Complexity of indefinite elliptic problems

Abstract:

This paper deals with the approximate solution of a linear regularly-elliptic 2mth-order boundary-value problem $Lu = f$, with $f \in {H^r}(\Omega )$ for $r \geq - m$. Suppose that the problem is indefinite, i.e., the variational form of the problem involves a weaklycoercive bilinear form. Of particular interest is the quality of the finite element method (FEM) of degree k using n inner products of f. The error of the approximation is measured in the Sobolev l-norm $(0 \leq l \leq m)$; we assume that $k \geq 2m - 1 - l$. We assume that an a priori bound is known for either the Sobolev r-norm or for the Sobolev r-seminorm of f. We first consider the normed case. We find that the FEM has minimal error if and only if $k \geq 2m - 1 + r$. Regardless of the values of k and r, there exists a linear combination (called the spline algorithm) of the inner products used by the FEM which does have minimal error. For the seminormed case, we give a very restrictive condition which is necessary and sufficient for the error of the FEM to have a bound which is independent of f. When this condition holds, we find that the FEM has minimal error if and only if $k \geq 2m - 1 + r$. However, we once again find that the spline algorithm (using the same inner products as does the FEM) has minimal error, no matter what values k and r have and regardless of whether the FEM has uniformly bounded error. We also show that the inner products used by the FEM is the best set of linear functionals to use.