A New Error Estimate for a Primal-Dual Crank-Nicolson Mixed Finite Element Using Lowest Degree Raviart-Thomas Spaces for Parabolic Equations

Abstract

We consider the heat equation as a model for parabolic equations. We establish a fully discrete scheme based on the use of Primal-Dual Lowest Order Raviart-Thomas Mixed method combined with the Crank-Nicolson method. We prove a new convergence result with convergence rate towards the “velocity” $P(t)=-\nabla u(t)$ in the norm of $L^2(H_\mathrm{div})$, under assumption that the solution is smooth. The order is proved to be two in time and one in space. This result is obtained thanks to a new well developed discrete a priori estimate. The convergence result obtained in this work improves the existing one for PDMFEM (Primal-Dual Mixed Finite Element Method) for Parabolic equations which states the convergence towards the velocity in only the norm of $L^\infty \left( \left( L^2\right) ^d\right)$, see [6, Theorem 2.1, p. 54].

This work is an extension of [1] which dealt with new error estimates of a MFE scheme of order one in time. It is also motivated by the work [9] in which a full discrete Crank Nicolson scheme based on another MFE approach, different from the one we use here, is established in the two dimensional space.

Supported by MCS team (LAGA Laboratory) of the “Université Sorbonne- Paris Nord”.