Abstract
When using boundary integral equation methods, we represent solutions of a linear partial differential equation as layer potentials. It is well-known that the approximation of layer potentials using quadrature rules suffer from poor resolution when evaluated closed to (but not on) the boundary. To address this challenge, we provide modified representations of the problem’s solution. Similar to Gauss’s law used to modify Laplace’s double-layer potential, we use modified representations of Laplace’s single-layer potential and Helmholtz layer potentials that avoid the close evaluation problem. Some techniques have been developed in the context of the representation formula or using interpolation techniques. We provide alternative modified representations of the layer potentials directly (or when only one density is at stake). Several numerical examples illustrate the efficiency of the technique in two and three dimensions.
Highlights
One can represent the solution of partial differential boundary-value problems using boundary integral equation methods, which involves integral operators defined on the domain’s boundary called layer potentials
We have provided modified representations for Laplace and Helmholtz layer potentials to address the close evaluation problem in several boundary value problems
A similar technique has been used in the context of BRIEF and density interpolation
Summary
The goal of this paper is to use (1) with well-chosen v to modify the representation of the solution of boundary value problems associated with L
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