Convergence analysis of the local defect correction method for diffusion equations

Summary. This paper is concerned with the convergence analysis of the local defect correction (LDC) method for diffusion equations. We derive a general expression for the iteration matrix of the method. We consider the model problem of Poisson's equation on the unit square and use standard five-point finite difference discretizations on uniform grids. It is shown via both an upper bound for the norm of the iteration matrix and numerical experiments, that the rate of convergence of the LDC method is proportional to H ² with H the grid size of the global coarse grid. Mathematics Subject Classification (2000):65N22, 65N50     

Accurate estimates of solutions of second order recursions

Two important types of two dimensional matrix-vector and second order scalar recursions are studied. Both types possess two kinds of solutions (to be called forward and backward dominant solutions). For the directions of these solutions sharp estimates are derived, from which the solutions themselves can be estimated.     

Multirate methods for the transient analysis of electrical circuits

Multirate integration is an important tool to increase the speed of the transient analysis of circuits. This paper shows an approach for the “Compound-Fast” multirate algorithm how to control the errors at the coarse and the refined time-grid by means of the independent stepsizes of these grids. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stable computation of solutions of unstable linear initial value recursions

An algorithm is given for approximating dominated solutions of linear recursions, when some initial conditions are given. The stability of this algorithm is investigated and expressions for the truncation and rounding errors are derived. A number of practical questions concerning the algorithm is considered, and several numerical examples sustain the theory.     

Characterizations of dominant and dominated solutions of linear recursions

Summary Dominated solutions of a linear recursion (i.e. solutions which are outgrown by other ones) cannot be computed in a stable way by forward recursion. We analyze this dominance phenomenon more closely and give practically significant characterizations for dominated and dominant solutions. For a dominated solution, in particular, this leads to a stable computational method.     

Efficient solution of Maxwell’s equations for geometries with repeating patterns by an exchange of discretization directions in the aperiodic Fourier modal method

The aperiodic Fourier modal method in contrast-field formulation is a numerical discretization and solution technique for solving scattering problems in electromagnetics. Typically, spectral discretization is used in the finite periodic direction and spatial discretization in the orthogonal direction. In the light of the fact that the structures of interest often have a large width-to-height ratio and that the two discretization approaches have different computational complexities, we propose exchanging the directions for spatial and spectral discretization. Moreover, if the scatterer has repeating patterns, swapping the discretization directions facilitates the reuse of previous computations. Therefore, the new method is suited for scattering from objects with a finite number of periods, such as gratings, memory arrays, metamaterials, etc. Numerical experiments demonstrate a considerable reduction of the computational costs in terms of time and memory. For a specific test case considered in this paper, the new method (based on alternative discretization) is 40 times faster and requires 100 times less memory than the method based on classical discretization.     

On solving boundary value problems for multi-scale systems using asymptotic approximations and multiple-shooting

This paper describes a method for obtaining the numerical solution of certain singularly perturbed boundary value problems for linear systems of ordinary differential equations whose solutions involve dynamics with multiple time scales. A technique to decouple different time scales using a Riccati transformation is presented. For the slow modes, which provide smooth solutions, regular perturbation expansion and multiple shooting strategies are combined. Fast modes, which are assumed to be significant only in endpoint layers, are computed after appropriate stretchings have been made in the system. Advantages of this approach include adaptability, flexible use of output points, and automatic determination of layer thicknesses.Dedicated to Professor Germund Dahlquist, with the hope that he will long continue to make significant and singular perturbations to the numerical analysis of differential equations.This research was supported in part by the National Science Foundation under Grant Number MCS-8301665 and by the U.S. Army Research Office under Grant Number DAAG29-82-K-0197.     

A local defect correction technique for time‐dependent problems

In this article a local defect correction technique for time‐dependent problems is presented. The method is suitable for solving partial differential equations characterized by a high activity, which is mainly located, at each time, in a small part of the physical domain. The problem is solved at each time step by means of a global uniform coarse grid and a local uniform fine grid. Local and global approximation are improved iteratively. Results of numerical experiments illustrate the accuracy, the efficiency, and the robustness of the method. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Local defect correction with slanting grids

The local defect correction (LDC) method is used to solve a convection‐diffusion‐reaction problem that contains a high‐activity region in a relatively small part of the domain. The improvement of the solution on a coarse grid is obtained by introducing a correction term computed from a local fine‐grid solution. This article studies problems where the high‐activity region is covered with a rectangular fine grid not aligned with the axes of the global domain. This study shows that the resulting method is less expensive than both a uniform refinement and tensor product grid method. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 1–17, 2004.