Realization of \(hp\)-Galerkin BEM in \(\mathbb{R}^ 3\). (English) Zbl 0881.65115

Sauter, Stefan A.; Schwab, Christoph

Hackbusch, Wolfgang (ed.) et al., Boundary elements: implementation and analysis of advanced algorithms. Proceedings of the 12th GAMM-Seminar, Kiel, Germany, January 19-21, 1996. Wiesbaden: Vieweg. Notes Numer. Fluid Mech. 54, 194-206 (1996).

Summary: It is well known that Galerkin discretizations based on \(hp\)-finite element spaces are converging exponentially with respect to the degrees of freedom for elliptic problems with piecewise analytic data. However, the question whether these methods can be realized for general situations such that the exponential convergence is preserved also with respect to the computing time is very essential.
We show how the numerical quadrature can be realized in order that the resulting fully discrete \(hp\)-boundary element method (BEM) converges exponentially with algebraically growing work. The key point is to approximate the integrals constituting the stiffness matrix by exponentially converging cubature methods.
For the entire collection see [Zbl 0871.00042].

MSC:

65N38	Boundary element methods for boundary value problems involving PDEs
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N12	Stability and convergence of numerical methods for boundary value problems involving PDEs

Keywords:

Galerkin method; Laplace equation; finite elements; \(hp\)-boundary element method; exponential convergence; numerical quadrature; cubature methods