Recovery Type A Posteriori Error Estimate Of FEMs For The Stokes-Darct System

In this paper, we consider the a posteriori error estimate for the finite element approximation of the Stokes-Darcy system.The paper consists of four chapters.In chapter1, the preliminaries are given. In§1.1, we give some basic concepts of the finite element method and the Galerkin variational principle including its definition, the existence theorem and the equivalent theorem of the variational principle we used in this paper. In§1.2, the definitions of a posteriori error estimate and two important kinds of posteriori error estimate§-residual type and recovery type. The procedure of the adaptive mesh refine-ment by utilizing a posteriori error estimate is also presented in this section. In§1.3, we give a short introduction about super convergence and postprocessing techniques which are the theoretical basis of a posteriori error estimate as well as some common superconvergence principles and Hood-Taylor element’s in-terpolation postprocessing. This chapter is the basis of the following chapters in the paper.In chapter2, first we consider the Stokes-Darcy flow problem, which is stated in§2.1, and its corresponding weak formulation. The model problem is as follows.The Stokes-Darcy flow equations satisfy the Beavers-Joseph-Saffman-Jones interface boundary conditionand the exterior boundaryFor the velocity and the pressure in the fluid region we use the Hood-Taylor element, and for the pressure in porous media region we use conforming piece-wise quadratic element. Then, the superclose result of finite element method for Stokes-Darcy equations are presented in§2.2, refer to [18],In chapter3, based on superclose result over the uniform isosceles right-triangle meshes, the results of superconvergence and asymptotical exactness of a posteriori error estimate are derived by two postprocessing techniques.First Method. Based on the whole superconvergence between recon-structive solution and exact solution, for the uniform isosceles right-triangle partition we introduce recovery postprocessing operators Ⅱ2h*and P2h*to con-struct the exact solution (μ, p,φ) and get the first recovery type of a posteriori estimator ηgSecond Method. Based on the whole superconvergence between recon-structive solution and exact solution, for the uniform isosceles right-triangle partition we introduce two new recovery postprocessing operator Gh to con-struct the exact solution (▽μ,▽φ) and get the second recovery type of a posteriori estimator ηgFurthermore, we can prove the the recovery type of a posteriori error estimator is asymptotically exact when the finite element method satisfies the global superconvergence condition.In chapter4, a numerical example is tested and it shows that the two posteriori error estimates are effective.