Some Convergence Results of a Multidimensional Finite Volume Scheme for a Semilinear Parabolic Equation with a Time Delay

Delay differential equations occur in many applications such as ecology and biology. They have long played important roles in the literature of theoretical population dynamics, and they have been continuing to serve as useful models. There is a huge literature on the approximation of ODDEs (Ordinary Delay Differential Equations) whereas a few contributions, w.r.t. ODDEs, dealt with DPDEs (Delay Partial Differential Equations). Some of these works which dealt with the numerical approximation of DPDEs consider only the one dimensional case. In this contribution we construct a linearized implicit scheme, in which the space discretization is performed using a general class of nonconforming finite volume meshes, to approximate a semilinear parabolic equation with a time delay. We prove the existence and uniqueness of the discrete solution. We derive a discrete a priori estimate which allows to derive error estimates in discrete seminorms of L-infinity (H-0(1)) and W-1,W-2 (L-2).