Adaptive FE–BE coupling for an electromagnetic problem in ℝ³—A residual error estimator

Abstract

We construct a reliable and efficient residual-based local a posteriori error estimator for a Galerkin method coupling finite elements and boundary elements for an eddy current problem in a three-dimensional polyhedral domain. For the proof of the efficiency of the error estimator, we assume that the boundary mesh is quasi-uniform and that the boundary surface and the boundary data satisfy certain smoothness assumptions. The Galerkin method uses lowestorder Nédélec elements in the interior domain and vectorial surface rotations of continuous, piecewise bilinear functions on the boundary. Singular, weakly singular and hypersingular boundary integral operators appearing in the variational formulation show up in terms of the error estimator as well. The estimator is derived from the defect equation using a Helmholtz decomposition and Green's formulas. The decomposed parts of the Galerkin error are approximated by local interpolation operators. Numerical tests underline reliability and efficiency of the residual error estimator.

Space-Time Residual-Based A Posteriori Estimators for the A−ϕ Magnetodynamic Formulation of the Maxwell System

“…The term (65) is equal to zero thanks to the semi-discrete weak formulation (18) with A ′ = 0, as shown in the proof of the Lemma 4.8 (see the relation (62)). Using the Cauchy-Schwarz inequality and the definitions (23) and (26) of some parts of the estimator, the right-hand side of the previous identity can be estimated as follows :…”

“…where the last deduction is due to the semi-discrete weak formulation (18) with A ′ = 0 applied for all the discrete time steps q = 1, . .…”

“…derived from the semi-discrete weak formulation (18) with A ′ = 0 (see (62)), and the property that b T = 0 over ∂T , we can estimate r m T :…”

“…Some works devoted to the potential formulations were also performed in [15,33]. An estimator for a coupling of the Boundary and Finite element methods was introduced in [18], as well as in [27] in the context of a Discontinuous Galerkin Method. Very recently, an adaptive h − p finite element algorithm was proposed for time-harmonic Maxwell's equations [17].…”

Coupled BEM–FEM for the convected Helmholtz equation with non-uniform flow in a bounded domain

“…Coupled BEM-FEM can be traced back to McDonald and Wexler [34], Zienkiewicz, Kelly and Bettess [45], Johnson and Nédélec [26] and Jin and Liepa [25]. Over the last decade, such methods have been investigated, among others, for electromagnetic scattering [21,29,30], elasticity [11], and fluid-structure [13] or solid-solid interactions [33,44]. Coupled BEM-FEM for the classical Helmholtz equation can present resonant frequencies, leading to infinitely many solutions.…”