A Gradient Recovery-based A Posteriori Error Estimators For The Ciarlet-Raviart Mixed FEM Of The Biharmonic Equations

The main result of this paper is the a posteriori error estimate of gradient recovery type for the Ciarlet -Raviart mixed finite element methods for the biharmonic equations. This paper is a introduction for the Mixed FEM, we discussed some well-know results of the solutions and the a priori-error estimators for the biharmonic problems at first (in the Ch1-Ch3).It proposes a posteriori error estimator of gradient recovery type for the Ciarlet-Raviart formulation of the first biharmonic equation in the Ch4. By the appropriate modification of Weighted Cle'ment -type interpolation, we give the proper scaling of the gradient recovery leading to both lower and upper estimation on the non-uniform meshes. Moreover, it is proved that a posteriori error estimate is also asymptotically exact on the uniform meshes if the solution is smooth enough.In the Ch5, using the Ciarlet-Raviart formulation methods, the second biharmonic problem can be decomposed into two Poisson equations with Dirichlet boundary conditions. We derive a posteriori error estimator of gradient recovery type for the second biharmonic problem. We show that a posteriori error estimate provide both upper and lower bounds for the discretization error on the non-uniform meshes. Moreover, it is proved that a posteriori error estimate is also asymptotically exact on the uniform meshes if the solution is smooth enough.At last, the numerical results demonstrating the theoretical results are also presented by two examples in the Ch6.