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

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- by Arthur G. Werschulz PDF
- Math. Comp.
**38**(1982), 401-413 Request permission

## 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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## Additional Information

- © Copyright 1982 American Mathematical Society
- Journal: Math. Comp.
**38**(1982), 401-413 - MSC: Primary 65N30
- DOI: https://doi.org/10.1090/S0025-5718-1982-0645658-4
- MathSciNet review: 645658