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We treat the time discretization of an initial-value problem for a homogeneous abstract parabolic equation by rst using a representation of the solution as an integral along the boundary of a sector in the right half of the complex plane, then transforming this into a real integral on the nite interval 0; 1], and nally applying a standard quadrature formula(More)
We study the numerical approximation of an integro-differential equation which is intermediate between the heat and wave equations. The proposed discretization uses convolution quadrature based on the first-and second-order backward difference methods in time, and piecewise linear finite elements in space. Optimal-order error bounds in terms of the initial(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)
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)
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 study the spatially semidiscrete lumped mass method for the model homogeneous heat equation with homogeneous Dirichlet boundary conditions. Improving earlier results we show that known optimal order smooth initial data error estimates for the standard Galerkin method carry over to the lumped mass method whereas nonsmooth initial data estimates require(More)