The hp-BEM with quasi-uniform meshes for the electric field integral equation on polyhedral surfaces: a priori error analysis

This paper presents an a priori error analysis of the hp-version of the boundary element method for the electric field integral equation on a piecewise plane (open or closed) Lipschitz surface. We use H(div)-conforming discretisations with Raviart-Thomas elements on a sequence of quasi-uniform meshes of triangles and/or parallelograms. Assuming the regularity of the solution to the electric field integral equation in terms of Sobolev spaces of tangential vector fields, we prove an a priori error estimate of the method in the energy norm. This estimate proves the expected rate of convergence with respect to the mesh parameter h and the polynomial degree p

Adaptive BEM with optimal convergence rates for the Helmholtz equation

We analyze an adaptive boundary element method for the weakly-singular and hypersingular integral equations for the 2D and 3D Helmholtz problem. The proposed adaptive algorithm is steered by a residual error estimator and does not rely on any a priori information that the underlying meshes are sufficiently fine. We prove convergence of the error estimator with optimal algebraic rates, independently of the (coarse) initial mesh. As a technical contribution, we prove certain local inverse-type estimates for the boundary integral operators associated with the Helmholtz equation

Multilevel methods for the h-, p-, and hp-versions of the boundary element method

AbstractIn this paper we give an overview on the definition of finite element spaces for the h-, p-, and hp-version of the BEM along with preconditioners of additive Schwarz type. We consider screen problems (with a hypersingular or a weakly singular integral equation of first kind on an open surface Γ) as model problems. For the hypersingular integral equation and the h-version with piecewise bilinear functions on a coarse and a fine grid we analyze a preconditioner by iterative substructuring based on a non-overlapping decomposition of Γ. We prove that the condition number of the preconditioned linear system behaves polylogarithmically in H/h. Here H is the size of the subdomains and h is the size of the elements. For the hp-version and the hypersingular integral equation we comment in detail on an additive Schwarz preconditioner which uses piecewise polynomials of high degree on the fine grid and yields also a polylogarithmically growing condition number. For the weakly singular integral equation, where no continuity of test and trial functions across the element boundaries has to been enforced, the method works for nonuniform degree distributions as well. Numerical results supporting our theory are reported

IgA-BEM for 3D Helmholtz problems using conforming and non-conforming multi-patch discretizations and B-spline tailored numerical integration

An Isogeometric Boundary Element Method (IgA-BEM) is considered for the numerical solution of Helmholtz problems on 3D bounded or unbounded domains, admitting a smooth multi-patch representation of their finite boundary surface. The discretization spaces are formed by C0 inter-patch continuous functional spaces whose restriction to a patch simplifies to the span of tensor product B-splines composed with the given patch NURBS parameterization. Both conforming and non-conforming spaces are allowed, so that local refinement is possible at the patch level. For regular and singular integration, the proposed model utilizes a numerical procedure defined on the support of each trial B-spline function, which makes possible a function-by-function implementation of the matrix assembly phase. Spline quasi-interpolation is the common ingredient of all the considered quadrature rules; in the singular case it is combined with a B-spline recursion over the spline degree and with a singularity extraction technique, extended to the multi-patch setting for the first time. A threshold selection strategy is proposed to automatically distinguish between nearly singular and regular integrals. The non-conforming C0 joints between spline spaces on different patches are implemented as linear constraints based on knot removal conditions, and do not require a hierarchical master-slave relation between neighbouring patches. Numerical examples on relevant benchmarks show that the expected convergence orders are achieved with uniform discretization and a small number of uniformly spaced quadrature nodes