Approximation of Integrals for Boundary Element Methods

A new method for approximating two-dimensional integrals B f (x) µ(dx) over surfaces B ⊂ R 3 is introduced where µ is the standard measure of surface area. Such integrals typically occur in boundary element methods. The algorithm is based on triangulations T := T i approximating B. Under the assumption that the surface B is given implicitly by an equation H(x) = 0, a retraction P : U ⊃ B → B is used to obtain a curved subdivision B = B i via B i := P T i. Except in very special cases, this retraction is not analytically accessible, but is generated by a subroutine. Hence standard multiple integral techniques are not available. Thus, the approach given here differs from the usual panel method. It is shown how to calculate the integrals as precisely as wished. Two numerical examples are given. The first integrand f (x) ≡ 1 is regular, and it is shown that a very accurate extrapolation method can be used. The second integrand f (x) ∼ ||x − x 0 || −1 is singular, and an adaptive refinement procedure is displayed. 1. Motivation. Let us motivate the following discussion by considering a well-known integral equation related to Laplace's equation: (1.1) ∆u = 0 in D, u = g on B, where D ⊂ R 3 is a bounded open nonempty domain with sufficiently regular boundary B. We introduce the fundamental solution s(x, y) := 1 4π 1 ||x − y|| , and consider the following double layer potential ansatz for a solution u of (1.1): (1.2) u(x) = B ∂s(x, y) ∂ν(y) σ(y) µ(dy), where ν(y) for y ∈ B is the unit normal vector pointing out of D and σ is an unknown density function on B to be determined by the boundary integral method. Here and in the following, the symbol µ denotes the standard measure of surface area; µ(dy) is often called the surface element. It turns out, see, e.g., [9–12], that σ satisfies the following integral equation of the second kind: (1.3) σ(x) − 2 B ∂s(x, y) ∂ν(y) σ(y) µ(dy) = −2g(x) .