A seamless reduced basis element method for 2D Maxwell's prolem: An Introduction.


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Abstract: We present a reduced basis element method (RBEM) for the time-harmonic Maxwell's equation. The RBEM is a Reduced Basis Method (RBM) with parameters describing the geometry of the computational domain, coupled with a domain decomposition method. The basic idea is to first decompose the computational domain into a series of subdomains, each of which is deformed from some reference domain, and then to associate with each reference domain precomputed solutions to the same governing partial differential equation, but with different choices of deformations. Finally one seeks the approximation on a new shape as a linear combination of the corresponding precomputed solutions on each subdomain. Unlike the work on RBEM for thermal fin and fluid flow problems, we do not need a mortar type method to “glue” the various local functions. This “gluing” is done “automatically” thanks to the use of a discontinuous Galerkin method. We present the rationale for the method together with numerical results showing exponential convergence for the simulation of a metallic pipe with both ends open. Some theoretical techniques for the a posteriori error estimate for RBEM are also discussed.

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