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All well-posed problems have uniformly stable and convergent discretizations

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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 functions from their data. Well-posedness of the problem means that this recovery is continuous. Discretization recovers restricted trial functions 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, provided that sufficiently many test data are used. The solutions of discretized problems converge when enlarging the trial spaces, and the convergence rate is determined by how well the data of the function solving the analytic problem can be approximated by the data of the trial functions. This allows new and 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.

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  1. Institut für Numerische und Angewandte Mathematik, Universität Göttingen, Lotzestraße 16-18, 37083, Göttingen, Germany

    Robert Schaback

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Schaback, R. All well-posed problems have uniformly stable and convergent discretizations. Numer. Math. 132, 597–630 (2016). https://doi.org/10.1007/s00211-015-0731-8

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  • DOI: https://doi.org/10.1007/s00211-015-0731-8

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