Questions: Finite Element Method (Mathematical Foundations)

Question 1

Cea's lemma states that the Galerkin approximation error is bounded by:

• A. A constant times the best approximation error from V_h
• B. The mesh size h
• C. The number of elements in the mesh
• D. The condition number of the stiffness matrix

Answer: (A)

Cea's lemma: ||u - u_h||_H ≤ (M/α) inf_{v_h ∈ V_h} ||u - v_h||_H. The FEM solution is the best approximation from V_h up to a constant M/α (the ratio of continuity to coercivity constants). The approximation-theoretic question of how well V_h approximates u is then separate from the PDE question.

Question 2

The Galerkin orthogonality condition states a(u - u_h, v_h) = 0 for all v_h ∈ V_h.

• T. True
• F. False

Answer: True

Since a(u,v_h) = F(v_h) and a(u_h,v_h) = F(v_h), subtraction gives a(u - u_h, v_h) = 0. The error u - u_h is orthogonal to V_h with respect to the bilinear form a. This means u_h is the a-projection of u onto V_h.

Question 3

For piecewise linear elements on a quasi-uniform mesh of size h, what is the convergence rate in H¹ for a smooth solution?

Model answer: ||u - u_h||_{H¹} = O(h), first-order convergence

Piecewise linear functions on a mesh of size h approximate smooth functions to order h in H¹ and h² in L² (by the Aubin-Nitsche trick). Higher-order polynomials give faster convergence: piecewise polynomials of degree k give O(h^k) in H¹.

Question 4

The stiffness matrix in FEM is always symmetric positive definite for the Poisson equation with Dirichlet conditions.

• T. True
• F. False

Answer: True

The stiffness matrix K_{ij} = a(φ_j, φ_i) = ∫∇φ_j·∇φ_i dx inherits symmetry from the symmetry of a, and positive definiteness from the coercivity of a on V_h. This allows efficient solution by the conjugate gradient method.