The double‐layer potential operator over polyhedral domains II: Spline Galerkin methods

Abstract

We examine the numerical approximation of the integral equation (A -K)u =f, where K is the double layer (harmonic) potential operator on a closed polyhedral surface in R3 and I, lAl 2 1, is a complex constant. The solution is approximated by Galerkin's method, which is based on piecewise polynomials of arbitrary degree on graded triangulations. By utilizing spline spaces which are modified in that the trial functions vanish on some of the triangles closest to the vertices and edges, we investigate the stability of this method in Lz. Furthermore, the use of suitably graded meshes leads to the same quasioptimal error estimates as in the case of a smooth surface.

Finite elements on degenerate meshes: inverse-type inequalities and applications

“…In this case the generalization to less smooth surfaces (e.g. polyhedra) is still an open problem-see Elschner (1992). Subject to the further restriction that ω > 0, the hypersingular operator (4.4c) satisfies (4.5) in…”

Approximate moment matrix decomposition in wavelet Galerkin BEM

“…The triangulation has cosine spacing, that is, the distance of the i-th row of triangles from an edge is O((i/n) 2 ), where n is the total number of rows in a linear direction. For this kind of discretization it is known that the full Galerkin scheme is stable in L 2 , if a fixed number of rows of triangles is deleted from an edge, see Elschner [17]. Furthermore, the error is O(1/n) and the number of panels is N = O(n 2 ).…”

Multiwavelets for Second-Kind Integral Equations

“…Moreover, (2.6) and (2.7) ensure stability and quasi-optimal convergence of Galerkin discretizations of (2.3) ( [17,21] The space L 2 (Γ) can be embedded into the scale H s (Γ), s ∈ R, of Sobolev spaces on the manifold Γ which are defined in the usual way (see [24], [31], [32]). The corresponding norms will be denoted by · s .…”