Novel analysis approach for the convergence of a second order time accurate mixed finite element scheme for parabolic equations

We design a fully discrete MFE (Mixed Finite Element) scheme, based on a PDM (Primal-Dual Mixed) formulation, combined with the Crank-Nicolson method for multidimensional parabolic equations in which the finite element spaces are included in H-div and L-2. We state and prove novel convergence results with convergence rate towards the "velocity" p(t) = -del u(t) and "pressure" u(t) in, respectively, the L-infinity(H-div) and W-1,W-infinity(L-2) norms, under assumption that the solution is smooth, in a general setting of finite element spaces which require only the known discrete assumption of inf -sup and a coerciveness hypothesis similar to that developed in [15, Theorem 7.4.1, Page 249] for the case of elliptic equations. The order is proved to be two in time and is optimal in space. These results are obtained thanks to a new well-developed discrete a priori estimate. We apply these results to two known families of finite element spaces. The first one is the RTl (Raviart-Thomas finite elements of an arbitrary order l) and the second one has been proposed by Brezzi, Douglas and Marini in dimension d = 2 and by Brezzi, Douglas, Duran and Fortin when d = 3. For these two families, it is proved that the order is respectively k(2) + h(l+1) and k(2) + h(l) in the L-infinity(H-div) x W-1,W-infinity(L-2)- norm (L-infinity(H-div) for velocity and W-1,W-infinity(L-2) for pressure) when using respectively RTl and spaces of piecewise polynomials of degree less than or equal l, where h (resp. k) is the mesh size of the space (resp. time) discretization. Some other possible second order time accurate PDMFE schemes are also discussed. This work is an extension and improvement of two previous works. The first one is [3] in which similar estimates are proved for first order time (order one in time) accurate PDMFE under a restrictive hypothesis between the finite elements spaces. This hypothesis is a particular case of the one considered here. The second work is [4] in which a new convergence result with convergence rate towards only the "velocity" p(t) = del u(t) in only the norm of L(2)2(H-div) is proved using the particular case of the lowest degree Raviart-Thomas Spaces RT0 as discretization in space combined with the use of Crank-Nicolson method as discretization in time. This contribution is motivated by two pioneer works in MFEMs (Mixed Finite Element Methods) for two dimensional parabolic equations. The first one is [13] in which the convergence of semi-discrete MFE (discretization is only in space) schemes is proved towards velocity and pressure in only L-infinity((L-2)(2)) and L(infinity)8(L-2) norms, see [13, Theorem 2.1]. The second work is [18] in which a fully discrete scheme, based on a MFE formulation which is different from that we consider here, is established. The finite element spaces considered in [18] are included in (L-2)(2) and H-0(1). In addition to the difference in the spaces considered here and in [18], the formulation of the scheme presented here is simpler than that of [18].