×

zbMATH — the first resource for mathematics

Resolvent estimates in \(l_p\) for discrete Laplacians on irregular meshes and maximum-norm stability of parabolic finite difference schemes. (English) Zbl 0987.65093
This article is concerned with maximum-norm stability estimates for linear parabolic problems discretised in space on a non-uniform grid. The parabolic problem \[ u_t= \Delta u \] with homogeneous Dirichlet boundary and prescribed initial data is considered in a domain \(\Omega\) that is either an interval or a convex plane domain with smooth boundary: It is then discretized with piecewise linear finite elements using mass lumping, which can be interpreted as a finite difference scheme.
Using energy techniques and the duality mapping, resolvent estimates for the discrete Laplacian are derived in \(l^p\) for \(p\in [1,\infty]\). This proves the generation of an analytic semigroup by the solution operator of the semidiscrete problem. In contrast to previous work, the results are not restricted to quasi-uniform meshes: There is no restriction in the one-dimensional case. In two dimensions, the triangulation has to be of Delaunay type (having two triangles with a common edge then the sum of the two opposite angles is less or equal \(\pi\)).
As the authors state at the end of Section 3, the results remain true even in the three-dimensional case if the triangulation satisfies some properties that are more restrictive than those of a Delaunay triangulation. Moreover, non-conforming finite elements might be considered – under an additional assumption – as well.
In the last section, the authors also derive stability estimates in \(l^\infty(l^p)\) \((p\in [1,\infty])\) for the fully discrete problem, using a single-step method with an A-acceptable rational function.

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K05 Heat equation
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI EMIS EuDML