DG discretization of optimized Schwarz methods for Maxwell's equations
نویسندگان
چکیده
In the last decades, Discontinuous Galerkin (DG) methods have seen rapid growth and are widely used in various application domains (see [13] for an historical introduction). This is due to their main advantage of combining the best of finite element and finite volume methods. For the time-harmonic Maxwell equations, once the problem is discretized with a DG method, finding robust solvers is a difficult task since one has to deal with indefinite problems. From the pioneering work of Després [5] where the first provably convergent domain decomposition (DD) algorithm for the Helmholtz equation was proposed and then extended to Maxwell’s equations in [6], other studies followed. Preliminary attempts to obtain better algorithms for this kind of equations were given in [3, 4, 12], where the first ideas of optimized Schwarz methods can be found. Then, the advantage of the optimization process was used for the second order Maxwell system in [1]. Later on, an entire hierarchy of optimized transmission conditions for the first order Maxwell’s equations was proposed in [9, 11] . For the second order or curl-curl Maxwell’s equations second order optimized transmission conditions can be found in [14, 15, 16, 17]. We study here optimized Schwarz DD methods for the time-harmonic Maxwell equations discretized by a DG method. Due to the particularity of the latter, DG discretization applied to more sophisticated Schwarz methods is not straightforward. In this work we show a strategy of discretization and prove the equivalence between multi-domain and single-domain solutions. The proposed discrete framework is then illustrated by some numerical results in the two-dimensional case. We consider time-harmonic Maxwell’s equations in a homogeneous medium written as a first order system (see [10] for more details)
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