Optimal error properties of finite element methods for second order elliptic Dirichlet problems
Arthur G. Werschulz
Math. Comp. 38 (1982), 401-413
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Abstract: We use the informational approach of Traub and Woźniakowski  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 2m: 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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