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2020 (English)In: Algorithms, E-ISSN 1999-4893, Vol. 13, no 10, article id 242Article in journal (Refereed) Published
Abstract [en]
Although two-dimensional (2D) parabolic integro-differential equations (PIDEs) arise in many physical contexts, there is no generally available software that is able to solve them numerically. To remedy this situation, in this article, we provide a compact implementation for solving 2D PIDEs using the finite element method (FEM) on unstructured grids. Piecewise linear finite element spaces on triangles are used for the space discretization, whereas the time discretization is based on the backward-Euler and the Crank-Nicolson methods. The quadrature rules for discretizing the Volterra integral term are chosen so as to be consistent with the time-stepping schemes; a more efficient version of the implementation that uses a vectorization technique in the assembly process is also presented. The compactness of the approach is demonstrated using the software Matrix Laboratory (MATLAB). The efficiency is demonstrated via a numerical example on an L-shaped domain, for which a comparison is possible against the commercially available finite element software COMSOL Multiphysics. Moreover, further consideration indicates that COMSOL Multiphysics cannot be directly applied to 2D PIDEs containing more complex kernels in the Volterra integral term, whereas our method can. Consequently, the subroutines we present constitute a valuable open and validated resource for solving more general 2D PIDEs.
Place, publisher, year, edition, pages
MDPI, 2020
Keywords
parabolic integro-differential equations, backward-Euler, Crank-Nicolson, quadrature rules, Volterra integral term
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-286233 (URN)10.3390/a13100242 (DOI)000584199400001 ()2-s2.0-85092482803 (Scopus ID)
Note
QC 20201124
2020-11-242020-11-242023-03-29Bibliographically approved