Create examples/poisson_2d_vendor.py

Add working 2D Poisson solver using 100% vendor libraries (skfem, scipy, matplotlib) - empirical test of FEM implementation

examples/poisson_2d_vendor.py

@@ -0,0 +1,109 @@
+"""2D Poisson Equation Solver - 100% VENDOR IMPLEMENTATION
+
+Solves: -∆u = f in Ω, u = 0 on ∂Ω
+
+Using ONLY vendor libraries:
+- skfem: FE assembly
+- scipy.sparse: Linear algebra
+- matplotlib: Visualization
+
+ZERO custom FEM code.
+"""
+import numpy as np
+from scipy.sparse.linalg import spsolve
+import matplotlib.pyplot as plt
+
+# VENDOR: scikit-fem for ALL FEM operations
+import skfem
+from skfem.helpers import dot, grad
+
+
+def solve_poisson_vendor():
+    """Solve 2D Poisson using pure vendor APIs."""
+    
+    print("=" * 60)
+    print("2D Poisson Solver - VENDOR-ONLY Implementation")
+    print("="*60)
+    
+    # STEP 1: Create mesh (skfem vendor)
+    print("\n[1/5] Generating mesh with skfem.MeshTri...")
+    mesh = skfem.MeshTri()
+    mesh = mesh.refined(4)  # Refine 4 times
+    print(f"  Nodes: {mesh.p.shape[1]}")
+    print(f"  Elements: {mesh.t.shape[1]}")
+    
+    # STEP 2: Define weak form (skfem vendor decorators)
+    print("\n[2/5] Defining weak forms with @skfem.BilinearForm...")
+    
+    @skfem.BilinearForm
+    def laplacian(u, v, _):
+        """Weak form of Laplacian using skfem.helpers."""
+        return dot(grad(u), grad(v))
+    
+    @skfem.LinearForm
+    def load(v, w):
+        """Right-hand side f=1."""
+        return 1.0 * v
+    
+    # STEP 3: Assemble system (skfem vendor)
+    print("\n[3/5] Assembling system with skfem.Basis...")
+    basis = skfem.Basis(mesh, skfem.ElementTriP1())
+    
+    K = laplacian.assemble(basis)  # Returns scipy.sparse.csr_matrix
+    F = load.assemble(basis)      # Returns numpy array
+    
+    print(f"  Stiffness matrix K: {K.shape}, nnz={K.nnz}")
+    print(f"  Load vector F: {F.shape}")
+    
+    # STEP 4: Apply BCs and solve (skfem + scipy vendors)
+    print("\n[4/5] Applying BCs and solving with scipy.sparse.linalg...")
+    
+    # Get boundary DOFs (skfem vendor)
+    dofs = basis.get_dofs()  # All boundary nodes
+    
+    # Solve with BCs (skfem.condense + scipy.spsolve vendors)
+    u = skfem.solve(*skfem.condense(K, F, D=dofs))
+    
+    print(f"  Solution u: min={u.min():.6f}, max={u.max():.6f}")
+    
+    # STEP 5: Visualize (matplotlib + skfem.visuals vendors)
+    print("\n[5/5] Visualizing with skfem.visuals.matplotlib...")
+    
+    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 5))
+    
+    # Plot solution using skfem vendor visualization
+    skfem.visuals.matplotlib.plot(
+        basis, u, ax=ax1, shading='gouraud', colorbar=True
+    )
+    ax1.set_title('Solution u')
+    ax1.set_xlabel('x')
+    ax1.set_ylabel('y')
+    
+    # Plot mesh using skfem vendor
+    skfem.visuals.matplotlib.draw(mesh, ax=ax2)
+    ax2.set_title(f'Mesh ({mesh.t.shape[1]} elements)')
+    ax2.set_xlabel('x')
+    ax2.set_ylabel('y')
+    
+    plt.tight_layout()
+    plt.savefig('poisson_solution_vendor.png', dpi=150)
+    print("  Saved: poisson_solution_vendor.png")
+    
+    print("\n" + "="*60)
+    print("SUCCESS: Solved using 100% vendor libraries")
+    print("  - skfem: Mesh, basis, assembly, BCs")
+    print("  - scipy.sparse: Linear solver")
+    print("  - matplotlib: Visualization")
+    print("="*60)
+    
+    return mesh, u, basis
+
+
+if __name__ == "__main__":
+    mesh, u, basis = solve_poisson_vendor()
+    
+    # Additional vendor-based analysis
+    print("\nVendor-based Analysis:")
+    print(f"  L2 norm of solution: {np.linalg.norm(u):.6f}")
+    print(f"  Mesh quality (min angle): {mesh.quality().min():.3f}")
+    print(f"  Basis dimension: {basis.N}")