Three Numerical Methods For Fourth Order Elliptic Singular Perturbation Problems

In this thesis,three numerical methods for solving the fourth order elliptic singular perturbation problems are studied,which are the Hellan-Herrmann-Johnson(HHJ)mixed finite element method,the reduced local C0 discontinuous Galerkin method(reduced LCDG method)and the Helmholtz decomposition based mixed element method.First of all,the HHJ mixed finite element method is considered.By introducing the second-order tensor,the fourth-order elliptic singular perturbation problem can be transformed into a second-order system.Then,the well-posedness of the HHJ mixed finite element method is obtained.Finally,the error analysis of HHJ hybrid finite element method is carried out.Then the reduced LCDG method is used to solve the fourth-order singular elliptic perturbation problem.The reduced LCDG method is a local C~0discontinuous finite element method.If requiring the normal-normal part of the stress tensor to be continuous across all the interior edges additionally,then the reduced LCDG method is transformed into Hellan-Herrmann-Johnson method.Therefore,the theoretical technology of the latter method is taken for reference,and the well-posedness of the reduced LCDG method is deduced,and the prior error estimate is carried out as well.In the end,for the fourth-order elliptic singular perturbation problem,the commutative diagram and Helmholtz decomposition are used to reduce it,and the mixed variational form of the decomposition is obtained.At the same time,the coordinated finite element is used to discrete the mixed variational form,then the well-posedness of the continuous problem and the discrete problem is proved,and the error is estimated.