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

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.