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Variable transformations for nearly singular integrals in the boundary element method. (English) Zbl 1100.65109
Summary: We review variable transformation methods for evaluating nearly singular integrals over curved surfaces, which were proposed by the author and co-workers.
The rest of the paper is organized as follows. Section 2 gives a brief explanation of the boundary element formulation of the three-dimensional potential problem. In section 3, we analyze the nature of integral kernels occuring in such a formulation. In section 4, we present the outline of the projection and angular $ radial transformation method proposed by the author. In section 5, we treat the radial variable transformation, which is particularly important in the method. In section 6, we perform an error analysis of the method using complex function theory, which yields insight regarding the optimal radial variable transformation. In section 7, we mention the use of the double exponential transformation in the radial variable transformation.

MSC:
65N38 Boundary element methods for boundary value problems involving PDEs
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N15 Error bounds for boundary value problems involving PDEs
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