Study on Matrix Analysis For Steady Problems |
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  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 . Its extension to finite elements in 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.