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Fig.3 Comparison of error assessment method based on h-refinement and k-refinement: (a) isogeometric solution surface with control points; (b) error surface obtained by three h-refinement steps; (c) error surface obtained by three k-refinement steps; (d) exact error color map; (e) color map of error surface in Fig. 3 (b); (f) colormap of error surface in Fig. 3 (c).

$\begin{align*} \Delta U_h ={}& \frac{\partial^2 U_h}{\partial^2 x} + \frac{\partial^2 U_h}{\partial^2 y} \\ &= [(x_{\xi}^2 - y_{\xi}^2)(T_{\eta\eta} - \frac{\partial U_h}{\partial x}x_{\eta\eta} - \frac{\partial U_h}{\partial y}y_{\eta\eta}) - \\ & \qquad (x_{\eta}^2 - y_{\eta}^2)(T_{\xi\xi} - \frac{\partial U_h}{\partial x}x_{\xi\xi} - \frac{\partial U_h}{\partial y}y_{\xi\xi})] / K \\ &= (JL_{\xi}T_{\eta\eta} - JL_{\eta}T_{\xi\xi} + L_{\xi\xi}T_{\eta} - L_{\eta\eta}T_{\xi}) / (JK). \end{align*}$

where

$\begin{align*} L_{\xi} &= x_{\xi}^{2} - y_{\xi}^{2}, & L_{\eta} &= x_{\eta}^{2} - y_{\eta}^{2}, \\ L_{\xi\xi} &= (L_{\eta} y_{\xi\xi} - L_{\xi} y_{\eta\xi}) x_{\xi} - (L_{\eta} x_{\xi\xi} - L_{\xi} x_{\eta\xi}) y_{\xi}, \\ L_{\eta\eta} &= (L_{\eta} y_{\xi\xi} - L_{\xi} y_{\eta\xi}) x_{\eta} - (L_{\eta} x_{\xi\xi} - L_{\xi} x_{\eta\xi}) y_{\eta}. \end{align*}$

This completes the proof. □

The approximation error surface e from (4) also has a B-spline form. Different from the method in [16], we perform several k -refinement operations to achieve more accurate results. Though it is much more expensive, we can use it as an error assessment method for isogeometric simulation solutions.

As a summary, the procedure of error assessment method for model problem (1) can be described as follows,

Input: the isogeometric solution $U_h(\mathbf{x})$ over computational domain $\Omega$

Output: the error surface e

Compute $\Delta U_h(\mathbf{x})$ according to Proposition 1;
Solve the isogeometric analysis problem (4) with several $k$-refinement steps;
Output the error surface $e$.

IV. EXAMPLES AND COMPARISONS

In this paper, we test the error assessment methods based on h-refinement and k-refinement for the heat conduction problem (1) with source term

$f(x, y) = - \frac{4\pi^2}{9} \sin\left(\frac{\pi x}{3}\right) \sin\left(\frac{\pi y}{3}\right).$

For this problem with boundary condition $U_0(x) = 0$, the exact solution over the computational domain $[0, 3] \times [0, 3]$ is

$U(x, y) = 2 \sin\left(\frac{\pi x}{3}\right) \sin\left(\frac{\pi y}{3}\right).$

Fig. 3 illustrates an example over the computational domain $[0,3] \times [0,3]$, which has exact solution for problem (1). The isogeometric solution surface with control points are shown in Fig. 3(a), the error surface obtained by three h-refinement steps is illustrated in Fig. 3 (b), Fig. 3 (c) shows the error