Computation of resonance frequencies for Maxwell equations in non-smooth domains
We address the computation by finite elements of the non-zero eigenvalues of the (curl, curl) bilinear form with perfect conductor boundary conditions in a polyhedral cavity. One encounters two main difficulties: (i) The infinite dimensional kernel of thi
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Summary. We address the computation by finite elements of the non-zero eigenvalues of the (curl, curl) bilinear form with perfect conductor boundary conditions in a polyhedral cavity. One encounters two main difficulties: (i) The infinite dimensional kernel of this bilinear form (the gradient fields). (ii) The unbounded singularities of the eigen-fields near comers and edges of the cavity. We first list possible variational spaces with their functional properties and provide a short description of the edge and comer singularities. Then we address different formulations using a Galerkin approximation by edge elements or nodal elements. After a presentation of edge elements, we concentrate on the functional issues connected with the use of nodal elements. In the framework of conforming methods, nodal elements are mandatory if one regularises the bilinear form (curl, curl) in order to get rid of the gradient fields. A plain regularisation with the (div, div) bilinear form converges to a wrong solution if the domain has reentrant edges or comers. But remedies do exist. We will present the method of addition of singular functions, and the method of regularisation with weight, where the (di v, div) bilinear form is modified by the introduction of a weight which can be taken as the distance to reentrant edges or corners.
Introduction Computing Maxwell eigenfrequencies has been an interesting challenge for the numerical analysis community for many years. Besides its many obvious and very important practical applications, ranging from signal processing over heart and brain biology to nuclear fusion, the Maxwell eigenvalue problem has been attractive because of some mathematical features that set it apart from the standard fields of elliptic eigenvalue problems. There is, on one hand, its simplicity as one of the very basic problems in partial differential equations. On this basic level there is the relation between the Maxwell equations and the de Rham complex and algebraic topology. In the construction and analysis of special families of finite elements, based on Nedelec's edge elements or "Whitney elements", this relation plays an important role. Major progress has been made in the theory of these special elements in recent years and many questions have M. Ainsworth et al. (eds.), Topics in Computational Wave Propagation © Springer-Verlag Berlin Heidelberg 2003
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Martin Costabel and Monique Dauge
found satisfactory answers, but some questions concerning the approximation of the eigenvalue problem remain open. On the other hand, the Maxwell eigenvalue problem has several rather irritating peculiarities: (i) The gauge invariance allows for many different variational formulations. The simplest of these are non-elliptic, have a non-empty essential spectrum, and the energy space is not compactly embedded into £2. The effect is that straightforward discretisation do not generally give any useful approximation of the eigenvalues. Some Galerkin schemes may produce convergence of the numerical eigenvalues, but the limits ma
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