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In 1973, H. Fujii investigated discrete versions of the maximum principle for the model heat equation using piecewise linear finite elements in space. In particular, he showed that the lumped mass method allows a maximum principle when the simplices of the triangulation are acute, and this is known to generalize in two space dimensions to triangulations of… (More)

Let Ω be a convex domain with smooth boundary in R d. 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… (More)

The numerical solution of a parabolic equation with memory is considered. The equation is first discretized in time by means of the discontin-uous Galerkin method with piecewise constant or piecewise linear approximating functions. The analysis presented allows variable time steps which, as will be shown, can then efficiently be selected to match… (More)

We consider finite element methods for a model second-order el-liptic equation on a general bounded convex polygonal or polyhedral domain. Our first main goal is to extend the best approximation property of the error in the í µí± 1 ∞ norm, which is known to hold on quasi-uniform meshes, to more general graded meshes. We accomplish it by a novel proof… (More)

We consider finite element operators defined on " rough " functions in a bounded polyhedron Ω in R N. Insisting on preserving positivity in the approximations, we discover an intriguing and basic difference between approximating functions which vanish on the boundary of Ω and approximating general functions which do not. We give impossibility results for… (More)

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