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

*(English)*Zbl 0951.65128Summary: The aim of the paper is two-fold. 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 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.

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 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.

##### MSC:

65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

35J25 | Boundary value problems for second-order elliptic equations |