Gradient recovery-based adaptive stabilized mixed FEM for the convection–diffusion–reaction equation on surfaces

Abstract In this article, we consider a gradient recovery-based adaptive stabilized mixed finite element method (FEM) for solving the convection–diffusion–reaction equation on surfaces. By defining a new total flux on the surface, we firstly transform the original equation into an equivalent mixed first-order formulation. Then, by using both the convection–convection term and the div–div term to stabilize the weak formulation, we prove the discrete inf–sup condition and the error estimate for the mixed FEM. In this new method, the equal order Lagrangian elements are allowed for the primal variable and the total flux, and a high accurate numerical solution for the flux in the H 1 space is obtained. The impact of the curvature on the algorithm is also investigated. Additionally, to overcome the low resolution and the non-physical oscillation in simulating the convection-dominated problem, we study a gradient recovery-based adaptive stabilized mixed FEM. Finally, we show some numerical experiments to verify the theoretical prediction and efficiency of the studied method.