Mixed discontinuous Galerkin approximation of the Maxwell operator: The indefinite case

Abstract

Abstract. We present and analyze an interior penalty method for the numerical discretization of the indefinite time-harmonic Maxwell equations in mixed form. The method is based on the mixed discretization of the curl-curl operator developed in [Houston et al., J. Sci. Comp. 22 (2005) 325-356] and can be understood as a non-stabilized variant of the approach proposed in [Perugia et al., Comput. Methods Appl. Mech. Engrg. 191 (2002) 4675-4697]. We show the well-posedness of this approach and derive optimal a priori error estimates in the energy-norm as well as the L 2 -norm. The theoretical results are confirmed in a series of numerical experiments.

A Locally Divergence-Free Interior Penalty Method for Two-Dimensional Curl-Curl Problems

“…Many alternative approaches have been developed which include H(curl; Ω) conforming edge element methods [41,42,13,33,38,40], H 1 conforming nodal finite methods with weighted regularization [27,29], the singular complement/field method [32,5], and interior penalty methods [43,36,37,35,34].…”

Computational Performance of LDG Methods Applied to Time Harmonic Maxwell Equation in Polyhedral Domains

“…Several variations of the proposed method have also been recently analyzed in [14][15][16][17]. Assuming a scalar electric permittivity tensor, the general idea is to introduce a Lagrange multiplier, ϕ, obtained from a Helmholtz orthogonal decomposition of the vector field, E = U + ε −1 ω −2 ∇ϕ.…”

“…Since then other DG formulations based on a different spatial discretization, namely the Interior Penalty (IP) [1], have been theoretically and numerically investigated in a series of articles. For instance, in [22], a Lagrange multiplier technique based on a Helmholtz orthogonal decomposition of the vector field was combined, for the first time, with the IP spatial discretization to control the divergence free constraint of the electric field; and more recently in a series of articles [14][15][16][17].…”

Precondicionamiento del método LDG para la ecuación vectorial de Helmholtz

“…En la última década se han propuesto una serie de métodos numéricos, basados en discretizaciones espaciales discontinuas, para aproximar la solución de la ecuación de Helmholtz a bajas frecuencias, también conocido como problema de corrientes de Foucault, (o corrientes de Eddy en la literatura anglosajona), [28,27,23,22,7,17]. Este tipo de métodos se caracteriza principalmente por no imponer ningún tipo de continuidad entre celdas adyacentes; por lo que su formulación es naturalmente apropiada para refinamiento en espacio y en el grado de aproximación.…”