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Quadrature for $$hp$$-Galerkin BEM in $$\mathbb{R}^3$$. (English) Zbl 0901.65069
The authors describe and analyze an $$hp$$-type method for the discretization of integral operators with Cauchy singular kernels on two-dimensional piecewise analytic manifolds with edges and corners. An exponential rate of convergence can be ensured under certain assumptions concerning the behavior of the kernel functions, the mesh grading and the polynomial order of the trial functions.
The total numerical effort, however, grows algebraically with the number of degrees of freedom. To this end, special quadrature techniques are necessary which involve coordinate transformations, splitting of the integration domain and regularization techniques. These strategies are described in each detail. Several cases must be distinguished and handled separately. Furthermore, a priori estimates for the necessary number of quadrature sample points are given. All these things are not only of interest for the $$hp$$-Galerkin method. They are applicable for other versions of the boundary element method (BEM) as well.

##### MSC:
 65N38 Boundary element methods for boundary value problems involving PDEs 65R20 Numerical methods for integral equations 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 45E05 Integral equations with kernels of Cauchy type 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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