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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)
Acknowledgements Ivo Babuška and Greg Rodin have gone far beyond the call of duty as thesis advisors. The thesis could certainly not have been written without them and I sometimes doubt whether the other vissicitudes of graduate school would have been overcome either. Jerry Bona has throughout my years at UT-Austin been generous with his time and offered(More)
We consider the discretization in time of a parabolic equation, using a representation of the solution as an integral along a smooth curve in the complex left half plane. The integral is then evaluated to high accuracy by a quadrature rule. This reduces the problem to a finite set of elliptic equations, which may be solved in parallel. The procedure is(More)
The stability of the Z.2-projection onto some standard finite element spaces Vh, considered as a map in Lp and W^, 1 ^ p < oo, is shown under weaker regularity requirements than quasi-uniformity of the triangulations underlying the definitions of the Vh. 0. Introduction. The purpose of this paper is to show the stability in Lp and Wp, for 1 < p < oo, of the(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)
In a previous paper, McLean and Thomée (to appear), we studied three numerical methods for the discretization in time of a fractional order evolution equation, in a Banach space framework. Each of the methods applied a quadrature rule to a contour integral representation of the solution in the complex plane, where for each quadrature point an elliptic(More)