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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
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