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Volume 21, Issue 4
Mixed Virtual Element Method for Linear Parabolic Integro-Differential Equations

Meghana Suthar & Sangita Yadav

DOI: 10.4208/ijnam2024-1020

Int. J. Numer. Anal. Mod., 21 (2024), pp. 504-527.

Published online: 2024-06

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  • Abstract

This article develops and analyses a mixed virtual element scheme for the spatial discretization of linear parabolic integro-differential equations (PIDEs) combined with backward Euler’s temporal discretization approach. The introduction of mixed Ritz-Volterra projection significantly helps in managing the integral terms, yielding optimal convergence of order $O(h^{k+1})$ for the two unknowns $p(x, t)$ and $\sigma(x, t).$ In addition, a step-by-step analysis is proposed for the super convergence of the discrete solution of order $O(h^{k+2}).$ The fully discrete case has also been analyzed and discussed to achieve $O(\tau)$ in time. Several computational experiments are discussed to validate the proposed schemes computational efficiency and support the theoretical conclusions.

  • Keywords

Mixed virtual element method, parabolic integro-differential equation, error estimates, super-convergence.

  • AMS Subject Headings

65M15, 65N12, 65N15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-21-504, author = {Suthar , Meghana and Yadav , Sangita}, title = {Mixed Virtual Element Method for Linear Parabolic Integro-Differential Equations}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2024}, volume = {21}, number = {4}, pages = {504--527}, abstract = {

This article develops and analyses a mixed virtual element scheme for the spatial discretization of linear parabolic integro-differential equations (PIDEs) combined with backward Euler’s temporal discretization approach. The introduction of mixed Ritz-Volterra projection significantly helps in managing the integral terms, yielding optimal convergence of order $O(h^{k+1})$ for the two unknowns $p(x, t)$ and $\sigma(x, t).$ In addition, a step-by-step analysis is proposed for the super convergence of the discrete solution of order $O(h^{k+2}).$ The fully discrete case has also been analyzed and discussed to achieve $O(\tau)$ in time. Several computational experiments are discussed to validate the proposed schemes computational efficiency and support the theoretical conclusions.

}, issn = {2617-8710}, doi = {https://doi.org/10.4208/ijnam2024-1020}, url = {http://global-sci.org/intro/article_detail/ijnam/23200.html} }
TY - JOUR T1 - Mixed Virtual Element Method for Linear Parabolic Integro-Differential Equations AU - Suthar , Meghana AU - Yadav , Sangita JO - International Journal of Numerical Analysis and Modeling VL - 4 SP - 504 EP - 527 PY - 2024 DA - 2024/06 SN - 21 DO - http://doi.org/10.4208/ijnam2024-1020 UR - https://global-sci.org/intro/article_detail/ijnam/23200.html KW - Mixed virtual element method, parabolic integro-differential equation, error estimates, super-convergence. AB -

This article develops and analyses a mixed virtual element scheme for the spatial discretization of linear parabolic integro-differential equations (PIDEs) combined with backward Euler’s temporal discretization approach. The introduction of mixed Ritz-Volterra projection significantly helps in managing the integral terms, yielding optimal convergence of order $O(h^{k+1})$ for the two unknowns $p(x, t)$ and $\sigma(x, t).$ In addition, a step-by-step analysis is proposed for the super convergence of the discrete solution of order $O(h^{k+2}).$ The fully discrete case has also been analyzed and discussed to achieve $O(\tau)$ in time. Several computational experiments are discussed to validate the proposed schemes computational efficiency and support the theoretical conclusions.

Meghana Suthar & Sangita Yadav. (2024). Mixed Virtual Element Method for Linear Parabolic Integro-Differential Equations. International Journal of Numerical Analysis and Modeling. 21 (4). 504-527. doi:10.4208/ijnam2024-1020
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