High Accuracy Finite Element Method For Nonlinear Evolution Partial Differential Equations

Researches on efficient numerical methods for nonlinear partial differential equations always been regarded as one of hot and difficult issues in computational mathematics field.This thesis aims to make an intensive and systematic study of finite element methods for some nonlinear evolution partial differential equations possessing important physical background,such as nonlinear Benjamin-Bona-Mahoney(BBM)equation、nonlinear parabolic equation,nonlinear Poisson-Nernst-Planck(PNP)equations,nonlinear Joule heating equations(including time integer and fractional types).The suitable finite elements are selected and different high precision numerical approximation schemes or algorithms are constructed for the above equations,such as Galerkin finite element method,mixed finite element method,H1-Galerkin finite element method,discontinuous finite element method,two-grid finite element method and so on.Based on new analysis approaches and sharper estimating techniques,the corresponding super-closeness and super-convergence results are derived.The main innovations of this thesis are as follows:(1)For the nonlinear BBM equation,by employing quasi-Wilson element、bilinear element and zero order Raviart-Thomas(R-T)element,nonconforming constrained rotated Q1(CNQ1rot)element and piecewise constant Q0 ×Q0 element,three kinds of finite element methods are developed,respectively.By use of special characters of elements and new analysis techniques,the super-closeness and super-convergence properties for the considered semi-discrete and fully-discrete schemes are derived without the restriction on the ratio between the space mesh size h and the time step τ.(2)For the nonlinear parabolic equation,a new discontinuous finite element method is proposed with the famous nonconforming Wilson element.The convergence analysis are discussed for the considered semi-discrete scheme,the linearized Euler and BDF2 fullydiscrete schemes are investigated,and optimal error estimates of order O(h2)are obtained in the new norm,which improve the result of Wilson element with optimal order of O(h)in the classical energy norm.(3)For the nonlinear PNP equations,since the gradient of electrostatic potential ψ appeared in the coupled terms P1▽ψ and P2▽ψ,there exist two deficiencies in the existing studies:One is that only the suboptimal error estimates in L2-norm of the concentration of particles p1 and p2 are derived.The other one is that the regularity requirement of solutionψ is very strong(belongs to H3(Q)∩ W2,∞(Ω))so as to get the high accuracy estimates of bilinear element on rectangular meshes.We pay more attention to discuss the applications of conforming triangular linear element and nonconforming rotated Q1(EQ1rot)element for this equations,and to give the corresponding super-closeness and super-convergence results,respectively for the semi-discrete and fully-discrete schemes when ψ only belongs to H3(Ω),and high accuracy analysis tricks and new ideas are used.Thus the above two disadvantages are overcome and it provides a model for further investigation of other efficient methods.(4)For the nonlinear Joule heating equations(including time integer and fractional types),different from optimal error estimates and super-convergence analysis for rectangular bilinear element in the existing literature,by use of the mean-value and integral identity technique of linear element on the combined big element for uniform triangular meshes、novel ideas and approaches in the proofs skillfully,the difficulty caused by the coupled term δ(u)|▽φ|2 is overcome,and the super-closeness and super-convergence results for temperature u and electric potential φ are derived in the energy norm with lower regularity of φ only belongs to H3(Q)instead of H3(Ω)∩ W2,∞(Ω)and u belongs to H3(Ω)instead of H4(Ω).Thus the disadvantages in the previous works are overcome.It should be mentioned that numerical tests are carried out for the semi-discrete and fullydiscrete schemes considered in each part above,which verify the correctness of the theoretical analysis and the efficiency of the proposed methods.