Abstract.
We consider the discretization in time of an inhomogeneous parabolic integro-differential equation, with a memory term of convolution type, in a Banach space setting. The method is based on representing the solution as an integral along a smooth curve in the complex plane which is evaluated to high accuracy by quadrature, using the approach in recent work of López-Fernández and Palencia. This reduces the problem to a finite set of elliptic equations with complex coefficients, which may be solved in parallel. The method is combined with finite element discretization in the spatial variables to yield a fully discrete method. The paper is a further development of earlier work by the authors, which on the one hand treated purely parabolic equations and, on the other, an evolution equation with a positive type memory term.
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The authors acknowledge the support of the Australian Research Council.
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McLean, W., Sloan, I. & Thomée, V. Time discretization via Laplace transformation of an integro-differential equation of parabolic type. Numer. Math. 102, 497–522 (2006). https://doi.org/10.1007/s00211-005-0657-7
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DOI: https://doi.org/10.1007/s00211-005-0657-7