Varying coefficients in parallel shared-memory variational splitting solvers for non-stationary Maxwell equations

Abstract

Direction-splitting implicit solvers employ the regular structure of the computational domain augmented with the splitting of the partial differential operator to deliver linear computational cost solvers for time-dependent simulations. The finite difference community originallye mployed this method to deliver fast solvers for PDE-based formulations. Later, this method was generalized into so-called variationals plitting. The tensor product structure of basis functions over regular computational meshes allows us to employ the Kronecker product structureo f the matrix and obtain linear computational cost factorization for finite element method simulations. These solvers are traditionally usedf or fast simulations over the structures preserving the tensor product regularity. Their applications are limited to regular problems and regularm odel parameters. This paper presents a generalization of the method to deal with non-regular material data in the variational splitting method. Namely, we can vary the material data with test functions to obtain a linear computational cost solver over a tensor product grid with nonregularm aterial data. Furthermore, as described by the Maxwell equations, we show how to incorporate this method into finite element methods imulations of non-stationary electromagnetic wave propagation over the human head with material data based on the three-dimensional MRI scan.