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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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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 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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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