3 research outputs found
Preconditioned implicit time integration schemes for Maxwell’s equations on locally refined grids
In this paper, we consider an efficient implementation of higher-order implicit time integration schemes for spatially discretized linear Maxwell’s equations on locally refined meshes.
In particular, our interest is in problems where only a few of the mesh elements are small while the majority of the elements is much larger. We suggest to approximate the solution of the linear systems arising in each time step by a preconditioned Krylov subspace method, e.g., the quasi-minimal residual method by Freund and Nachtigal [13].
Motivated by the analysis of locally implicit methods by Hochbruck and Sturm [25], we show how to construct a preconditioner in such a way that the number of iterations required by the Krylov subspace method to achieve a certain accuracy is bounded independently of the diameter of the small mesh elements. We prove this behavior by using Faber polynomials and complex approximation theory.
The cost to apply the preconditioner consists of the solution of a small linear system, whose dimension corresponds to the degrees of freedom within the fine part of the mesh (and its next coarse neighbors). If this dimension is small compared to the size of the full mesh, the preconditioner is very efficient.
We conclude by verifying our theoretical results with numerical experiments for the fourth-order Gauß-Legendre Runge–Kutta method
Asymptotic Analysis for Overlap in Waveform Relaxation Methods for RC Type Circuits
Waveform relaxation (WR) methods are based on partitioning large circuits
into sub-circuits which then are solved separately for multiple time steps in
so-called time windows, and an iteration is used to converge to the global
circuit solution in each time window. Classical WR converges quite slowly,
especially when long time windows are used. To overcome this issue, optimized
WR (OWR) was introduced which is based on optimized transmission conditions
that transfer information between the sub-circuits more efficiently than
classical WR. We study here for the first time the influence of overlapping
sub-circuits in both WR and OWR applied to RC circuits. We give a circuit
interpretation of the new transmission conditions in OWR and derive closed-form
asymptotic expressions for the circuit elements representing the optimization
parameter in OWR. Our analysis shows that the parameter is quite different in
the overlapping case, compared to the nonoverlapping one. We then show
numerically that our optimized choice performs well, also for cases not covered
by our analysis. This paper provides a general methodology to derive optimized
parameters and can be extended to other circuits or system of differential
equations or space-time PDEs.Comment: 23 Pages, 15 Figure
On the nonlinear Dirichlet-Neumann method and preconditioner for Newton's method
The Dirichlet-Neumann (DN) method has been extensively studied for linear partial differential equations, while little attention has been devoted to the nonlinear case. In this paper, we analyze the DN method both as a nonlinear iterative method and as a preconditioner for Newton's method. We discuss the nilpotent property and prove that under special conditions, there exists a relaxation parameter such that the DN method converges quadratically. We further prove that the convergence of Newton's method preconditioned by the DN method is independent of the relaxation parameter. Our numerical experiments further illustrate the mesh independent convergence of the DN method and compare it with other standard nonlinear preconditioners