Gradient Recovery Based A Posteriori Error Estimation Of Discontinuous Galerkin Methods For Biharmonic Equation

Biharmonic equation is a typical fourth-order partial differential equation,and it is an important partial differential equation model in elastic thin plate,biophysics and other fields.Numerical methods of biharmonic equation have always been a hot and difficult research in related fields.Discontinuous finite element methods have become an important numerical method to solve kinds of partial differential equations and practical problems because of its high plasticity and adaptability.In this thesis,we introduce an intermediate variable to rewrite the biharmonic equation into two elliptic equations,and we present the corresponding discrete scheme for biharmonic equation based the interior penalty discontinuous Galerkin scheme of elliptic equation.Secondly,a gradient recovery based a posteriori error estimation of the discontinuous Galerkin method for elliptic equation is proposed,in which the gradient recovery is based on area harmonic average,and the reliability and efficiency of a posteriori error estimator are further proved.Then,a gradient recovery based a posteriori error estimation of discontinuous Galerkin methods for biharmonic equation is constructed,and the reliability and efficiency of the estimator are established.Finally,some numerical examples are given to show the performance of the proposed a posteriori error estimator in adaptive algorithm,it is show that the new recovery type posterior error estimates are asymptotically exact.