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

DOI:
https://doi.org/10.1090/S0025-5718-1982-0645658-4

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 2*m*: 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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© Copyright 1982
American Mathematical Society