Published by De Gruyter

Volume 15 Issue 4

Issue of Journal of Numerical Mathematics

Submit manuscript

Contents
Journal Overview

December 4, 2007

Overlapping additive Schwarz preconditioners for isotropic elliptic problems with degenerate coefficients

S. Beuchler, S. V. Nepomnyaschikh

Page range: 245-276

Abstract

The degenerate isotropic boundary value problem –∇(ω 2 ( x )∇ u ( x, y )) = f ( x, y ) on the unit square (0, 1) 2 is considered in this paper. The weight function is assumed to be of the form ω 2 (ξ) = ξ α , where α ≥ 0. This problem is discretized by piecewise linear finite elements on a triangular mesh of isosceles right triangles. The system of linear algebraic equations is solved by a preconditioned conjugate gradient method using a domain decomposition preconditioner with overlap. Two different preconditioners are presented and the optimality of the condition number for the preconditioned system is proved for α ≠ 1. The preconditioning operation requires O( N ) operations, where N is the number of unknowns. Several numerical experiments show the performance of the proposed method.

December 4, 2007

On the numerical solution of a semilinear elliptic eigenproblem of Lane–Emden type, II: Numerical experiments

F. J. Foss, R. Glowinski, R. H. W. Hoppe

Page range: 277-298

Abstract

In this second part of our two-part article, we present and discuss the corresponding numerical results from implementations of the numerical algorithms described in the first part. With these results, we observed that • operator splitting applied to the associated time-dependent problem is suitable for solving only the first eigenproblem, • among those tried, the perturbation and arclength continuation approach was the sole effective and robust approach for solving higher eigenproblems, • on the eigenproblems for which (undamped or damped) Newton's method converged, it was without question the most efficient.

December 4, 2007

Unified edge-oriented stabilization of nonconforming FEM for incompressible flow problems: Numerical investigations

S. Turek, A. Ouazzi

Page range: 299-322

Abstract

This paper deals with various aspects of edge-oriented stabilization techniques for nonconforming finite element methods for the numerical solution of incompressible flow problems. We discuss two separate classes of problems which require appropriate stabilization techniques: First, the lack of coercivity for nonconforming low order approximations for treating problems with the symmetric deformation tensor instead of the gradient formulation in the momentum equation (‘Korn's inequality’) which particularly leads to convergence problems of the iterative solvers for small Reynolds (Re) numbers. Second, numerical instabilities for high Re numbers or whenever convective operators are dominant such that the standard Galerkin formulation fails and leads to spurious oscillations. We show that the right choice of edge-oriented stabilization is able to provide simultaneously excellent results regarding robustness and accuracy for both seemingly different cases of problems, and we discuss the sensitivity of the involved parameters w.r.t. variations of the Re number on unstructured meshes. Moreover, we explain how efficient multigrid solvers can be constructed to circumvent the problems with the arising ‘non-standard’ FEM data structures, and we provide several examples for the numerical efficiency for realistic flow configurations with benchmarking character.

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

Article formats
Original research articles

Information on Submission Process