On the efficient implementation of the Galerkin-BEM

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We present efficient methods to approximate nearly singular surface integrals arising massively when discretizing boundary integral equations via the collocation method. The idea is to introduce local polar coordinates centred at a corner of the triangle. Thus it is possible to perform the inner integration analytically, where either the corresponding formulae can be evaluated numerically stable or can be replaced by simple (rational) approximation quite efficiently. We show that the outer integration can be performed by simple Gauss-Legendre quadrature and how to adapt the order of the Gauss formulae to a required order of consistency. Numerical tests will emphasize the efficiency of our method.

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The boundary element method (BEM) leads to a system of linear equations with a full matrix, while FEM yields sparse matrices. This fact seems to require much computational work for the definition of the matrix, for the solution of the system, and, in particular, for the matrix-vector multiplication, which always occurs as an elementary. In this paper a method for the approximate matrix-vector multiplication is described which requires much less arithmetical work. In addition, the storage requirements are strongly reduced.
Boundary integral operators arise in the reduction to the boundary method for solving elliptic boundary value problems. These are classical pseudodifferential operators of integer order on the boundary. In order to exploit these boundary integral operators for computational methods one needs explicit knowledge of the corresponding kernel properties in the framework of Hadamard finite part regularization, explicit representation based on local polar coordinates and explicit transformation under the change of the parametric representation of the boundary manifold. Here we present these properties for the special case of a (piecewise) smooth two-dimensional boundary surface immersed into ℝ3.