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Time discretization of an evolution equation via Laplace transforms
Authors:McLean  William; Thomee  Vidar
Institution: 1 School of Mathematics, The University of New South Wales, Sydney 2052, Australia 2 Department of Mathematics, Chalmers University of Technology, SE-41296 Göteborg, Sweden
Abstract:Following earlier work by Sheen, Sloan, and Thomée concerningparabolic equations we study the discretization in time of aVolterra type integro-differential equation in which the integraloperator is a convolution of a weakly singular function andan elliptic differential operator in space. The time discretizationis accomplished by using a modified Laplace transform in timeto represent the solution as an integral along a smooth curveextending into the left half of the complex plane, which isthen evaluated by quadrature. This reduces the problem to afinite set of elliptic equations with complex coefficients,which may be solved in parallel. Stability and error boundsof high order are derived for two different choices of the quadraturerule. The method is combined with finite-element discretizationin the spatial variables.
Keywords:evolution equation  memory term  Laplace transform  parallel algorithm  quadrature error
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