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Math Notes

University Mathematics

TheoremComplete

Cea's Lemma and Finite Element Error Estimates

Theorem8.2Cea's Lemma

Let B(⋅,⋅)B(\cdot,\cdot) be a continuous, coercive bilinear form on V=H01(Ω)V = H_0^1(\Omega): ∣B(u,v)∣≤M∥u∥∥v∥|B(u,v)| \leq M\|u\|\|v\| and B(v,v)≥α∥v∥2B(v,v) \geq \alpha\|v\|^2. Let u∈Vu \in V be the exact solution of B(u,v)=F(v)B(u,v) = F(v) for all v∈Vv \in V, and uh∈Vh⊂Vu_h \in V_h \subset V the Galerkin approximation. Then: ∥u−uh∥V≤Mαinf⁡vh∈Vh∥u−vh∥V\|u - u_h\|_V \leq \frac{M}{\alpha} \inf_{v_h \in V_h} \|u - v_h\|_V The Galerkin solution is a quasi-best approximation: its error is within a factor M/αM/\alpha of the best approximation from VhV_h.

Proof

Proof

Galerkin orthogonality. By definition, B(u,vh)=F(vh)B(u, v_h) = F(v_h) and B(uh,vh)=F(vh)B(u_h, v_h) = F(v_h) for all vh∈Vhv_h \in V_h. Subtracting: B(u−uh,vh)=0B(u - u_h, v_h) = 0 for all vh∈Vhv_h \in V_h. The error eh=u−uhe_h = u - u_h is BB-orthogonal to VhV_h.

Error estimate. For any vh∈Vhv_h \in V_h: α∥u−uh∥2≤B(u−uh,u−uh)\alpha \|u - u_h\|^2 \leq B(u - u_h, u - u_h) by coercivity. Write u−uh=(u−vh)+(vh−uh)u - u_h = (u - v_h) + (v_h - u_h) where vh−uh∈Vhv_h - u_h \in V_h: B(u−uh,u−uh)=B(u−uh,u−vh)+B(u−uh,vh−uh)B(u - u_h, u - u_h) = B(u - u_h, u - v_h) + B(u - u_h, v_h - u_h)

The second term vanishes by Galerkin orthogonality. For the first: B(u−uh,u−vh)≤M∥u−uh∥⋅∥u−vh∥B(u - u_h, u - v_h) \leq M\|u - u_h\| \cdot \|u - v_h\|

Combining: α∥u−uh∥2≤M∥u−uh∥∥u−vh∥\alpha\|u - u_h\|^2 \leq M\|u - u_h\|\|u - v_h\|. Dividing by ∥u−uh∥\|u - u_h\|: ∥u−uh∥≤Mα∥u−vh∥\|u - u_h\| \leq \frac{M}{\alpha}\|u - v_h\|

Since this holds for all vh∈Vhv_h \in V_h, take the infimum: ∥u−uh∥≤Mαinf⁡vh∈Vh∥u−vh∥\|u - u_h\| \leq \frac{M}{\alpha}\inf_{v_h \in V_h}\|u - v_h\|. □\square

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Consequences

ExampleError Estimate for P1 Elements

For P1 elements on a quasi-uniform triangulation with mesh size hh: the best approximation error from VhV_h satisfies inf⁡vh∈Vh∥u−vh∥H1≤Ch∣u∣H2\inf_{v_h \in V_h}\|u - v_h\|_{H^1} \leq Ch|u|_{H^2} (by the Bramble-Hilbert lemma and interpolation estimates). Cea's lemma then gives ∥u−uh∥H1≤MαCh∣u∣H2=O(h)\|u - u_h\|_{H^1} \leq \frac{M}{\alpha}Ch|u|_{H^2} = O(h). For the L2L^2 error, the Aubin-Nitsche duality argument gains one additional order: ∥u−uh∥L2≤Ch2∣u∣H2\|u - u_h\|_{L^2} \leq Ch^2|u|_{H^2}.

RemarkOptimality and Extensions

The constant M/α=κ(B)M/\alpha = \kappa(B) (the "condition number" of the bilinear form) is sharp in general. For symmetric positive definite BB, the Galerkin solution is the true best approximation in the energy norm ∥v∥B=B(v,v)\|v\|_B = \sqrt{B(v,v)}, and M/α=1M/\alpha = 1. For non-symmetric or indefinite problems, the inf-sup (Babuska-Brezzi) condition replaces coercivity, and the analog of Cea's lemma involves the inf-sup constant β\beta instead of α\alpha.

RemarkA Posteriori Error Estimation

While Cea's lemma gives an a priori bound (requiring knowledge of ∣u∣H2|u|_{H^2}), practical computation uses a posteriori error estimators computed from uhu_h. The residual-based estimator ηT2=hT2∥f+Δuh∥L2(T)2+hT∥[∇uh⋅n]∥L2(∂T)2\eta_T^2 = h_T^2\|f + \Delta u_h\|_{L^2(T)}^2 + h_T\|[\nabla u_h \cdot n]\|_{L^2(\partial T)}^2 (element residual + jump terms) satisfies C1η≤∥u−uh∥H1≤C2ηC_1 \eta \leq \|u - u_h\|_{H^1} \leq C_2 \eta, providing both lower and upper bounds for adaptive mesh refinement.