Maximum-Norm Stability, Smoothing and Resolvent Estimates for Parabolic Finite Element Equations

Maximum-norm Stability, Smoothing and Resolvent Estimates for Parabolic Finite Element Equations

We survey work on stability and smoothing estimates in maximum-norm for spatially semidiscrete finite element approximations of a model parabolic equation, and related such estimates for the resolvent of the corresponding discrete elliptic operator. We end with a short discussion of stability of fully discrete time stepping methods. Résumé. Nous présentons un bilan des résultats sur la stabilit...

Maximum-norm Resolvent Estimates for Elliptic Finite Element Operators on Nonquasiuniform Triangulations

In recent years several papers have been devoted to stability and smoothing properties in maximum-norm of finite element discretizations of parabolic problems. Using the theory of analytic semigroups it has been possible to rephrase such properties as bounds for the resolvent of the associated discrete elliptic operator. In all these cases the triangulations of the spatial domain has been assum...

Maximum-norm estimates for resolvents of elliptic finite element operators

Let Ω be a convex domain with smooth boundary in Rd. It has been shown recently that the semigroup generated by the discrete Laplacian for quasi-uniform families of piecewise linear finite element spaces on Ω is analytic with respect to the maximum-norm, uniformly in the mesh-width. This implies a resolvent estimate of standard form in the maximum-norm outside some sector in the right halfplane...

Interior Maximum-norm Estimates for Finite Element Methods, Part Ii

We consider bilinear forms A(-, •) connected with second-order elliptic problems and assume that for uh in a finite element space S¡,, we have A(u U),, x) = F(x) for x m Sh with local compact support. We give local estimates for u Uf, in L^ and W^ of the type "local best approximation plus weak outside influences plus the local size of F ".

Pointwise localized error estimates for parabolic finite element equations