Boundary and Finite Element Domain Decomposition Methods

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. For example, Finite Element Methods (FEM) and Boundary Element Methods (BEM) exhibit certain complementary properties. Therefore, it is not astonishing that the coupling of FEM and BEM within a DD framework has successfully been used in many practical applications. Among the DD methods, the so-called Finite Element Tearing and Interconnecting (FETI) methods are probably the most successful ones, at least, for large- scale parallel computations. Recently, the applicants 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. In this project, we propose to construct and analyze new DD solvers for large-scale FEM, BEM and coupled FEM- BEM DD equations derived from linear and non-linear magnetostatic problems as well as from linear and non- linear eddy current problems in the time and in the frequency domain. The numerical treatment of non-linear eddy current problems in the frequency domain is not straightforward. The 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 challenging. The new algorithms to be developed in this project will essentially contribute to a new software generation in Computational Electromagnetics.

Domain Decomposition (DD) methods are nowadays not only used for constructing parallel PDE (Partial Differential Equations) solvers, but also for coupling different physical fields and different discretization techniques. For example, Finite Element Methods (FEM) and Boundary Element Methods (BEM) exhibit certain complementary properties. Therefore, it is not astonishing that the coupling of FEM and BEM within a DD framework has successfully been used in many practical applications. Among the DD methods, the so-called Finite Element Tearing and Interconnecting (FETI) methods are probably the most successful ones, at least, for large-scale parallel computations. The proposers 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. In the finished FWF project, we have generalized the BETI methods to curl-curl problems that typically arise in electrotechnical applications. In one project part, one of the PhD students was able to construct and analyse new, stable and efficient BETI methods even for the more complicated Helmholtz-type problems in the frequency domain. The second part was devoted to eddy-current problems that are typical for low-frequency applications in electromagnetics. The second PhD student investigated the multiharmonic FEM not only for the simulation of linear and non-linear, time-periodic eddy-current problems, but also the optimal control of such kind of eddy-current problems. In the latter case, the multiharmonic FEM approach turned out to be a very effective method for solving the optimality system since the forward primal eddy-current problem is coupled with the adjoint problem that is backward in time. We were able to construct robust and, at the same time, asymptotically optimal or, at least, almost optimal (with respect to the complexity) preconditioners that then lead to highly efficient, parallel iterative solvers for the corresponding large-scale system of finite element equations. Similar results have been obtained for time-periodic parabolic problems and the corresponding optimal control problems. The Springer monograph on "Finite and Boundary Element Tearing and Interconnecting Solvers for Multiscale Problems by C. Pechstein makes fundamental contributions to the construction of preconditioners which even work in the case of very heterogeneous coefficients. The new numerical algorithms developed in the project were implemented not only for numerical testing, but, at least, partially they are used in application projects with the ACCM in Linz and with ABB in Switzerland. Both projects are devoted to the simulation and optimization of electrical devices such as electrical machines and power transformers.