Mixed virtual element method for linear parabolic integro-differential equations

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(hk+1) for the two unknowns p(x,t) and σ(x,t). In addition, a step-by-step analysis is proposed for the super convergence of the discrete solution of order O(hk+2). The fully discrete case has also been analyzed and discussed to achieve O(τ) in time. Several computational experiments are discussed to validate the proposed schemes computational efficiency and support the theoretical conclusions