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Int. J. Numer. Anal. Mod., 21 (2024), pp. 504-527.
Published online: 2024-06
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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} }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.