Abstract
A complex screen is an arrangement of panels that may not be even locally orientable because of junction lines. A comprehensive trace space framework for first-kind variational boundary integral equations on complex screens has been established in Claeys and Hiptmair (Integr Equ Oper Theory 77:167–197, 2013. https://doi.org/10.1007/s00020-013-2085-x) for the Helmholtz equation, and in Claeys and Hiptmair (Integr Equ Oper Theory 84:33–68, 2016. https://doi.org/10.1007/s00020-015-2242-5) for Maxwell’s equations in frequency domain. The gist is a quotient space perspective that allows to make sense of jumps of traces as factor spaces of multi-trace spaces modulo single-trace spaces without relying on orientation. This paves the way for formulating first-kind boundary integral equations in weak form posed on energy trace spaces. In this article we extend that idea to the Galerkin boundary element (BE) discretization of first-kind boundary integral equations. Instead of trying to approximate jumps directly, the new quotient space boundary element method employs a Galerkin BE approach in multi-trace boundary element spaces. This spawns discrete boundary integral equations with large null spaces comprised of single-trace functions. Yet, since the right-hand-sides of the linear systems of equations are consistent, Krylov subspace iterative solvers like GMRES are not affected by the presence of a kernel and still converge to a solution. This is strikingly confirmed by numerical tests.
Highlights
We are concerned with the scattering of acoustic or electromagnetic waves at objects like those displayed in Fig. 1, i.e. geometries composed of essentially twodimensional piecewise smooth surfaces joined together
We investigate the performance of quotient-space boundary element method (BEM) in a few numerical experiments, which were carried out using the BETL library [21]
For each of the boundary integral equations (BIEs) we report the dimensions of the discrete kernels, we compute the generalized condition numbers of the Galerkin matrices, and study the convergence of the Conjugate Gradient (GC) and Generalized Minimal Residual (GMRES) iterative solvers
Summary
We are concerned with the scattering of acoustic or electromagnetic waves at objects like those displayed in Fig. 1, i.e. geometries composed of essentially twodimensional piecewise smooth surfaces joined together. We recast the boundary value problems as variational boundary integral equations (BIEs) posed in spaces of functions on the surface of the scattering object For simple screens this is well established [31, Section 3.5.3]. We call a simple screen an orientable, piecewise smooth two-dimensional bounded manifold Γ embedded in 3D space R3 In this case, coercive variational first-kind boundary integral equations arise, known as weakly singular and hypersingular BIEs in the acoustic setting [14, 15,33], and as Electric Field Integral Equation (EFIE) for electromagnetics [4]. We are going to present an approach that will yield a Galerkin BEM discretization of the boundary integral equations for acoustic and electromagnetic scattering at general triangulated multi-screens. S10(T ) p.w. linear continuous functions on inflated screen, Page 14 S0−1(T ) p.w. constant functions on inflated screen, Page 14
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