High Accuracy Finite Element Analysis For Two Kinds Of Nonlinear Partial Differential Equations

In this paper,the mixed finite elements method is used to study the following two problems.In the first part,a mixed finite element method is investigated for the Maxwell's equations in Debye medium with the thermal effect.In particular,the zero order Nédélec element(Q₀₁ ×Q10),the piecewise constant space Q₀ element,and the bilinear element Q₁₁ are used to approximate the electric field E and the polarization electric field P,the magnetic field H,and the temper-ature field u,respectively.With the help of the known high accuracy results,mean-value technique and interpolation postprocessing approach,the O??+h²?order superclose estimates and global superconvergence results are obtained for the linearized backward Euler fully discrete scheme of the coupled model.Here h is the subdivision parameter and ? is the time step,? = O(h^1+?),?>0.At the same time,a numerical experiment is given to varifly the correctness of the theoretical analysis and the validity of our method.In the second part,a two-grid method?TGM?of the H¹-Galerkin mixed finite element method is established f'or the semilinear Sobolev equations with the bilinear element Q₁₁ and zero order Raviart-Thomas?R-T?element(Q10 ×Q₀₁).Similar to the analytical techniques in part one,the superclose estimates and global superconvergence results of the original variable u in H¹-norm and flux variable ???=?u in H?div?-norm are deduced for the backward Euler fully discrete TGM scheme.At the same time,the numerical results show that the computing cost of the proposed TGM is only half of the conventional ?1-Galerkin mixed finite element methods,which illustrates that the TGM is indeed a very effective high precision algorithm.