A stabilized mixed finite element method for steady and unsteady reaction–diffusion equations

In this paper, we propose a new mixed finite element method, called stabilized mixed finite element method, for the approximation of steady reaction–diffusion partial differential equations (PDEs). The method is obtained by translating the primal second-order PDEs into a first-order mixed system, and then adding some suitable elementwise residual terms multiplied by a stabilization parameter to the weak formulation. The new method is compatible, i.e., the added terms equal to zero in the continuous case. Furthermore, it is mesh-independent, i.e., the stabilization parameter is independent of the mesh size. We prove both coercive and continuous properties in a weighted norm for the corresponding new mixed bilinear formulation. These assure that the finite element function spaces do not require to satisfy the classical Ladyzhenkaya–Babuska–Brezzi (LBB) consistency condition. Therefore, the widely used Lagrange finite element can be adopted. A simple proof of a priori error estimate with lower order regularity requirement is discussed, and numerical experiments confirm the efficiency and reliability of the new stabilized mixed method. Finally, the method is applied to solving unsteady reaction–diffusion equations. Error estimates are also given, and numerical examples still support the theoretical analysis very well.