Approximation with harmonic and generalized harmonic polynomials in the partition of unity method

Abstract

The aim of the paper is twofold. In the first part, we present an analysis of the approximation properties of "complete systems", that is, systems of functions which satisfy a given differential equation and are dense in the set of all solutions. We quantify the approximation properties of these complete systems in terms of Sobolev norms. As a first step of the analysis, we consider the approximation of harmonic functions by harmonic polynomials. By means of the theory of Bergman and Vekua, the approximation results for harmonic polynomials are then extended to the case of general elliptic equations with analytic coefficients if the harmonic polynomials are replaced with their analogs, "generalized harmonic polynomials". In the second part of the paper, we present the Partition of Unity Method (PUM). This method has the feature that it allows for the inclusion of a priori knowledge about the local behavior of the solution in the ansatz space. Therefore, the PUM can lead to very effective and robust methods. We illustrate the PUM with an application to Laplace's equation and the Helmholtz equation.