Search

We study the convergence of multigrid schemes for the Helmholtz equation, focusing in particular on the choice of the coarse scale operators. Let G_c denote the number of points per wavelength at the coarse level. If the coarse scale solutions are to approximate the true solutions, then the oscillatory nature of the solutions implies the requirement G_c > 2. However, in examples the requirement is more like G_c >= 10, in a trade-off involving also the amount of damping present and the number of multigrid iterations. We conjecture that this is caused by the difference in phase speeds between the coarse and fine scale operators. Standard 5-point finite differences in 2-D are our first example. A new coarse scale 9-point operator is constructed to match the fine scale phase speeds. We then compare phase speeds and multigrid performance of standard schemes with a scheme using the new operator. The required G_c is reduced from about 10 to about 3.5, with less damping present so that waves propagate over > 100 wavelengths in the new scheme. Next we consider extensions of the method to more general cases. In 3-D comparable results are obtained with standard 7-point differences and optimized 27-point coarse grid operators, leading to an order of magnitude reduction in the number of unknowns for the coarsest scale linear system. Finally we show how to include PML boundary layers, using a regular grid finite element method. Matching coarse scale operators can easily be constructed for other discretizations. The method is therefore potentially useful for a large class of discretized high-frequency Helmholtz equations., Coarse scale operators are simplified and only standard smoothers used in v3; 5 figures, 12 tables

This paper addresses global error estimation and control for initial value problems forordinary differential equations. The focus lies on a comparison between a novel approach based onthe adjoint method combined with a small sample statistical initialization and the classical approachbased on the first variational equation. Control is achieved through tolerance proportionality. Bothapproaches are found to work well and to enable estimation and control in a reliable manner.Key words.Numerical integration for ODEs, global error estimation, global error control, defectsand local errors, tolerance proportionality, adjoint method, small sample statistical initialization

An implicit-explicit (IMEX) extension of the explicit Runge--Kutta--Chebyshev (RKC) scheme designed for parabolic PDEs is proposed for diffusion-reaction problems with severely stiff reaction terms. The IMEX scheme treats these reaction terms implicitly and diffusion terms explicitly. Within the setting of linear stability theory, the new IMEX scheme is unconditionally stable for reaction terms having a Jacobian matrix with a real spectrum. For diffusion terms the stability characteristics remain unchanged. A numerical comparison for a stiff, nonlinear radiation-diffusion problem between an RKC solver, an IMEX-RKC solver, and the popular implicit BDF solver VODPK using the Krylov solver GMRES illustrates the excellent performance of the new scheme.

In this paper we study a multigrid (MG) method for the solution of a linear second order elliptic equation, discretized by discontinuous Galerkin (DG) methods, and we give a detailed analysis of the convergence for different block-relaxation strategies. We find that pointwise block-partitioning gives much better results than the classical cellwise partitioning. Both for the Baumann--Oden method and for the symmetric DG method, with and without interior penalty (IP), the block-relaxation methods (Jacobi, Gauss--Seidel, and symmetric Gauss--Seidel) give excellent smoothing procedures in a classical MG setting. Independent of the mesh size, simple MG cycles give convergence factors of 0.075--0.4 per iteration sweep for the different discretization methods studied.

This paper explains some of the convergence behaviour of iterative implicit and defect-correction schemes for the solution of the discrete steady Euler equations. Such equations are also commonly solved by (pseudo) time integration, the steady solution being achieved as the limit (for $t \to \infty $) of the solution of a time-dependent problem. Implicit schemes are then often chosen for their favourable stability properties, permitting large timesteps for efficiency. An important class of implicit schemes involving first- and second-order accurate upwind discretisations is considered. In the limit of an infinite timestep, these implicit schemes approach defect-correction algorithms. Thus our analysis is informative for both types of construction.Simple scalar linear model problems are introduced for the one-dimensional and two-dimensional cases. These model problems are analyzed in detail by both Fourier and matrix analyses. The convergence behaviour appears to be strongly dependent on a parameter $\beta...