A computational tool for comparing all linear PDE solvers

Abstract

The paper provides a computational technique that allows to compare all linear methods for PDE solving that use the same input data. This is done by writing them as linear recovery formulas for solution values as linear combinations of the input data, and these formulas are continuous linear functionals on Sobolev spaces. Calculating the norm of these functionals on a fixed Sobolev space will then serve as a quality criterion that allows a fair comparison of all linear methods with the same inputs, including standard, extended or generalized finite–element methods, finite–difference– and meshless local Petrov–Galerkin techniques. The error bound is computable and yields a sharp worst–case bound in the form of a percentage of the Sobolev norm of the true solution. In this sense, the paper replaces proven error bounds by calculated error bounds. A number of illustrative examples is provided. As a byproduct, it turns out that a unique error–optimal method exists. It necessarily outperforms any other competing technique using the same data, e.g. those just mentioned, and it is necessarily meshless, if solutions are written “entirely in terms of nodes” (Belytschko et. al. Comput. Methods Appl. Mech. Eng., Spec. issue, 139, 3–47, 1996). On closer inspection, it turns out that it coincides with symmetric meshless collocation carried out with the kernel of the Hilbert space used for error evaluation, e.g. with the kernel of the Sobolev space used. This technique is around since at least 1998, but its optimality properties went unnoticed, so far. Examples compare the optimal method with several others listed above.