All Well--Posed Problems have Uniformly Stable and Convergent Discretizations

Abstract

This paper considers a large class of linear operator equations, including linear boundary value problems for partial differential equations, and treats them as linear recovery problems for objects from their data. Well-posedness of the problem means that this recovery is continuous. Discretization recovers restricted trial objects from restricted test data, and it is well-posed or stable, if this restricted recovery is continuous. After defining a general framework for these notions, this paper proves that all well-posed linear problems have stable and refinable computational discretizations with a stability that is determined by the well-posedness of the problem and independent of the computational discretization. The solutions of discretized problems converge when enlarging the trial spaces, and the convergence rate is determined by how well the full data of the object solving the full problem can be approximated by the full data of the trial objects. %of the discretization. This allows very simple proofs of convergence rates for generalized finite elements, symmetric and unsymmetric Kansa-type collocation, and other meshfree methods like Meshless Local Petrov-Galerkin techniques. It is also shown that for a fixed trial space, weak formulations have a slightly better convergence rate than strong formulations, but at the expense of numerical integration. Since convergence rates are reduced to those coming from Approximation Theory, and since trial spaces are arbitrary, this also covers various spectral and pseudospectral methods. All of this is illustrated by examples.