Layer potential identities and subtraction techniques
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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 establish new layer potential identities that can be used to modify the solution's representation. Similar to Gauss's law used to modify Laplace's double-layer potential, we provide new identities to modify Laplace's single-layer potential and Helmholtz layer potentials that avoid the close evaluation problem. Several numerical examples illustrate the efficiency of the technique in two and three dimensions.
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