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 on . For , where , a quasi-uniform finite element method using *n* linear functionals has -norm error . 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 , where , then there is a finite element method whose -norm error is for , and this is asymptotically optimal; thus, the optimal error improves as *m* increases. If the integrals are approximated by using *n* evaluations of *f*, then there is a finite element method with quadrature with -norm error where . We show that when , there is *no* method using *n* function evaluations whose error is better than ; thus for , the finite element method with quadrature is asymptotically optimal among *all* methods using *n* evaluations of *f*.

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

Article copyright:
© Copyright 1982
American Mathematical Society