Study on Matrix Analysis For Steady Problems

Analysis of Discrete Maximum Principle in Matrix Analysis for Steady Problems

Abstract

If the solutionof a given boundary value problem satisfies a maximum principle, then aproperly designed approximation should behave in the same way. A numericalscheme that does not generate spurious global extrema in the interior of thecomputational domain is said to satisfy a discrete maximum principle (DMP). As in the continuous case, theprecise formulation of this criterion is problem-dependent. In particular, thezero row sum property (second rule from Section 1.6.3) has the sameimplications as the constraint ·v _0 in continuous maximum principles. In the contextof finite difference approximations to linear elliptic problems, sufficientconditions of DMP were formulated and proven by Varga [340] as early as in1966. These conditions are related to the concept of monotone operators and, in particular, M-matrices which play an important role in numerical linearalgebra [339, 354]. A general approach to DMP analysis for finite differenceoperators was developed by Ciarlet [63]. Its extension to finite elements in[64] features a proof of uniformconvergence, as well as simple geometric conditions that ensure thevalidity of DMP for a piecewise-linear Galerkin discretization of the (linear)model problem.