Mixed Virtual Element Method for Linear Parabolic Integro-Differential Equations

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.