Optimal Error Properties of Finite Element Methods for Second Order Elliptic Dirichlet Problems

We use the informational approach of Traub and Woźniakowski [9] to study the variational form of the second order elliptic Dirichlet problem on <!-- MATH $\Omega \subset {{\mathbf{R}}^N}$ --> . For <!-- MATH $f \in {H^r}(\Omega )$ --> , where <!-- MATH $r \geqslant - 1$ --> , a quasi-uniform finite element method using n linear functionals <!-- MATH ${\smallint _\Omega }f{\psi _i}$ --> has <!-- MATH ${H^1}(\Omega )$ --> -norm error <!-- MATH $\Theta ({n^{ - (r + 1)/N}})$ --> . 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 <!-- MATH $f \in {H^r}(\Omega )$ --> , where <!-- MATH $r \geqslant - m$ --> , then there is a finite element method whose <!-- MATH ${H^\alpha }(\Omega )$ --> -norm error is <!-- MATH $\Theta ({n^{ - (2m + r - \alpha )/N}})$ --> for <!-- MATH $0 \leqslant \alpha \leqslant m$ --> , and this is asymptotically optimal; thus, the optimal error improves as m increases. If the integrals <!-- MATH ${\smallint _\Omega }f{\psi _i}$ --> are approximated by using n evaluations of f, then there is a finite element method with quadrature with <!-- MATH ${H^1}(\Omega )$ --> -norm error <!-- MATH $O({n^{ - r/N}})$ --> where N/2$">. We show that when , there is no method using n function evaluations whose error is better than <!-- MATH $\Omega ({n^{ - r}})$ --> ; thus for , the finite element method with quadrature is asymptotically optimal among all methods using n evaluations of f.