The Interior Penalty Morley-Wang-Xu Element Method For The Elliptic Singular Perturbation Problem

The finite element method is an important method for solving elliptic boundary value problems.The study of the numerical solution of the elliptic problem and the convergence of the discrete method has important theoretical significance and practical application value.In this paper,we study the second-order elliptic problem and fourth-order elliptic singular perturbation problem based on the Morley element discrete numerical method.The main contents are as follows:This paper first studies the Morley element method for second-order elliptic problems.Considering that the numerical method obtained by directly discretizing the second-order elliptic problem with Morley element is divergent.We modify the variational form using the interior penalty discontinuous finite element method.The second-order elliptic problem is then based on the Morley element discretization interior penalty discontinuous finite element method.Then we analyze the mandatory and boundedness of the new bilinear form.The a priori error analysis based on Morley discrete interior point penalty discontinuous finite element method is given.Using a special interpolation operator,the second-order elliptic problem is constructed based on Morley element discrete interior point penalty discontinuous finite element method for a posteriori error estimation,and the validity of posterior error estimation is proved.Secondly,a discrete interior penalty discontinuous finite element method based on Morley-Wang-Xu element for the fourth-order elliptic singular perturbation problem in arbitrary dimension space is proposed,namely the interior penalty Morley-Wang-Xu element method.The Morley-Wang-Xu element has the least local degree of freedom,and the discrete biharmonic operator are convergent.For the second-order elliptic operator in the fourth-order elliptic singular perturbation problem,the above-mentioned interior penalty discontinuous form is used.Then some a posteriori error estimates are established using the bubble function technique,and the optimal error estimate of the interior penalty Morley-Wang-Xu element method is given under the assumption that the exact solution has the minimum regularity.This a priori error analysis is convergence with grid size h and parameters ?.We also considered the boundary layer in the error analysis.For general parameters ?,the inner point penalty Morley-Wang-Xu element method can achieve the optimal half-order convergence.Further we improved the convergence order in the following two cases:1)The parameters ? have consistent upper and lower bounds on the mesh size h;2)?(?)h?,with ?>1.