TY - JOUR
T1 - A compact FEM implementation for parabolic integro-differential equations in 2D
AU - Reddy, Gujji Murali Mohan
AU - Seitenfuss, Alan B.
AU - Medeiros, Débora de Oliveira
AU - Meacci, Luca
AU - Assunção, Milton
AU - Vynnycky, Michael
N1 - Publisher Copyright:
© 2020 by the authors. Licensee MDPI, Basel, Switzerland.
PY - 2020/10
Y1 - 2020/10
N2 - 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.
AB - 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.
KW - Backward-Euler
KW - Crank–Nicolson
KW - Parabolic integro-differential equations
KW - Quadrature rules
KW - Volterra integral term
UR - http://www.scopus.com/inward/record.url?scp=85092482803&partnerID=8YFLogxK
U2 - 10.3390/a13100242
DO - 10.3390/a13100242
M3 - Article
AN - SCOPUS:85092482803
SN - 1999-4893
VL - 13
JO - Algorithms
JF - Algorithms
IS - 10
M1 - 242
ER -