Mixed virtual element method for integro-differential equations of parabolic type

This article presents and analyzes a mixed virtual element approach for discretizing parabolic integro-differential equations in a bounded subset of \(\mathbb {R}^2\) ${\mathbb{R}}^2$ , in addition to the backward Euler approach for temporal discretization. With the help of the intermediate projection along with Fortin and \(L^2\) $L^2$ projections, we effectively tackle the treatment of integral terms in both the fully discrete and semi-discrete analysis. This inclusion leads to the derivation of optimal a priori error estimates with an order of \(O(h^{k+1})\) $O(h^{k+1})$ for the two unknowns. Furthermore, we present a systematic analysis that outlines the step-by-step process for achieving super convergence of the discrete solution, with an order of \(O(h^{k+2})\) $O(h^{k+2})$ . Several computational experiments are discussed to validate the proposed scheme’s computational efficiency and support the theoretical conclusions.