Time discretization of an evolution equation via Laplace transforms

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Abstract
Following earlier work by Sheen, Sloan, and Thomée concerning parabolic equations we study the discretization in time of a Volterra type integro‐differential equation in which the integral operator is a convolution of a weakly singular function and an elliptic differential operator in space. The time discretization is accomplished by using a modified Laplace transform in time to represent the solution as an integral along a smooth curve extending into the left half of the complex plane, which is then evaluated by quadrature. This reduces the problem to a finite set of elliptic equations with complex coefficients, which may be solved in parallel. Stability and error bounds of high order are derived for two different choices of the quadrature rule. The method is combined with finite‐element discretization in the spatial variables.

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