Two New Error Estimates of a Fully Discrete Primal-Dual Mixed Finite Element Scheme for Parabolic Equations in Any Space Dimension

Abstract

We consider a fully discrete implicit scheme in which the discretization in space is performed using the PDMFEM (Primal-Dual Mixed Finite Element Method) defined in Ciarlet (The finite element method for elliptic problems. Classics in applied mathematics, 40. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, pp 415–416, 2002) and Quarteroni and Valli (Numerical approximation of partial differential equations. Springer, Berlin, p 230, 2008) for the heat equation as a model of parabolic equations in any space dimension. The time discretization is performed using a uniform mesh. The scheme is formulated in a general setting in which the pairs of finite element spaces $(V_{h}^\mathrm{div}, W_{h})\subset H_{\mathrm{div}}(\varOmega ) \times L^2(\varOmega )$ are arbitrary but satisfy the $inf-sup$ hypothesis and another known condition. The discrete unknowns of the considered scheme are the set of couples $\rho _h^n:= (p_h^n, u_h^n) \in V_h^\mathrm{div}\times W_h$ which are expected to approximate $\rho (t_n):= (\nabla u(t_n), u(t_n))$ where $\left( t_n\right) _n$ are the mesh points of the time discretization and u is the exact solution. We justify rigorously the existence and uniqueness of the discrete solution. We then prove the new convergence results which state the optimal estimate for the error $\left( \nabla u(t_n)-p_h^n\right) _n$ (resp. $\left( u_t(t_n)-\partial ^1u_h^n\right) _n$, where $\partial ^1$ denotes a discrete time derivative) in $L^\infty (H_\mathrm{div}(\varOmega ))$ (resp. $L^\infty (L^2(\varOmega ))$), under assumption that the exact solution is smooth. These results are obtained thanks to some new well developed discrete a priori estimates. We applied these results to the particular case when the spaces $(V_h^\mathrm{div}, W_h)$ are defined using the well-known Raviart–Thomas spaces of of order $l\ge 0$ and we justified that the error $\left( \rho _h^n-\rho (t_n)\right) _n$ is of order $k+h^{l+1}$ in the discrete norms of $L^\infty (H_\mathrm{div}(\varOmega ))\times W^{1,\infty }(L^2(\varOmega ))$ under assumption that the solution u is satisfying $u\in \mathscr {C}^3\left( [0,T]; \,\, H^{l+3}(\varOmega )\right)$. The convergence results obtained in this work improve the known existing ones for PDMFEM for parabolic equations which state the convergence towards $(-\nabla u(t),u(t))$ in the discrete norms of $L^\infty (L^2(\varOmega )^d)\times L^\infty (L^2(\varOmega ))$, see Johnson and Thomee (Error estimates for some mixed finite element methods for parabolic type problems. RAIRO Anal. Numér. 15(1):41–78, 1981).