Domain Decomposition (DD) methods are nowadays not only used for constructing parallel solvers for Partial Differential Equations (PDE), but also for coupling different physical fields and different discretization techniques. Since Finite Element Methods (FEM) and Boundary Element Methods (BEM) exhibit certain complementary properties, it is sometimes very useful to couple these discretization techniques and thus benefit from the advantages of both worlds. This concerns not only the treatment of unbounded domains (BEM), but also the right handling of singularities (BEM), moving parts (BEM), air regions in electromagnetics (BEM), source terms (FEM), non-linearities (FEM), etc. Therefore, it is not astonishing that the coupling of FEM and BEM within a DD framework is successfully used in many practical applications. Among the DD methods, the so-called Finite Element Tearing and Interconnecting (FETI) methods are probably the most successful, at least for large-scale parallel computations. Recently, the proposers have introduced data-sparse Boundary Element Tearing and Interconnecting (BETI) methods as boundary element counterparts of the well-established FETI methods, as well as coupled BETI-FETI methods for some model problems such as the potential equation and the linear elasticity system. The advantage of these tearing and interconnecting methods is not only the nearly optimal asymptotical behavior of the iteration numbers with respect to the discretization parameter h and the subdomain scaling parameter H, but also the robustness with respect to large coefficient jumps and the excellent scalability on massively parallel computers.
In this project, we propose to construct and analyze new DD solvers, including DD solvers based on the tearing and interconnecting technique, for large-scale finite element (FE), boundary element (BE), and coupled FE-BE DD equations derived from linear and non-linear magnetostatic problems as well as from linear and non-linear eddy current problems in both the time and frequency domains. The numerical treatment of non-linear eddy current problems in the frequency domain is not at all straightforward. A multiharmonic approach that is based on Fourier series is one possible technique to treat such problems. The construction of fast solvers, in particular, efficient DD solvers for the resulting large-scale system of non-linear equations, is quite challenging. The new DD algorithms to be developed in this project will essentially contribute to a new generation software in Computational Electromagnetics.
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