Numerische Mathematik

, Volume 102, Issue 3, pp 497–522 | Cite as

Time discretization via Laplace transformation of an integro-differential equation of parabolic type

  • William McLeanEmail author
  • Ian H. Sloan
  • Vidar Thomée


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.

Mathematics Subject Classification (2000)

 45K05 65M12 44A10 65D32 65M12 


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© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.School of MathematicsThe University of New South WalesSydneyAustralia
  2. 2.Department of MathematicsChalmers University of TechnologySweden

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