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A new variant of the Adaptive Cross Approximation (ACA) for approximation of dense block matrices is presented. This algorithm can be applied to matrices arising from the Boundary Element Methods (BEM) for elliptic or Maxwell systems of partial differential equations. The usual interpolation property of the ACA is generalised for the matrix valued case. Some numerical examples demonstrate the efficiency of the new method. The main example will be the electromagnetic scattering problem, that is, the exterior boundary value problem for the Maxwell system. Here, we will show that the matrix valued ACA method works well for high order BEM, and the corresponding high rate of convergence is preserved. Another example shows the efficiency of the new method in comparison with the standard technique, whilst approximating the smoothed version of the matrix valued fundamental solution of the time harmonic Maxwell system. Copyright © 2016 John Wiley & Sons, Ltd.

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The electromagnetic scattering at a perfectly conducting object is usually initiated by an incoming electromagnetic field. It is well known that the classical boundary element implementations solving for the scattered electric field are not uniformly stable with respect to the frequency of the incoming signal. The subject of this article is to develop a stabilized boundary element formulation that does not suffer from the so-called low-frequency breakdown. The mathematical theory is verified by numerical examples.Ã .(The second order Equation (1) can be derived from the harmonic Maxwell equations under the assumption that ! > 0. This assumption is crucial because the second order equation is wrong for the limit ! D 0, where the electric field turns into an electrostatic field. However, even if ! > 0, the numerical solution for Equation (1) may fail to converge. The low-frequency problem, for instance, denotes the numerical instability that appears when the frequency ! is very small compared with the dimension of the scatterer [5,6]. The reason is that, in this case, the implicit built-in of the Gauß law div ."E/ .x/ D 0 for x 2 c is not sufficiently imposed by Equation (1). L. WEGGLERFor industrial applications, it is important to have a robust numerical solver for all frequencies ! > 0. This means, however, that a stabilized formulation for low frequencies must be developed. It is the goal of this paper to present the mathematical theory and the numerical verification of a stabilized boundary element formulation that is robust for all frequencies ! > 0. This paper is organized as follows. The function spaces that are needed to formulate boundary integral variational equations for the Maxwell equations are introduced in Section 2. The class of energy spaces defined in Subsection 2.1 contain functions or vector fields that are square integrable on open Lipschitz polyhedra. Recent results on the regularity of vector fields defined on boundaries of Lipschitz polyhedra lead to a characterization of their traces [7][8][9]. The trace spaces are discussed in Subsection 2.2.The fundamental results leading to the classical boundary element method are stated in Section 3. By the distributional setting, which we use to derive the second order Equation (1), the continuity equation appears as an additional condition characterizing the unknown normal trace n E of the electric field. This is the starting point to explain the mathematical and physical origin of the low-frequency problem, and it will lead to the stabilized boundary element formulation analyzed in Section 4. In Section 5, the stabilized formulation is further considered for the electrostatic case ! D 0. The relation between the electrostatic case and the stabilized formulation is fundamental to understand the numerical results presented in Section 6. u, v 2 L 2 loc c : Z K vu dx Ä 1 , for all compact subsets K c . The subscript loc is skipped in the following.

Trace spaces of the energy space H.curl, / appear naturally in variational formulations that result from the Maxwell equations. Their characterization is a topic that has been intensively studied in the past 10 years. The first contributions treat the case of smooth manifolds [1-3] whereas Lipschitz domains are considered more recently in [4]. Although focus is made on the smooth case, Nédélec's monograph is a path-leading reference as most basic definitions of tangent vector spaces and surface differential operators are presented. And thus, the generalizations presented in [4] build up nicely on Nédélec's exposition. The characterization of trace spaces of H.curl, / for general Lipschitz domains as presented in [4] is rather abstract and there was the need to find yet another access leading to an intuitive understanding of the regularity in these function spaces. In the pioneering paper [5], explicit characterizations of the trace spaces of H.curl, / are given under the assumption that be the boundary of a Lipschitz polyhedral domain . Explicit means that the regularity of the functionals is captured by integral expressions that allow for numerical evaluation. The current developments regarding high-order methods require generalizations of the explicit characterizations for curvilinear Lipschitz polyhedral domains .Let be a curvilinear Lipschitz polyhedral domain with piecewise smooth , that is, for any x 2 , there exists a neighbourhood U.x/ \ that is C 1 -diffeomorphic to a neighbourhood of a boundary point of a polyhedron [6]. Owing to this property, there exists a decomposition of into simplicial subdomains (vertices, edges and elements), and the lack of global smoothness may occur only at vertices and along the common edges of neighbouring elements. Definition 1Let be the boundary of an open curvilinear Lipschitz polyhedral domain. We assume that is given by