Preconditioning for time-harmonic Maxwell's equations using the Laguerre transform

A method of numerically solving the Maxwell equations is considered for modeling harmonic electromagnetic fields. The vector finite element method makes it possible to obtain a physically consistent discretization of the differential equations. However, solving large systems of linear algebraic equations with indefinite ill-conditioned matrices is a challenge. The high order of the matrices limits the capabilities of the Gaussian method to solve such systems, since this requires large RAM and much calculation. To reduce these requirements, an iterative preconditioned algorithm based on integral Laguerre transform in time is used. This approach allows using multigrid algorithms and, as a result, needs less RAM compared to the direct methods of solving systems of linear algebraic equations.Comment: 12 pages, 4 figure

Parallel dichotomy algorithm for solving tridiagonal SLAEs

A parallel algorithm for solving a series of matrix equations with a constant tridiagonal matrix and different right-hand sides is proposed and studied. The process of solving the problem is represented in two steps. The first preliminary step is fixing some rows of the inverse matrix of SLAEs. The second step consists in calculating solutions for all right-hand sides. For reducing the communication interactions, based on the formulated and proved main parallel sweep theorem, we propose an original algorithm for calculating share components of the solution vector. Theoretical estimates validating the efficiency of the approach for both the common- and distributed-memory supercomputers are obtained. Direct and iterative methods of solving a 2D Poisson equation, which include procedures of tridiagonal matrix inversion, are realized using the mpi technology. Results of computational experiments on a multicomputer demonstrate a high efficiency and scalability of the parallel sweep algorithm.Comment: 18 page