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Section: New Results

Multiscale finite element methods for time-domain wave models

Participants : Marie-Helene Lallemand Tenkes, Stéphane Lanteri, Claire Scheid, Frédéric Valentin [LNCC, Petrópolis, Brazil] .

Mathematical (partial differential equation) models embedding multiscale features occur in a wide range of natural situations and industrial applications involving wave propagation. This is for instance the case of electromagnetic or seismic wave propagation in heterogenous media. Although the related applications take place at the macro-scale, it is well known that the parameters describing the macro-scale processes are eventually determined by the solution behavior at the micro-sacle. As a result, each stage of the modeling of the underlying problem is driven by distinct sets of PDEs with highly heterogeneous coefficients and embedded high-contrast interfaces. Because of the huge difference in physical scales in heterogenous media it is not computationally feasible to fully resolve the micro-scale features directly. Macroscopic models or upscaling techniques have therefore to be developed that are able to accurately capture the macroscopic behavior while significantly reducing the computational cost. In this context, researchers at LNCC have recently proposed a new family of finite element methods [51] - [50] , called Multiscale Hybrid-Mixed methods (MHM), which is particularly adapted to be used in high-contrast or heterogeneous coefficients problems. Particularly, they constructed a family of novel finite element methods sharing the following properties: (i) stable and high-order convergent; (ii) accurate on coarse meshes; (iii) naturally adapted to high-performance parallel computing; (iv) induce a face-based a posteriori error estimator (to drive mesh adaptativity); (v) locally conservative. We have started this year a new reserach direction aiming at the design of similar MHM methods for solving PDE models of time-domain electromagnetic and seismic wave propagation.