The p-version of the boundary element method for hypersingular operators on piecewise plane open surfaces

Abstract

We prove an optimal a priori error estimate for the p-version of the boundary element method with hypersingular operators on piecewise plane open surfaces. The solutions of problems on open surfaces typically exhibit a singular behavior at the edges and corners of the surface which prevent an approximation analysis in H 1 . We analyze the approximation by polynomials of typical singular functions in fractional order Sobolev spaces thus giving, as an application, the optimal rate of convergence of the p-version of the boundary element method. This paper extends the results of [C. Schwab, M. Suri, The optimal p-version approximation of singularities on polyhedra in the boundary element method, SIAM J. Numer. Anal., 33 (1996), pp. 729-759] who only considered closed surfaces where the solution is in H 1 .

An iterative substructuring method for the hp‐version of the BEM on quasi‐uniform triangular meshes

“…This is the practically relevant case and the other preconditioner leading to P D behaves analogously. In the plot we also give the curve of (1 + log p) 4 , and the numerical results behave similarly in the given range of p. Exact numbers are listed in Table I together with the condition numbers of the un-preconditioned stiffness matrix. We also give the iteration numbers of the conjugate gradient method that are needed to reduce the l 2 -norm of the initial residuals of the preconditioned/un-preconditioned linear systems by a factor of 10 −8 .…”

“…For our model problem we choose the domain = (−1/2, 1/2) 2 × {0}, use uniform triangular meshes of the type given in Figure 5, and select the right-hand side function f = 1 in (1). For this right-hand side function the hp-version with quasi-uniform meshes converges like O(h 1/2 p −1 ) in the energy norm, see [4,5]. FIG.…”

“…For the analysis, the preconditioned stiffness matrix can be considered as the additive Schwarz operator P corresponding to the underlying subspace decomposition with given bilinear forms. The main result of this study states that the condition number of the preconditioned matrix P is bounded by O((1 + log p) 4 ) with a constant that is independent of the mesh parameter h.…”

“…In this study, we follow Bicȃ's construction to define an iterative substructuring method for the hp-version of the BEM with triangular quasiuniform meshes. Our preconditioner is quasi-optimal in the sense that the condition number of the preconditioned stiffness matrix is bounded by O((1 + log p) 4 ) with a constant that is independent of the mesh parameter h. In this way we improve the bound O((1 + log p) 7 ) by Cao and Guo [8] (for a slightly different construction and only for the p-version without mesh refinement) which, to our knowledge, is so far the only available theoretical result for additive Schwarz type preconditioners for the p-BEM with triangular meshes.…”

Thehp-version of the boundary element method with quasi-uniform meshes in three dimensions

“…In our previous paper [11] we studied the p-version of the BEM for hypersingular operators on open surfaces. The strongest singularities of typical solutions are edge singularities which behave like y 1/2 where y denotes the distance to an edge of the surface.…”