Abstract

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.

Highlights

  • As regards existing software platforms which could be used for a finite-element implementation without having to resort to lengthy programming from first principles, the main choices are freefem++ [29], deal.II [30], iFEM [31], FEniCS [32], DUNE [33], FEATool [34], Matrix Laboratory (MATLAB)’s Partial Differential Equation Toolbox [35] and COMSOL Multiphysics [36]; it is not immediately obvious how to solve parabolic integro-differential equations (PIDEs) with any of these

  • This is perhaps surprising in the case of MATLAB, which has become increasingly popular in recent years for the numerical solutions of various partial differential equations (PDEs) in structural mechanics and heat transfer [37,38,39,40,41,42,43,44,45,46,47], and has proven to be an excellent tool for academic education in general [40]

  • We have presented a compact implementation using the finite-element method for solving linear PIDEs in arbitrary 2D geometries, both in terms of sets of pseudocode and in terms of a MATLAB code, which we have made openly available; we note that such a code is not available in other competing FEM software, such as freefem++ or deal.II

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Summary

Introduction

Parabolic integro-differential equations (PIDEs) arise in various physical contexts, such as heat conduction in materials with memory [1,2,3], the compression of poro-viscoelastic media [4], nuclear reactor dynamics [5], epidemic phenomena in biology [6] and drug absorption/release [7,8].Existing and unique results from such kinds of problems can be found in [9,10,11,12].Inevitably, such equations have to be solved numerically and various approaches have been developed for this, such as spectral methods, spline and collocation, the method of lines and finite-element methods [13,14,15,16,17,18,19,20,21]. As regards existing software platforms which could be used for a finite-element implementation without having to resort to lengthy programming from first principles, the main choices are freefem++ [29], deal.II [30], iFEM [31], FEniCS [32], DUNE [33], FEATool [34], MATLAB’s Partial Differential Equation Toolbox [35] and COMSOL Multiphysics [36]; it is not immediately obvious how to solve PIDEs with any of these This is perhaps surprising in the case of MATLAB, which has become increasingly popular in recent years for the numerical solutions of various partial differential equations (PDEs) in structural mechanics and heat transfer [37,38,39,40,41,42,43,44,45,46,47], and has proven to be an excellent tool for academic education in general [40]

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