Scalable solvers for complex electromagnetics problems

Abstract

In this work, we present scalable balancing domain decomposition by constraints methods for linear systems arising from arbitrary order edge finite element discretizations of multi-material and heterogeneous 3D problems. In order to enforce the continuity across subdomains of the method, we use a partition of the interface objects (edges and faces) into sub-objects determined by the variation of the physical coefficients of the problem. For multi-material problems, a constant coefficient condition is enough to define this sub-partition of the objects. For arbitrarily heterogeneous problems, a relaxed version of the method is defined, where we only require that the maximal contrast of the physical coefficient in each object is smaller than a predefined threshold. Besides, the addition of perturbation terms to the preconditioner is empirically shown to be effective in order to deal with the case where the two coefficients of the model problem jump simultaneously across the interface. The new method, in contrast to existing approaches for problems in curl-conforming spaces does not require spectral information whilst providing robustness with regard to coefficient jumps and heterogeneous materials. A detailed set of numerical experiments, which includes the application of the preconditioner to 3D realistic cases, shows excellent weak scalability properties of the implementation of the proposed algorithms.

D3.1 Report on nonlinear domain decomposition preconditioners and release of the solvers

“…A relaxed definition of the PB subdomains, where we only require that the maximal contrast of the two physical coefficients is smaller than a predefined thresholds, allows one to extend the range of applicability of the preconditioner to truly heterogeneous materials. The extension of PB-BDDC to electromagnetics applications has been recently developed in [2].…”

“…The interested reader is referred to [1,2,5] for a detailed exposition of BDDC, PB-BDDC, and relaxed PB-BDDC (rPB-BDDC) methods, resp., which is not the aim of this deliverable.…”

“…Initial conditions are simply u 0 = 0. The partition Θ r pb is obtained in all simulations for r = 10 2 (see [2] for details). The nonlinear scheme is stopped when the L 2 -norm of the nonlinear residual (Eq.…”

“…Next, we present average counters for the total number of linear solver applications for the simulation of the whole time interval [0, 5] ms in Table 3, for a local problem size of H/h = 10 and different partitions. The resulting aggregation of cells into subsets based on their physical coefficient α (see [2]) for t = 4 ms is depicted in Fig. 2c.…”

A tutorial-driven introduction to the parallel finite element library FEMPAR v1.0.0

“…For grad-conforming (i.e., H 1 -conforming) FE spaces, as those required for the discretization of the Poisson PDE, the coarse DOFs of a FE function u h are defined as the value of the function at vertices, or the mean values of the function on coarse edges/faces. These concepts have been generalized for div-and curl-conforming FE spaces as well; see, e.g., [35], and references therein, for the latter kind of spaces.…”