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Optimal error properties of finite element methods for second order elliptic Dirichlet problems

Author: Arthur G. Werschulz
Journal: Math. Comp. 38 (1982), 401-413
MSC: Primary 65N30
MathSciNet review: 645658
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Abstract: We use the informational approach of Traub and Woźniakowski [9] to study the variational form of the second order elliptic Dirichlet problem $ Lu = f$ on $ \Omega \subset {{\mathbf{R}}^N}$. For $ f \in {H^r}(\Omega )$, where $ r \geqslant - 1$, a quasi-uniform finite element method using n linear functionals $ {\smallint _\Omega }f{\psi _i}$ has $ {H^1}(\Omega )$-norm error $ \Theta ({n^{ - (r + 1)/N}})$. We prove that it is asymptotically optimal among all methods using any information consisting of any n linear functionals. An analogous result holds if L is of order 2m: if $ f \in {H^r}(\Omega )$, where $ r \geqslant - m$, then there is a finite element method whose $ {H^\alpha }(\Omega )$-norm error is $ \Theta ({n^{ - (2m + r - \alpha )/N}})$ for $ 0 \leqslant \alpha \leqslant m$, and this is asymptotically optimal; thus, the optimal error improves as m increases. If the integrals $ {\smallint _\Omega }f{\psi _i}$ are approximated by using n evaluations of f, then there is a finite element method with quadrature with $ {H^1}(\Omega )$-norm error $ O({n^{ - r/N}})$ where $ r > N/2$. We show that when $ N = 1$, there is no method using n function evaluations whose error is better than $ \Omega ({n^{ - r}})$; thus for $ N = 1$, the finite element method with quadrature is asymptotically optimal among all methods using n evaluations of f.

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Article copyright: © Copyright 1982 American Mathematical Society