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When is the error in the \(h\)-BEM for solving the Helmholtz equation bounded independently of \(k\)? (English) Zbl 1320.65187

The authors consider solving the sound-soft scattering problem for the Helmholtz equation in two or three dimensions using the standard second-kind combined-field integral equations. To solve the integral equations, the authors consider the \(h\)-version of the Galerkin method, i.e., \(\nu_N\) consists of piecewise polynomials of degree \(p\) for some fixed \(p\geq 0 \). In the majority of the paper \(\Gamma\) is \(C^2\), in which case \(\nu_N\) will be the space of piecewise polynomials of degree \(p\) for some fixed \(p\geq 0 \) on shape regular meshes of diameter \(h\) decreasing to zero. The authors also consider the case when \(\Gamma\) is the boundary of a 2-d polygon, and in this case \(\nu_N\) will consist of piecewise polynomials on a mesh appropriately graded towards the corners.

MSC:

65N38 Boundary element methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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