50380cbcf584de6a8f | subsection | 20 | 29 | Convergence to KKT Points | In this case, the pointwise convergence
implies that w^k(t)+\rho _k g(x^{k+1})(t)<0 for sufficiently large k and,
hence, v^k(t)=0 for all such k.Case 2. g(\bar{x})(t)=0. Then consider a fixed k \in \mathcal {K} . If g(x^{k+1})(t)\ge 0, it follows again from the definition
of v^k that v^k(t)=0. On the other hand, if g(x... | {
"cite_spans": []
} | 10.1137/16M1107103 | 1807.04467 | An Augmented Lagrangian Method for Optimization Problems in Banach
Spaces | [
"Christian Kanzow",
"Daniel Steck",
"Daniel Wachsmuth"
] | [
"math.OC"
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70af1a901cdb4cafee0cfcac519981af64dffeec | subsection | 21 | 29 | Applications | We now give some applications and numerical results for Algorithm REF . To this end, we consider some standard problems from the literature. Apart from the first example, we place special emphasis on nonlinear and nonconvex problems since the appropriate treatment of these is one of the focal points of our method.All o... | {
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"raw": "M. Hintermüller and K. Kunisch. Feasible and noninterior path-following in constrained minimization with low multiplier regularity. SIAM J. Control Optim., 45(4):1198–1221, 2006.",
"source_re... | 10.1137/16M1107103 | 1807.04467 | An Augmented Lagrangian Method for Optimization Problems in Banach
Spaces | [
"Christian Kanzow",
"Daniel Steck",
"Daniel Wachsmuth"
] | [
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a73de8f9aedd471f31e69207c3fc86d18c2c9bca | subsection | 22 | 29 | The Obstacle Problem | We consider the well-known obstacle problem , . To this end, let \Omega \subseteq \mathbb {R}^d be a bounded domain, and let X:=Y:=H_0^1(\Omega ), Z:=L^2(\Omega ). The obstacle problem considers the minimization problem\min \ f(u) \quad \text{s.t.}\quad u\ge \psi ,where f(u):=\Vert \nabla u\Vert _{L^2(\Omega )}^2 and \... | {
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"doi": "10.1007/bf01445164",
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"raw": "K. Ito and K. Kunisch. An augmented Lagrangian technique for variational inequalities. Appl. Math. Optim., 21(3):223–241, 1990.",
"source_ref_i... | 10.1137/16M1107103 | 1807.04467 | An Augmented Lagrangian Method for Optimization Problems in Banach
Spaces | [
"Christian Kanzow",
"Daniel Steck",
"Daniel Wachsmuth"
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ffe856df2a17cb446a3467092265670dd719c894 | subsection | 23 | 29 | The Obstacle Problem | The subproblems occurring in Algorithm REF are unconstrained minimization problems which we solve by means of a standard semismooth Newton method.
[Table: Numerical results for the obstacle problem.]Table REF contains the inner and outer iteration numbers together with the final penalty parameter for different values o... | {
"cite_spans": []
} | 10.1137/16M1107103 | 1807.04467 | An Augmented Lagrangian Method for Optimization Problems in Banach
Spaces | [
"Christian Kanzow",
"Daniel Steck",
"Daniel Wachsmuth"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
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1119e6d5f35b84a1d3c8ffd6105c1261f7611cf7 | subsection | 24 | 29 | The Obstacle Bratu Problem | Let us briefly consider the obstacle Bratu problem , , which we simply refer to as Bratu problem. This is a non-quadratic and nonconvex problem which differs from (REF ) in the choice of objective function. To this end, letf(u):=\Vert \nabla u\Vert _{L^2(\Omega )}^2 - \alpha \int _{\Omega } e^{-u(x)} dxfor some fixed \... | {
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"doi": "10.1007/b97544",
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"... | 10.1137/16M1107103 | 1807.04467 | An Augmented Lagrangian Method for Optimization Problems in Banach
Spaces | [
"Christian Kanzow",
"Daniel Steck",
"Daniel Wachsmuth"
] | [
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] | 2,018 | en | Mathematics | [
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97e93280043e0fde1926042c7051ce875f8d7b2c | subsection | 25 | 29 | The Obstacle Bratu Problem | Let us writee^{u+h}-e^u-e^uh = \int _0^1 \int _0^1 e^{u+sth}sh^2 \ dt \ ds,which implies\Vert e^{u+h}-e^u-e^uh\Vert _{L^1(\Omega )} \le \frac{1}{2}\Vert e^{u+|h|}\Vert _{L^2(\Omega )} \Vert h\Vert _{L^4(\Omega )}^2
\le c\Vert e^{u+|h|}\Vert _{L^2(\Omega )} \Vert h\Vert _{H^1(\Omega )}^2.Using the boundedness property o... | {
"cite_spans": []
} | 10.1137/16M1107103 | 1807.04467 | An Augmented Lagrangian Method for Optimization Problems in Banach
Spaces | [
"Christian Kanzow",
"Daniel Steck",
"Daniel Wachsmuth"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
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1baafd76f0eddb44db7af031439bb357afb9a074 | subsection | 26 | 29 | Optimal Control Problems | We now turn to a class of optimal control problems subject to a semilinear elliptic equation. Let \Omega \subseteq \mathbb {R}^d, d=2,3, be a bounded Lipschitz domain.
The control problem we consider consists of minimizing the functionalJ(y,u):=\frac{1}{2}\Vert y-y_d\Vert _{L^2(\Omega )}^2 + \frac{\alpha }{2}\Vert u\Ve... | {
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"arxiv_id": "",
"doi": "10.1090/gsm/112",
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Spaces | [
"Christian Kanzow",
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"Daniel Wachsmuth"
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fb241414526e157ac87934060856fcc3f3e603f9 | subsection | 27 | 29 | Optimal Control Problems | By reintroducing the state variable y, we can write these subproblems as\min \ J(y,u)+\frac{\rho _k}{2}\left\Vert \left( y_c-y+\frac{w^k}{\rho _k} \right)_+ \right\Vert ^2
\quad \text{s.t.}\quad y=S(u).Hence, we have transformed (REF ) into a sequence of optimal control problems which include the state equation but not... | {
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Spaces | [
"Christian Kanzow",
"Daniel Steck",
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11a9fd845fa22e5520c608139fcf032888497abc | subsection | 28 | 29 | Final Remarks | We have presented an augmented Lagrangian method for the solution of optimization problems in Banach spaces, which is essentially a generalization of the modified augmented Lagrangian method from . Furthermore, we have shown how the method can be applied to well-known problem classes, and the corresponding numerical re... | {
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8e2a35eef5f22c41da49cbdd7b273937fbf6b21a | abstract | 0 | 126 | Abstract | This article introduces the DPG-star (from now on, denoted DPG$^*$) finite
element method. It is a method that is in some sense dual to the discontinuous
Petrov-Galerkin (DPG) method. The DPG methodology can be viewed as a means to
solve an overdetermined discretization of a boundary value problem. In the same
vein, th... | {
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} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
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b2f189e8ceee333ce023dfc5d8c6736ea464e78b | subsection | 1 | 126 | Introduction | The ideal Discontinuous Petrov–Galerkin (DPG)
Method with Optimal Test Functions
, admits three
interpretations . First, it can be
viewed as a Petrov–Galerkin (PG) discretization in which optimal test functions are computed on the fly. Here, the word
“optimal” refers to the fact that the test functions realize the
supr... | {
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"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
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"math.NA"
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2657b69c20822b49793b8b1b2186aca3a9d52b68 | subsection | 2 | 126 | Operator equations | Central to this paper are the twin relatives of the operator equationB u = \ell ,given in [eq:dpgA]eq:dpgA and [eq:dpg*A]eq:dpg*A below. Here
B : U\rightarrow {\prime } is a bounded linear operator from a Hilbert space
U to the dual of a Hilbert space , is given, and
u U is to be found. All spaces here are over R, the ... | {
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11824a0fd89ef3e8b79e7f965f289c354b190e20 | subsection | 3 | 126 | Operator equations | Due to
the structural similarity, both formulations can be viewed at once as
instantiations of the following general saddle-point problem& \text{Find } v \in {V}\text{ and } w \in {U}\text{ satisfying}
&&
\left\lbrace
\begin{}{3}
\operatorname{{R}}_{V}&v + {B}w &&= F \,, \\
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67e9a9085f21f1e826b8074523e2985e98ad76f7 | subsection | 4 | 126 | Operator equations | Here and throughout, for any Banach space X, the right annihilator of a subset Y \operatorname{\subseteq }X and the
left annihilator of a Z \operatorname{\subseteq }X^{\prime } are defined byY^\perp & = \lbrace E \in X^{\prime }: \langle { E, y} \rangle _X=0\text{ for all }y\in Y\rbrace ,
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2ffb1de3eba5b5c72dfe54819c13192f7696e408 | subsection | 5 | 126 | Operator equations | Under the same
assumption [eq:Bbddbelow]eq:Bbddbelow, consider{B}^{\prime } v = G.Assumption [eq:Bbddbelow]eq:Bbddbelow implies that {B}^{\prime } is surjective,
so [eq:adjointproblem]eq:adjointproblem is always solvable, but its solution
need not be unique in general. Thus, [eq:adjointproblem]eq:adjointproblem may be ... | {
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"arxiv_id": "",
"doi": "10.5860/choice.34-1602",
"end": 679,
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"raw": "Å. Björck, Numerical methods for least squares problems, SIAM, 1996.",
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... | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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07a9bd2651635b693ae7a897ab4b33e0679ba3af | subsection | 6 | 126 | Operator equations | Thus, while (\mathrm {Null}\,{B}^{\prime })^\perp is a subspace of
{V}^{\prime }, the notation (\mathrm {Null}\, {B}^{\prime })_\perp indicates the
subspace of {V} defined by(\mathrm {Null}\,{B}^{\prime })_\perp = \lbrace v \in {V}: {(v, \nu _0)_{V}= 0},\; \forall \, \nu _0 \in \mathrm {Null}\, {B}^{\prime }\rbrace
.O... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.01439889520406723,
-0.0... |
afc820d34a7f8ede20be76a482c6ab781f63847e | subsection | 7 | 126 | Operator equations | First, note that one may also decompose F into orthogonal components:F = F^0 + F^\perp , \qquad F^0 \in {R}_{V}(\mathrm {Null}\, {B}^{\prime }), \quad F^\perp \in {R}_{V}(\mathrm {Null}\, {B}^{\prime })_\perp = (\mathrm {Null}\,{B}^{\prime })^\perp .Second, note that when [eq:Bbddbelow]eq:Bbddbelow
holds, \mathopen {|\... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.018397901207208633,
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0.... |
a47ba7ba77995128c3a6ad77ebd316af327cdb16 | subsection | 8 | 126 | Operator equations | Then the following identities hold:\Vert v_0 \Vert _{V}^2 + \Vert {R}_{V}v_\perp + {B}w \Vert _{{V}^{\prime }}^2
= \Vert F \Vert _{{V}^{\prime }}^2,
\\
\Vert v_0 \Vert _{V}^2 + \Vert {B}w\Vert _{{V}^{\prime }}^2
= \Vert F - {R}_{V}v_\perp \Vert _{{V}^{\prime }}^2.Moreover, v_0 = {R}_{V}^{\raisebox {.2ex}{\scriptscript... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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... |
84618d773a4a9b7016c5351f2a190b2931fa2602 | subsection | 9 | 126 | Operator equations | Hence {R}_{V}^{\raisebox {.2ex}{\scriptscriptstyle -1}}{B}w is in (\mathrm {Null}\,{B}^{\prime })_\perp .
Therefore, when the first equation of [eq:Mixedgeneral]eq:Mixedgeneral is
rewritten asv_0 + (v_\perp + {R}_{V}^{\raisebox {.2ex}{\scriptscriptstyle -1}}{B}w) = {R}_{V}^{\raisebox {.2ex}{\scriptscriptstyle -1}}F,an ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/978-3-642-36519-5",
"end": 1419,
"openalex_id": "https://openalex.org/W592911714",
"raw": "D. Boffi, M. Fortin, and F. Brezzi, Mixed finite element methods and applications, Springer series in computational mathematics, Springer, Be... | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.032072775065898895,
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-... |
c4add9a659c382c7f3f4ecbdd470c9913f596458 | subsection | 10 | 126 | Operator equations | To prove [eq:identity3]eq:identity3, we begin by noting that the isometry induced by
{R}_{V} implies\Vert v_\perp \Vert _{V}=
\sup _{\nu _\perp \in (\mathrm {Null}\,{B}^{\prime })_\perp }
\frac{(\nu _\perp , v_\perp )_{V}}{ \Vert \nu _\perp \Vert _{V}}
=
\sup _{\nu _\perp \in (\mathrm {Null}\,{B}^{\prime })_\perp }
\fr... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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-0.021002795547246933,
0.03483166918158531,
-0... |
6fd1d6e21ed98476ac89c0761ad2b61e40e7c746 | subsection | 11 | 126 | Operator equations | Therefore, v_0=0 and [eq:identity5]eq:identity5 follows
from [eq:identity3]eq:identity3.Identities like [eq:identity4]eq:identity4 have often been referred to by
the name hypercircle identities and their
use in a posteriori error estimation is now standard. We shall
return to this in sec:aposteriorierrorcontrol. | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 258,
"openalex_id": "",
"raw": "S. Repin, S. Sauter, and A. Smolianski, Two-sided a posteriori error estimates for mixed formulations of elliptic problems, SIAM J. Numer. Anal., 45 (2007), pp. 928–945.",
"source_ref_id": "cd... | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.0018280413933098316... |
72b099a7b87695d696778b7663526e6017e90a22 | subsection | 12 | 126 | Forms and discretization | It is traditional to write mixed systems
using a bilinear form defined byb(\mu , \nu ) = \langle { {B}\mu , \nu } \rangle _{V}for all \mu \in {U}, \nu \in {V}. In terms of b,
the mixed problem [eq:Mixedgeneral]eq:Mixedgeneral is to
find v \in {V} and w \in {U} satisfying\left\lbrace
\begin{}{5}
&(v, \nu )_{V}+
b(w, \n... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/j.camwa.2016.05.004",
"end": 1170,
"openalex_id": "https://openalex.org/W1955273710",
"raw": "C. Carstensen, L. Demkowicz, and J. Gopalakrishnan, Breaking spaces and forms for the DPG method and applications including Maxwell equati... | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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25e498312a1d6064d5e434f7ff98031819be3342 | subsection | 13 | 126 | Forms and discretization | In
both cases, we must typically find {V}_h with
\text{dim}({V}_h) > \text{dim}({U}_h) with provable discrete
stability.A key feature of [eq:Mixed-General-form-discrete]eq:Mixed-General-form-discrete is that the
the top left form, (v, \nu )_{V}, being an inner product, is always
coercive. Hence the discrete stability
o... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1090/s0025-5718-2013-02721-4",
"end": 834,
"openalex_id": "https://openalex.org/W2119759746",
"raw": "J. Gopalakrishnan and W. Qiu, An analysis of the practical DPG method, Math. Comput., 83 (2014), pp. 537–552.",
"source_ref_id": ... | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.0... |
bbf3e326eff148ec7b0c3d0889d9383b80eb78ba | subsection | 14 | 126 | Forms and discretization | In that case, inverting \mathsf {G} is computationally feasible and the Schur complement of {eq:Mixed-General-form-matrix} (cf. {eq:normal_equation,eq:normal_equation2}) may be used to solve for the vector \mathsf {w} in a much smaller system, independent of \mathsf {v}:
\begin{equation}
\mathsf {B}^{\raisebox {.2ex}{\... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.030055256560444832,
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0.017377125099301338,
-0... |
3f3d2cd531b91aab2501dc5576508dd066924f42 | subsection | 15 | 126 | Ultraweak formulations | Many PDEs originate in the following strong form:\mathcal {L}u
=
f
\,,where \mathcal {L} is a linear differential operator and f is a prescribed
function. It is possible to give many general DPG and DPG* formulations for
such operator equations using the framework
of (which generalizes the
Friedrichs systems framework... | {
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{
"arxiv_id": "",
"doi": "10.1137/16m1099765",
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"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.0... |
8076de042d03fd1f49cb42a1d6d339e843ad5eb0 | subsection | 16 | 126 | Ultraweak formulations | Likewise, all L^2-inner products restricted to a measurable subset K\operatorname{\subseteq } will be denoted (\cdot ,\cdot )_{{\scriptscriptstyle K}}.The action of \mathcal {L}^* on v: \rightarrow \mathbb {R}^l is given by[\mathcal {L}^*v ]_j = \sum _{i=1}^l \sum _{|\alpha | \le k} (-1)^{|\alpha |}
{a_{ij\alpha }}\, \... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.01555989682674408,
0.0... |
c45b798d0507252852d1c5c0ca94b271b84ba49e | subsection | 17 | 126 | Ultraweak formulations | Also define linear operators \hspace{-2.5pt}{D}\hspace{2.9pt}: H(\mathcal {L}) \rightarrow H(\mathcal {L}^\ast )^\prime and \hspace{-2.5pt}{D}\hspace{2.9pt}^\ast : H(\mathcal {L}^\ast ) \rightarrow H(\mathcal {L})^\prime such that\langle {\hspace{-2.5pt}{D}\hspace{2.9pt}u, v} \rangle _{H(\mathcal {L}^\ast )}
= (\mathca... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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... |
6cd78cb7e19cb3a87af4fb11047158299d56dee8 | subsection | 18 | 126 | Ultraweak formulations | Define H_0(\mathcal {L})\operatorname{\subseteq }H(\mathcal {L}) and H_0(\mathcal {L}^\ast )\operatorname{\subseteq }H(\mathcal {L}^\ast ) to be two subspaces satisfyingH_0(\mathcal {L}) = \mathop {\mmlmultiscripts{\mathop {\hspace{-2.5pt}{D}\hspace{2.9pt}^\ast (H_0(\mathcal {L}^\ast ))}\mmlprescripts {\mmlnone }{\perp... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.030676281079649925,
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0.01605544611811638,
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0.03778829425573349,
0.0... |
581cb0672029e95b04887c09609283d565010720 | subsection | 19 | 126 | Ultraweak formulations | The natural inner products on these spaces, induced by these graph norms, are defined(u,\widetilde{u})_{H(\mathcal {L}_h)}
=
(\mathcal {L}_h u,\mathcal {L}_h \widetilde{u})_{\scriptscriptstyle }+ (u,\widetilde{u})_{\scriptscriptstyle },
\qquad (v,\widetilde{v})_{H(\mathcal {L}^\ast _h)}
=
(\mathcal {L}^\ast _h v,\mathc... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.014594039879739285,
-... |
2e8c5428486cfdcbe6c049ba6c5d71ce6ed25c6c | subsection | 20 | 126 | Ultraweak formulations | Finally, letQ(\mathcal {L}_h)
& = \lbrace p \in H(\mathcal {L}_h)^\prime : \text{ there is a $v \in H_0(\mathcal {L}^\ast )$ such that }
p = \hspace{-2.5pt}{D}\hspace{2.9pt}^\ast _hv\rbrace ,
\\
Q(\mathcal {L}_h^\ast ) & = \lbrace q \in H(\mathcal {L}_h^\ast )^\prime : \text{ there is a $u \in H_0(\mathcal {L})$ such t... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.... |
d64d3e364524ee0bcfebc918537393f31a35b8fe | subsection | 21 | 126 | Ultraweak formulations | Given any F \in H(\mathcal {L}_h^\ast )^\prime , find u\in L^2
and q \in Q(\mathcal {L}_h^\ast ) such that(u, \mathcal {L}^*_h \nu )_{\scriptscriptstyle }+ \langle { q,\nu } \rangle _h = F(\nu )
\qquad \forall \, \nu \in H(\mathcal {L}_h^\ast ).Similarly proceeding with eq:bvp* and setting F(\nu ) = (g, \nu )_{\script... | {
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"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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... |
c536a4aacf3428f3f3a1060b108cfcbf3113d27b | subsection | 22 | 126 | Ultraweak formulations | For instance, the adjoint of the ultraweak formulation eq:uwprob is the following: Given any G \in (L^2\times Q(\mathcal {L}_h^\ast ))^\prime , find v \in H(\mathcal {L}_h^\ast ) such that(\mu , \mathcal {L}^*_h v)_{\scriptscriptstyle }+ \langle { \rho ,v } \rangle _h = G(\mu ,\rho )
\qquad \forall \, \mu \in L^2,\, \r... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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... |
1ca16f52e2e551a15dc0043af64dcd0724922b74 | subsection | 23 | 126 | Ultraweak formulations | Therefore, we prove only the first statement.Let the operator {B}: L^2 \times Q(\mathcal {L}_h^\ast ) \rightarrow H(\mathcal {L}_h^\ast )^\prime be defined
\langle { {B}(\mu , \rho ), \nu } \rangle _{H(\mathcal {L}_h^\ast )} = (\mu , \mathcal {L}^*_h \nu )_{\scriptscriptstyle }+ \langle { \rho ,\nu } \rangle _h, for al... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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01cc359439fb38663c4c98cd3d987863bdaa4ca7 | subsection | 24 | 126 | Ultraweak formulations | Moreover, since \mathcal {D} is densely contained in L^2, this shows
that \mathcal {L}^\ast v = \mathcal {L}_h^\ast v = g.
Thus v \in H(\mathcal {L}^\ast ).
Using [{eq:rhotu}]{\textup {{\ref *{eq:rhotu}}}} again, observe (cf. \cite [Lemma~A.3]{demkowicz2016spacetime}) that
\begin{equation}
0 = \langle {\rho , v } \rang... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
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9357294f32d3041a15c47a723214e6a1f43cbf78 | subsection | 25 | 126 | Ultraweak formulations | Then, given a
G \in (L^2 \times Q(\mathcal {L}_h))^\prime , the problem of finding a function u \in H(\mathcal {L}_h)
satisfying\left\lbrace
\begin{aligned}& (u, \nu )_{V}\;-\; b( (\lambda , \sigma ), \nu ) &&=
0
\quad && \forall \, \nu \in H(\mathcal {L}_h),
\\
& b( (\mu , \rho ), u) &&=
G((\mu , \rho ))
\quad && \fo... | {
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e4bd53a012888c6da9b976add7ea1683837af8c7 | subsection | 26 | 126 | Ultraweak formulations | For brevity, we will not expand on the intricate details here, but simply act to remind the reader that ultraweak variational formulations are not a prerequisite for any DPG-type method coming from eq:Mixed-General-form-discrete.Example 2.7 (Poisson equation)
In this example, which resurfaces throughout the document,... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
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660c9438a1b12f33f987c26c94ffbc21d7ecc49e | subsection | 27 | 126 | Ultraweak formulations | Along the lines of [eq:DDtPoisson]eq:DDtPoisson, we also have\langle {\hspace{-2.5pt}{D}\hspace{2.9pt}^\ast _h(\vec{\sigma },\mu ),(\vec{p},v)} \rangle _h
& =
\sum _{K \in _h}
\bigg [\langle { \vec{\sigma }\cdot \vec{n}, v} \rangle _{H^{1/2}(\partial K)}
+ \langle {\vec{p}\cdot \vec{n}, \mu } \rangle _{H^{1/2}(\partial... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
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7b31e74d7098269f4c2695b2a7273883089c2845 | subsection | 28 | 126 | Ultraweak formulations | Then set
that
\begin{equation}
H^{{\raisebox {.4ex}{{\protect \scalebox {0.5}{-}}}}{}(\partial _h) = \operatorname{\mathrm {tr}}_n( H(\operatorname{div}, )),
\qquad H^{_0(\partial _h) = \operatorname{\mathrm {tr}}( H_0^1()).
}
Clearly,
Q(\mathcal {L}_h) = H^{{\raisebox {.4ex}{{\protect \scalebox {0.5}{-}}}}{}(\partia... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
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438c2dbb3ca83092f1e789decd8853a868563ac0 | subsection | 29 | 126 | Ultraweak formulations | In order to shorten the
notation for later discussions, we shall denote
(\operatorname{grad}_h \mu , \vec{\sigma })_{\scriptscriptstyle }+ (\mu , \operatorname{div}_h \vec{\sigma })_{\scriptscriptstyle }
by \langle {\mu , \vec{\sigma }\cdot \vec{n}} \rangle _h or
\langle {\vec{\sigma }\cdot \vec{n}, \mu } \rangle _h w... | {
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} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
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"Brendan Keith"
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d8d529974fea479fb418e252ab9318e1c43a29b2 | subsection | 30 | 126 | Related methods | Let {V}= H(\mathcal {L}_h^\ast ).
For any F\in H(\mathcal {L}_h^\ast )^\prime , the ultraweak DPG formulation defined by eq:1uw can be restated as the following system of variational equations:\left\lbrace
\begin{}{5}
& (\varepsilon , \nu )_{V}\;+\; (u, \mathcal {L}_h^\ast \nu )_{\scriptscriptstyle }+ \langle { p,\nu ... | {
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8093f37d050f0c42587073ee32fac86e8addb053 | subsection | 31 | 126 | Least-squares methods | Let {V}= L^2 and {U}= H_0(\mathcal {L}).
It is well-known that least-squares finite element methods follow from the following saddle-point formulation (cf. eq:dpgA,eq:DPGformulationExpanded):\left\lbrace
\begin{}{5}
&(\varepsilon , \nu )_{\scriptscriptstyle }+ (\mathcal {L}u,\nu )_{\scriptscriptstyle }&&= F(\nu )\,,\q... | {
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65d1bd8622a2094212608d916fe1de7242a143d7 | subsection | 32 | 126 | Body | Let {V}= L^2 and {U}= H_0(\mathcal {L}^\ast ).
Contrary to eq:LeastSquares, so-called \mathcal {L}\mathcal {L}^\ast methods relate to the following system (cf. eq:dpg*A,eq:DPG*formulationExpanded):\left\lbrace
\begin{}{5}
&(v,\nu )_{\scriptscriptstyle }- (\mathcal {L}^\ast \lambda ,\nu )_{\scriptscriptstyle }&&= 0\,,\... | {
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"Leszek Demkowicz",
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98479a993e25e084af69ecd7ac195e7a93c6688d | subsection | 33 | 126 | Weakly conforming least-squares methods | A weakly conforming least squares method for the primal problem [eq:bvp]eq:bvp seeks a minimizer of the least squares functionalw\mapsto \Vert \mathcal {L}w- f \Vert ^2_{L^2}\, ,under the conformity constraint\langle w, \rho \rangle _h = 0\,, \quad \forall \, \rho \in Q(\mathcal {L}_h)
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"Leszek Demkowicz",
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"Brendan Keith"
] | [
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3255279043819b5c3fce0afe8528cce91d1cc16f | subsection | 34 | 126 | Weakly conforming least-squares methods | Therefore, the first equation can be rewritten as(f-\lambda ,\mathcal {L}_h \nu )_{\scriptscriptstyle }+ \alpha ( v,\nu )_{\scriptscriptstyle }= \langle \sigma ,\nu \rangle _h \,,\quad \forall \, \nu \in H(\mathcal {L}_h)\,.Testing only with \nu \in H(\mathcal {L}), so that the term \langle \sigma ,\nu \rangle _h vanis... | {
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3030bd392c5a032d77f36152b842878ee6a75a5a | subsection | 35 | 126 | Solving the primal and dual problems simultaneously | In eq:Mixedgeneral, we may hypothetically consider any F\in {V}^\prime and G\in {U}^\prime we wish:\left\lbrace
\begin{}{3}
\operatorname{{R}}_{V}&v+ {B}w &&= F \,, \\
{B}^\prime &v&&= G \, .
\end{}
\right.Let {B} to be an isomorphism and define F = \operatorname{{R}}_{V}({B}^\prime )^{\raisebox {.2ex}{\scriptscriptst... | {
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} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
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6df6773f17fe7ca23ad1c8d49abfe97b64db62a9 | subsection | 36 | 126 | Solving the primal and dual problems simultaneously | Substituting \mu = \mathcal {L}^\ast \nu into eq:PrimalDualDPG and canceling terms in the first equation, we immediately find that ( w,\mathcal {L}^\ast \nu )_{\scriptscriptstyle }= (f,\nu )_{\scriptscriptstyle }.
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3b075ac7bad6df2a590cb820819e649db9754cae | subsection | 37 | 126 | Solving the primal and dual problems simultaneously | Clearly, w\rightarrow u as \alpha \rightarrow 0. | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
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443eb40aab4deaa2f7efcb76db8d980f5ee12bd8 | subsection | 38 | 126 | General results | Having explained the connections between the DPG* method and the mixed
formulation [eq:Mixedgeneral]eq:Mixedgeneral, it should not be a surprise that
its error analysis reduces to standard mixed theory. To state the
result, let v \in {V} and \lambda \in {U} satisfy\left\lbrace
\begin{}{5}
&(v, \nu )_{V}- b(\lambda , \... | {
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7f08b69c51f25f3172a3a4d6638918fb8f4fec08 | subsection | 39 | 126 | General results | Consider \varepsilon \in {V} and u \in {U}
satisfying(\varepsilon , \nu )_{V}+ b(u, \nu )
&= F( \nu ),
\\
b(\mu , \varepsilon )
&= 0,for all \nu \in {V}, \mu \in {U}.
To conduct the duality argument, we suppose that there is a positive
c_0(h) that goes to 0 as h\rightarrow 0 satisfying\operatornamewithlimits{\vphantom... | {
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aae51ad14ae84b6ba2e5db9fc79ded1ad3749ea3 | subsection | 40 | 126 | General results | \end{aligned}The proof is completed by applying thm:apriori.It is interesting to note that the duality argument for the DPG* method
uses a DPG formulation: the system [eq:DPGexact]eq:DPGexact is
clearly a DPG formulation. Vice versa, the duality argument for DPG
methods uses DPG* formulations, as can be seen from the d... | {
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"Leszek Demkowicz",
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0.0029998430982232094,
-0.007254547439515591,
-0.06297954171895981,
-0.013990368694067001,
0.023830922320485115,
... |
36b6a67a7457af59a20f90c7f2b3283a4068a7c3 | subsection | 41 | 126 | General results | Then there
is a constant C such that the complete DPG* solution (v,\lambda )\in {V}\times {U} satisfies the error
estimate\Vert v - v_h \Vert _{V}+ \Vert \lambda - \lambda _h \Vert _{U}\le C
\bigg [
\operatornamewithlimits{\vphantom{p}inf}_{\nu \in {V}_h}
\Vert v - \nu \Vert _{V}+
\operatornamewithlimits{\vphantom{p}in... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.03629500791430473,
-0.02123059518635273,
-0.01619386114180088,
0.005574748385697603,
-0.02333686500787735,
-0.04178962484002113,
0.07204055041074753,
0.023672647774219513,
0.027015207335352898,
0.014377584680914879,
-0.041667524725198746,
-0.016697535291314125,
0.02689310535788536,
0.00... |
18d7fcdf15159ad9ddedb9f79689fd603d776b1e | subsection | 42 | 126 | General results | Then for any \mu \in {U}_h, we have\begin{aligned}F( v - v_h )
&
= ( \varepsilon , v-v_h)_{V}+ b( u, v-v_h)
\quad && \text{by [{eq:DPGexact-1}]{\textup {{\ref *{eq:DPGexact-1}}}},}
\\
&
= b( u, v-v_h)
&& \text{since $\varepsilon = 0$,}
\\
&
= b( u -\mu , v-v_h)
&& \text{by~[{eq:DPGstarExact}]{\textup {{\ref *{eq:DPGsta... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/j.camwa.2014.08.004",
"end": 881,
"openalex_id": "https://openalex.org/W2038386729",
"raw": "T. Bouma, J. Gopalakrishnan, and A. Harb, Convergence rates of the DPG method with reduced test space degree, Computers and Mathematics wit... | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.0037479200400412083,
-0.033935364335775375,
-0.03222639113664627,
-0.03189069777727127,
-0.047027334570884705,
-0.00234793359413743,
0.056701354682445526,
0.02633652836084366,
0.011428771540522575,
-0.007015190552920103,
-0.07220420241355896,
0.0030174092389643192,
0.021209603175520897,
... |
d86f5cd1bfb0e227e1d1a8b9e1f526d9f49438f1 | subsection | 43 | 126 | Application to the Poisson example | Given f \in L^2(), consider approximating the Dirichlet solution v-\Delta v = f \quad \text{in } ,
\qquad v=0 \quad \text{ on } \partial ,by the DPG* method. We follow the setting of eg:poisson.
Accordingly, we set{U}= L^2()^d \times L^2() \times H^{{\raisebox {.4ex}{{\protect \scalebox {0.5}{-}}}}{}(\partial _h)\times... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.030341612175107002,
0.006700185127556324,
-0.025579750537872314,
-0.014247432351112366,
-0.009302420541644096,
-0.018650628626346588,
0.04539031907916069,
0.013926922343671322,
0.040994755923748016,
0.018406430259346962,
-0.04999955743551254,
0.025182928889989853,
0.015056338161230087,
... |
47822d1915a0a67e4611c26f20f40a42fcd9d5fe | subsection | 44 | 126 | Application to the Poisson example | Define
P_p(\partial K) = \lbrace \mu : \mu |_E \in P(E)\; \text{ for all
codimension-one sub-simplices } E \text{ of } K \rbrace and
\widetilde{P}_p(\partial K) = P_p(\partial K) \cap C^0(\partial K),
where C^0(D) denotes the set of all continuous functions on a domain
D. Set
\begin{gather}
P_p(_h) = \prod _{K \in _h}... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.00024799874518066645,
0.0040061334148049355,
-0.034399330615997314,
0.009629981592297554,
-0.01742858812212944,
-0.03433828800916672,
0.05326249822974205,
0.012491505593061447,
0.010156502015888691,
0.04114489629864693,
-0.014330511912703514,
-0.011262957938015461,
0.009660504758358002,
... |
1eed9e80592d9c4fc276c28b95b4f064ce74f9ec | subsection | 45 | 126 | Application to the Poisson example | A Fortin operator satisfying~[{eq:Fortin}]{\textup {{\ref *{eq:Fortin}}}} for the case
\begin{align}
{U}_h
&=
\lbrace (\vec{\sigma }, \mu ,\widehat{\sigma }_n,\widehat{\mu }) \in {U}:
\vec{\sigma }\in P_p(_h)^d, \;
\mu \in P_p(_h),\;
\widehat{\sigma }_n \in \widehat{P}_p(\partial _h), \;
\widehat{\mu } \in \widetilde{P... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.02711401879787445,
-0.012969563715159893,
-0.03341570124030113,
-0.0062559074722230434,
0.004775851033627987,
-0.02778538316488266,
0.03143211826682091,
0.017501281574368477,
0.035887546837329865,
0.03555186465382576,
-0.03265278413891792,
-0.0002312587748747319,
0.022216100245714188,
0... |
c974d9b7117287d10dbd7abfacd691a906b3d834 | subsection | 46 | 126 | Application to the Poisson example | We follow the latter approach
in the next proof.
}\begin{}
The solution components \vec{\zeta }, \lambda , \widehat{\zeta }_n,\widehat{\lambda } of the
system~[{eq:dpgstar-poisson}]{\textup {{\ref *{eq:dpgstar-poisson}}}} can be characterized using
the remaining solution components, \vec{p} and v, and the function f a... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.0005473798373714089,
-0.0158530343323946,
-0.008178274147212505,
0.004047177731990814,
0.021635808050632477,
-0.040555696934461594,
0.06026899814605713,
0.0038450094871222973,
0.03445250540971756,
0.011176466010510921,
-0.029341084882616997,
0.00978799071162939,
-0.03091265633702278,
0.... |
6df54b82b12806726a123a65cbf32931d5c2cfd7 | subsection | 47 | 126 | Application to the Poisson example | \end{equation}
Next, we manipulate the first term of~[{eq:dpgstar-poisson-a}]{\textup {{\ref *{eq:dpgstar-poisson-a}}}}
as follows:
\begin{align}
((\vec{p},v), (\vec{\tau }, \nu ))_{V}& = (\vec{p}, \vec{\tau })_{\scriptscriptstyle }+ ( \operatorname{div}\vec{p}, \operatorname{div}\vec{\tau })_{\scriptscriptstyle }+ (v,... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
0.0012985654175281525,
0.014766653068363667,
0.00936645083129406,
-0.031104549765586853,
-0.006162941921502352,
-0.02077704668045044,
0.03810650482773781,
0.04713735356926918,
0.01169280894100666,
0.030296046286821365,
-0.014835298992693424,
-0.013378465548157692,
-0.013592032715678215,
-0... |
18472a95852675f2a235c21f66469c6865b22351 | subsection | 48 | 126 | Application to the Poisson example | Moreover, (\vec{r}, e)\in H_0(\mathcal {L}^\ast ) satisfies \mathcal {L}^* (\vec{r},e) = (0, v+2f), and
\vec{r}\cdot \vec{n}|_{\partial K} = \widehat{r}_n|_{\partial K}, e|_{\partial K} = \widehat{e}|_{\partial K} on all mesh element boundaries. Thus,
((\vec{p},v), (\vec{\tau }, \nu ))_{V}=
b( (\vec{p}+ \vec{r}, f+ e,... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
0.00816583726555109,
-0.002287579234689474,
-0.009799004532396793,
0.000007154646937124198,
-0.021093808114528656,
-0.019506430253386497,
0.02280329167842865,
0.0011848094873130322,
-0.0020986965391784906,
0.04899502173066139,
-0.04005075991153717,
0.0013116853078827262,
0.013340077362954617... |
aa780c06e51de3fde3e84da43cea75b82454112c | subsection | 49 | 126 | Application to the Poisson example | Therefore, our proof proceeds by showing that \operatornamewithlimits{\vphantom{p}inf}_{\vec{\nu } \in {V}_h}\Vert \vec{v} - \vec{\nu } \Vert _{V}^2 + \operatornamewithlimits{\vphantom{p}inf}_{\vec{\mu } \in {U}_h} \Vert \vec{\lambda } - \vec{\mu } \Vert _{U} is bounded from above by the right-hand side of eq:DPG*Poiss... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1090/s0025-5718-2011-02536-6",
"end": 1752,
"openalex_id": "https://openalex.org/W2089907194",
"raw": "L. Demkowicz, J. Gopalakrishnan, and J. Schöberl, Polynomial extension operators. Part III, Math. Comput., 81 (2012), pp. 1289–1326.",... | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.05408746004104614,
0.021305330097675323,
0.0013983531389385462,
0.002382732694968581,
-0.036414410918951035,
0.0009295328054577112,
-0.002811967860907316,
0.01156836748123169,
0.05765869468450546,
0.00787503644824028,
-0.05054674297571182,
-0.026082241907715797,
0.0003345650329720229,
0... |
1cbdb1b1c656961d587ca1f2e1ea915204004ebe | subsection | 50 | 126 | Application to the Poisson example | Moreover, there exists constants C, depending on the polynomial degree p and the shape of the domain , such thatv-gradvH1() C hs|v|Hs+1(),(1/2< s p+1),p-divpH(div,) C hs|p|Hs+1(), (0 < s p+1),-L2() C hs||Hs(), (0 < s p+1).
Notice that \Vert \vec{\lambda }- \vec{\mu } \Vert _{U}^2 = \Vert \vec{\zeta }- \sigma \Vert _{... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.05871844291687012,
-0.007793774362653494,
-0.03213644400238991,
0.004982217215001583,
0.013672485016286373,
-0.03671428561210632,
0.02771119773387909,
0.02600213699042797,
0.0217752642929554,
0.001945582451298833,
-0.04464920982718468,
-0.00666075898334384,
-0.005802413448691368,
0.0104... |
ef8c44341405f539aea6793398055c6f8cdc6765 | subsection | 51 | 126 | Application to the Poisson example | Therefore, invoking {eq:HdivInterpolantAPriori} and {eq:LMSolutions}, we see that \operatornamewithlimits{\vphantom{p}inf}_{\widehat{\sigma }_n\in H^{{\raisebox {.4ex}{{\protect \scalebox {0.5}{-}}}}{}(\partial _h)}\Vert \widehat{\zeta }_n - \widehat{\sigma }_{n}\Vert _{H^{(\partial _h)} \le C h^s ( 2|\vec{p}|_{H^{1+s}... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.050530098378658295,
-0.012075656093657017,
-0.003114268183708191,
0.018384288996458054,
-0.004767709877341986,
-0.020108293741941452,
0.013891199603676796,
0.014356528408825397,
0.011030574329197407,
0.030253980308771133,
-0.08366759121417999,
0.016522975638508797,
0.0031733878422528505,
... |
7e044856941fc9905be1b350e3c3826a229e7d85 | subsection | 52 | 126 | Application to the Poisson example | \end{equation}
}We now need to verify~{eq:reg}, so let us consider the present analog of~{eq:DPGexact}, with the functional F in~{eq:LoadL2ErrorDPG*argument}:
\begin{equation}
\left\lbrace
\begin{}{3}
&(\vec{\varepsilon }, \vec{\nu })_{V}+ b(\vec{u}, \vec{\nu })
&&= F( \vec{\nu }),
\quad &&
\forall \, \vec{\nu } \in {... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.03796663507819176,
0.01062852144241333,
-0.019166436046361923,
-0.035708170384168625,
-0.01788460463285446,
-0.03494517505168915,
0.024736300110816956,
0.029070112854242325,
0.02525513619184494,
0.04086601734161377,
-0.03022986464202404,
-0.022355755791068077,
0.025407735258340836,
0.00... |
b6fb2884191664bdebe19d66cf3ec99d0242db6e | subsection | 53 | 126 | Application to the Poisson example | Then there exists a constant C, depending only on p and the shape regularity of _h, such that
\begin{equation}
\operatornamewithlimits{\vphantom{p}inf}_{\vec{\mu } \in {U}_h} \Vert \vec{u} - \vec{\mu } \Vert _{U}\le Ch^{p+1}\big (\Vert u\Vert _{H^{p+2}()} + \Vert \vec{q}\Vert _{H^{p+1}(_h)}\big )
.
\end{equation}
\end{... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.06549185514450073,
-0.024429744109511375,
-0.017273247241973877,
-0.009979420341551304,
-0.02941945381462574,
-0.013237228617072105,
0.03799504041671753,
0.02543684095144272,
0.023529458791017532,
-0.01695280708372593,
-0.05462740734219551,
0.0019140574149787426,
0.021744150668382645,
0... |
cb1fbf9cdcc7ff934fd19d2aec8dc731df3ffa77 | subsection | 54 | 126 | Application to the Poisson example | Accordingly, we set{U}= L^2()^d \times L^2() \times H^{{\raisebox {.4ex}{{\protect \scalebox {0.5}{-}}}}{}(\partial _h)\times H^{_0(\partial _h),
\qquad {V}= H(\operatorname{div},_h) \times H^1(_h),
where H^{{\raisebox {.4ex}{{\protect \scalebox {0.5}{-}}}}{}(\partial _h) and H^{_0(\partial _h) are defined in {eq:Inte... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.03445212543010712,
0.02548053488135338,
-0.04055524617433548,
0.002055989345535636,
-0.003917441237717867,
-0.039975449442863464,
0.009086023084819317,
0.03005787543952465,
0.026518065482378006,
0.022627325728535652,
-0.0451631024479866,
0.00019274899386800826,
0.033963873982429504,
0.0... |
d5c21370588a6276673537df5dbaab876af7df8d | subsection | 55 | 126 | Application to the Poisson example | Set
\begin{gather}
P_p(_h) = \prod _{K \in _h} P_p(K), \qquad P_p(\partial _h) = \prod _{K \in _h} P_p(\partial K),
\\
\widetilde{P}_p(\partial _h) = \operatorname{\mathrm {tr}}(P_p(_h) \cap H_0^1()),
\qquad \widehat{P}_p(\partial _h) = \operatorname{\mathrm {tr}}_n (P_p(_h)^d \cap H(\operatorname{div}, )).
\end{gath... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.009283154271543026,
0.0011804174864664674,
-0.03636982664465904,
0.014920171350240707,
-0.014447242021560669,
-0.002265486167743802,
0.024271976202726364,
0.020168166607618332,
0.035057827830314636,
0.04161782190203667,
-0.03841410204768181,
-0.010038316249847412,
0.024805929511785507,
... |
60a879d8aaabd336974d08bef9294ed59cb38086 | subsection | 56 | 126 | Application to the Poisson example | One way to do this is to write down the
boundary value problem that \vec{\lambda } satisfies, as done
in~\cite {BoumaGopalHarb14,fuhrer2017superconvergence}. An alternate
technique can be seen in~\cite {fuhrer2017superconvergent}, which
directly manipulates the variational
equation~[{eq:dpgstar-poisson-a}]{\textup {{\r... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.0183733981102705,
-0.009842892177402973,
-0.009331672452390194,
0.003971492405980825,
0.012566855177283287,
-0.033908382058143616,
0.047184839844703674,
-0.004440746735781431,
0.03406098484992981,
0.022859927266836166,
-0.03854751214385033,
0.013039925135672092,
-0.011025565676391125,
0... |
97639b08202a5497917953d9bdbad01a08d10602 | subsection | 57 | 126 | Application to the Poisson example | \end{equation}
Next, we manipulate the first term of~[{eq:dpgstar-poisson-a}]{\textup {{\ref *{eq:dpgstar-poisson-a}}}}
as follows:
\begin{align}
((\vec{p},v), (\vec{\tau }, \nu ))_{V}& = (\vec{p}, \vec{\tau })_{\scriptscriptstyle }+ ( \operatorname{div}\vec{p}, \operatorname{div}\vec{\tau })_{\scriptscriptstyle }+ (v,... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
0.0012985654175281525,
0.014766653068363667,
0.00936645083129406,
-0.031104549765586853,
-0.006162941921502352,
-0.02077704668045044,
0.03810650482773781,
0.04713735356926918,
0.01169280894100666,
0.030296046286821365,
-0.014835298992693424,
-0.013378465548157692,
-0.013592032715678215,
-0... |
621c30f885b71cd9bbb1f739bd141a8391322aed | subsection | 58 | 126 | Application to the Poisson example | Moreover, (\vec{r}, e)\in H_0(\mathcal {L}^\ast ) satisfies \mathcal {L}^* (\vec{r},e) = (0, v+2f), and
\vec{r}\cdot \vec{n}|_{\partial K} = \widehat{r}_n|_{\partial K}, e|_{\partial K} = \widehat{e}|_{\partial K} on all mesh element boundaries. Thus,
((\vec{p},v), (\vec{\tau }, \nu ))_{V}=
b( (\vec{p}+ \vec{r}, f+ e,... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
0.00816583726555109,
-0.002287579234689474,
-0.009799004532396793,
0.000007154646937124198,
-0.021093808114528656,
-0.019506430253386497,
0.02280329167842865,
0.0011848094873130322,
-0.0020986965391784906,
0.04899502173066139,
-0.04005075991153717,
0.0013116853078827262,
0.013340077362954617... |
734221825cf8b9698a468e95ff2a7bd17c9129ec | subsection | 59 | 126 | Application to the Poisson example | Therefore, our proof proceeds by showing that \operatornamewithlimits{\vphantom{p}inf}_{\vec{\nu } \in {V}_h}\Vert \vec{v} - \vec{\nu } \Vert _{V}^2 + \operatornamewithlimits{\vphantom{p}inf}_{\vec{\mu } \in {U}_h} \Vert \vec{\lambda } - \vec{\mu } \Vert _{U} is bounded from above by the right-hand side of eq:DPG*Poiss... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1090/s0025-5718-2011-02536-6",
"end": 1752,
"openalex_id": "https://openalex.org/W2089907194",
"raw": "L. Demkowicz, J. Gopalakrishnan, and J. Schöberl, Polynomial extension operators. Part III, Math. Comput., 81 (2012), pp. 1289–1326.",... | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.05408746004104614,
0.021305330097675323,
0.0013983531389385462,
0.002382732694968581,
-0.036414410918951035,
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0.00787503644824028,
-0.05054674297571182,
-0.026082241907715797,
0.0003345650329720229,
0... |
b28d7a007f9b266847dc1377068effcf3999ba3d | subsection | 60 | 126 | Application to the Poisson example | Moreover, there exists constants C, depending on the polynomial degree p and the shape of the domain , such thatv-gradvH1() C hs|v|Hs+1(),(1/2< s p+1),p-divpH(div,) C hs|p|Hs+1(), (0 < s p+1),-L2() C hs||Hs(), (0 < s p+1).
Notice that \Vert \vec{\lambda }- \vec{\mu } \Vert _{U}^2 = \Vert \vec{\zeta }- \sigma \Vert _{... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.05871844291687012,
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-0.00666075898334384,
-0.005802413448691368,
0.0104... |
3ba8be1175f7c86cf9bb065f2980702e62e1708a | subsection | 61 | 126 | Application to the Poisson example | Therefore, invoking {eq:HdivInterpolantAPriori} and {eq:LMSolutions}, we see that \operatornamewithlimits{\vphantom{p}inf}_{\widehat{\sigma }_n\in H^{{\raisebox {.4ex}{{\protect \scalebox {0.5}{-}}}}{}(\partial _h)}\Vert \widehat{\zeta }_n - \widehat{\sigma }_{n}\Vert _{H^{(\partial _h)} \le C h^s ( 2|\vec{p}|_{H^{1+s}... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.050530098378658295,
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... |
62662db2bb24b1387e01f90d36f471960fd4cc92 | subsection | 62 | 126 | Application to the Poisson example | \end{equation}
}We now need to verify~{eq:reg}, so let us consider the present analog of~{eq:DPGexact}, with the functional F in~{eq:LoadL2ErrorDPG*argument}:
\begin{equation}
\left\lbrace
\begin{}{3}
&(\vec{\varepsilon }, \vec{\nu })_{V}+ b(\vec{u}, \vec{\nu })
&&= F( \vec{\nu }),
\quad &&
\forall \, \vec{\nu } \in {... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.03796663507819176,
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-0.022355755791068077,
0.025407735258340836,
0.00... |
6269b081a4460b156edac87c7d38acacca648cae | subsection | 63 | 126 | Application to the Poisson example | Then there exists a constant C, depending only on p and the shape regularity of _h, such that
\begin{equation}
\operatornamewithlimits{\vphantom{p}inf}_{\vec{\mu } \in {U}_h} \Vert \vec{u} - \vec{\mu } \Vert _{U}\le Ch^{p+1}\big (\Vert u\Vert _{H^{p+2}()} + \Vert \vec{q}\Vert _{H^{p+1}(_h)}\big )
.
\end{equation}
\end{... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.06557189673185349,
-0.02231520414352417,
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0.0268026664853096,
0.... |
2f824de1314404c3dcaea4c5b45cb762b4af631b | subsection | 64 | 126 | A posteriori error control | In this section, we will present an abstract a posteriori error
estimator valid for all ultraweak DPG* formulations (see
sub:ultraweakformulations). We then proceed to work out the
example of the Poisson problem in full detail. Note that abstract
ultraweak formulations encompass many physical models besides the
Poisson... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1137/120862065",
"end": 500,
"openalex_id": "https://openalex.org/W2060065779",
"raw": "L. Demkowicz and N. Heuer, Robust DPG method for convection-dominated diffusion problems, SIAM J. Numer. Anal., 51 (2013), pp. 2514–2537.",
"so... | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.0065669831819832325,
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-0.0197810810059309,
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0.04404953867197037,
-0.02103266306221485,
-0.016560550779104233,
0.00035534592461772263,
0.0... |
70019af4e4e7803fe08b377394eff94ddf59da14 | subsection | 65 | 126 | Designing error estimators for general ultraweak DPG* formulations | Consider the general setting of sub:ultraweakformulations and
the broken ultraweak DPG* formulation which is proved to be well posed
in thm:dpgstaruw. Namely, with \mathcal {L} set to the general partial
differential operator in eq:StrongFormulation, the problem of
finding a v \in H_0(\mathcal {L}) satisfying \mathcal ... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.038538265973329544,
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-0.013784372247755527,
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0.02093210630118847,
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-0.008261469192802906,
0.022213663905858994,
0... |
50533658f3b07d33b12305b32387cb8fcf078973 | subsection | 66 | 126 | Designing error estimators for general ultraweak DPG* formulations | Then, for any v_h \in {V}_h (not necessarily equal to
the DPG* solution),\Vert {B}\Vert ^{\raisebox {.2ex}{\scriptscriptstyle -1}}\Vert G - {B}^{\prime } v_h\Vert _{{U}^\prime }
\le \Vert v - v_h \Vert _{V}\le \Vert {B}^{\raisebox {.2ex}{\scriptscriptstyle -1}}\Vert \Vert G - {B}^{\prime } v_h\Vert _{{U}^\prime }and, m... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.017856009304523468,
-0.007302649784833193,
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-0.014345853589475155,
-0.030462047085165977,
-0.0389169417321682,
0.025288382545113564,
0.... |
701cfac2c4a9adfe96e838e4e84887ecee11b381 | subsection | 67 | 126 | Designing error estimators for general ultraweak DPG* formulations | Indeed, \Vert {B}^{\raisebox {.2ex}{\scriptscriptstyle -1}}\Vert ^{\raisebox {.2ex}{\scriptscriptstyle -1}}\Vert \mu \Vert _{U}\le \mathopen {|\hspace{-0.83328pt}|\hspace{-0.83328pt}|}
\mu
\mathclose {|\hspace{-0.83328pt}|\hspace{-0.83328pt}|}
_{{U}} \le \Vert {B}\Vert \Vert \mu \Vert _{U}, for all \mu \in {U}.
The fi... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.05720764398574829,
0.04207431524991989,
0.012417872436344624,
-0.044606707990169525,
-0.004801628179848194,
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0.006399628706276417,
-0.0259646438062191,
-0.005537699908018112,
-0.0023188164923340082,
0... |
e0f35d72078025233ab494364eb7404bacc85273 | subsection | 68 | 126 | Designing error estimators for general ultraweak DPG* formulations | Namely, with \mathcal {L} set to the general partial
differential operator in eq:StrongFormulation, the problem of
finding a v \in H_0(\mathcal {L}) satisfying \mathcal {L}v = f is reformulated as
eq:1uw, where\langle {{B}(\mu , \rho ), \nu } \rangle _{{V}} =
b((\mu , \rho ), \nu ) = (\mu , \mathcal {L}_h \nu )_{\scrip... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.04296807944774628,
0.01708957739174366,
0.005699068773537874,
-0.017745694145560265,
-0.002067533554509282,
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0.011909298598766327,
-0.03915344178676605,
-0.004695819225162268,
0.029021764174103737,
0.03... |
f8f5664eb8ca5fd843593161339628c9db6711dd | subsection | 69 | 126 | Designing error estimators for general ultraweak DPG* formulations | Then, for any v_h \in {V}_h (not necessarily equal to
the DPG* solution),\Vert {B}\Vert ^{\raisebox {.2ex}{\scriptscriptstyle -1}}\Vert G - {B}^{\prime } v_h\Vert _{{U}^\prime }
\le \Vert v - v_h \Vert _{V}\le \Vert {B}^{\raisebox {.2ex}{\scriptscriptstyle -1}}\Vert \Vert G - {B}^{\prime } v_h\Vert _{{U}^\prime }and, m... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.017856009304523468,
-0.007302649784833193,
0.00593674136325717,
-0.04050414264202118,
-0.027791276574134827,
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0.014391638338565826,
0.028615398332476616,
-0.014345853589475155,
-0.030462047085165977,
-0.0389169417321682,
0.025288382545113564,
0.... |
54932fce1b63688b5451e697b14a9f27839a78fb | subsection | 70 | 126 | Designing error estimators for general ultraweak DPG* formulations | Indeed, \Vert {B}^{\raisebox {.2ex}{\scriptscriptstyle -1}}\Vert ^{\raisebox {.2ex}{\scriptscriptstyle -1}}\Vert \mu \Vert _{U}\le \mathopen {|\hspace{-0.83328pt}|\hspace{-0.83328pt}|}
\mu
\mathclose {|\hspace{-0.83328pt}|\hspace{-0.83328pt}|}
_{{U}} \le \Vert {B}\Vert \Vert \mu \Vert _{U}, for all \mu \in {U}.
The fi... | {
"cite_spans": []
} | 10.1016/j.camwa.2020.01.012 | 1809.03153 | The DPG-star method | [
"Leszek Demkowicz",
"Jay Gopalakrishnan",
"Brendan Keith"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.05558570846915245,
0.036141395568847656,
0.017185477539896965,
-0.03629402071237564,
-0.006562837865203619,
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0.035042501986026764,
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-0.007001631893217564,
-0.019047491252422333,
0.003991120960563421,
-0.0056814332492649555,
... |
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