Published by De Gruyter

Volume 27 Issue 1

Issue of Journal of Numerical Mathematics

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Journal Overview

Publicly Available February 28, 2019

Frontmatter

Page range: i-iii

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January 29, 2018

Preconditioning methods for eddy-current optimally controlled time-harmonic electromagnetic problems

Owe Axelsson, Dalibor Lukáš

Page range: 1-21

Abstract

Time-harmonic problems arise in many important applications, such as eddy current optimally controlled electromagnetic problems. Eddy current modelling can also be used in non-destructive testings of conducting materials. Using a truncated Fourier series to approximate the solution, for linear problems the equation for different frequencies separate, so it suffices to study solution methods for the problem for a single frequency. The arising discretized system takes a two-by-two or four-by-four block matrix form. Since the problems are in general three-dimensional in space and hence of very large scale, one must use an iterative solution method. It is then crucial to construct efficient preconditioners. It is shown that an earlier used preconditioner for optimal control problems is applicable here also and leads to very tight eigenvalue bounds and hence very fast convergence such as for a Krylov subspace iterative solution method. A comparison is done with an earlier used block diagonal preconditioner.

February 1, 2018

Convergence of explicitly coupled simulation tools (co-simulations)

Thilo Moshagen

Page range: 23-36

Abstract

In engineering, it is a common desire to couple existing simulation tools together into one big system by passing information from subsystems as parameters into the subsystems under influence. As executed at fixed time points, this data exchange gives the global method a strong explicit component. Globally, such an explicit co-simulation schemes exchange time step can be seen as a step of an one-step method which is explicit in some solution components. Exploiting this structure, we give a convergence proof for such schemes. As flows of conserved quantities are passed across subsystem boundaries, it is not ensured that system-wide balances are fulfilled: the system is not solved as one single equation system. These balance errors can accumulate and make simulation results inaccurate. Use of higher-order extrapolation in exchanged data can reduce this problem but cannot solve it. The remaining balance error has been handled in past work by recontributing it to the input signal in next coupling time step, a technique labeled balance correction methods . Convergence for that method is proven. Further, the lack of stability for co-simulation schemes with and without balance correction is stated.

February 1, 2018

Balanced-norm error estimates for sparse grid finite element methods applied to singularly perturbed reaction–diffusion problems

Stephen Russell, Martin Stynes

Page range: 37-55

Abstract

We consider a singularly perturbed linear reaction–diffusion problem posed on the unit square in two dimensions. Standard finite element analyses use an energy norm, but for problems of this type, this norm is too weak to capture adequately the behaviour of the boundary layers that appear in the solution. To address this deficiency, a stronger so-called ‘balanced’ norm has been considered recently by several researchers. In this paper we shall use two-scale and multiscale sparse grid finite element methods on a Shishkin mesh to solve the reaction–diffusion problem, and prove convergence of their computed solutions in the balanced norm.

About this journal

Objective
The Journal of Numerical Mathematics (formerly East-West Journal of Numerical Mathematics) contains high-quality papers featuring contemporary research in all areas of Numerical Mathematics. This includes the development, analysis, and implementation of new and innovative methods in Numerical Linear Algebra, Numerical Analysis, Optimal Control/Optimization, and Scientific Computing. The journal will also publish applications-oriented papers with significant mathematical content in computational fluid dynamics and other areas of computational engineering, finance, and life sciences.

Topics

Numerical Mathematics
Computational Mathematics
Applied Mathematics
Numerical Linear Algebra
Numerical Analysis
Optimal Control/Optimization
Scientific Computing
Computational Fluid Dynamics
Finance
Life Sciences

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