Researches On Finite Element Methods For The Nonlinear Unsteady RLW And Rosenau Equations

The main aim of this paper is to investigate the second-order RLW equation,fourth-order Rosenau and Rosenau-RLW equation with the Galerkin finite element methods（FEMs）and mixed finite element methods（MFEMs）.By means of the special characteristics of the selected elements（including the conforming and nonconforming elements）and various analysis techniques,the convergence,superclose and superconvergence behavior are deeply and systematically explored for the energy conservation or stable schemes,and further the corresponding numerical experiments are done as well.The main innovations are as follows:1)For the second-order nonlinear unsteady RLW equation,based on the stable time discrete finite element approximation scheme,by use of the properties of the bilinear element and through analyzing the stability of numerical solution,the superconvergence result is derived without the restriction of time and space step ratio,which improves the results with this restriction in the previous literature.In addition,combining the two-grid method with the above scheme,the computational efficiency of the numerical experiment is largely increased but without reducing the accuracy,which further enriches the related work;2)For the nonlinear unsteady fourth-order Rosenau equation,based on the conforming bilinear element and nonconforming Quasi-Wilson element,the mixed finite element approximation scheme and related modified scheme are constructed.With the help of the properties of the elements and characteristics of the constructed schemes,the superclose and superconvergence results are achieved as well,but there are only convergence result of the conforming element in the previous literature.Moreover,according to the boundedness of the numerical solution in the sense of some norms,the unconditional superconvergence results are deduced without the introduction of time discrete system,which makes up deficiency of conditional convergence in literature,and further enriches and develops the researches of the Rosenau equation with MFEMs.Significantly,for the common nonconforming Q₁^rot element,EQ₁^rot element and CNRQ₁ element,due to the lack of relevant special properties,up to now,the conserving scheme cannot be constructed and further the boundedness of the energy norm cannot be obtained,which plays a significant role in proving the existence and uniqueness of solution and analyzing error estimations.This further shows that it is very important to select the appropriate elements,construct the proper formats and make full use of its special properties,when we analyze the Rosenau equation with the nonconforming MFEMs;3)For the nonlinear unsteady fourth-order Rosenau equation,the conforming Galerkin finite element method is discussed.Firstly,for its parabolic type,a new fully implicit numerical approximation scheme is constructed,which keeps energy conservation.Through the high accuracy analysis of the bicubic Hermite element,the results of superclose is obtained,which improves the convergence results in the previous literature.Secondly,as to its hyperbolic type,the existence and uniqueness of solution and convergence analysis for the semi-discrete scheme are given.Moreover,a fully discrete approximation scheme is constructed and the stability analysis is given.The convergent result in H²-norm is obtained,which makes up for the deficiency of this problem in hyperbolic type without the numerical analysis until now;4)For the nonlinear unsteady fourth-order Rosenau-RLW equation,based on the bilinear element and low order Nedelec element,the fully-discrete scheme is firstly proposed,and the existence and uniqueness of its solution are demonstrated from the perspective of MFEMs.With the help of the high accuracy analysis and estimation techniques,the superclose and superconvergence results are achieved.In particular,the stability of the constructed scheme is analyzed and the boundedness of the numerical solution in H¹-norm is obtained,which is helpful to get the superconvergence results without the limitation of time and space step ratio,further enriching its connotation;Firstly,for the second-order nonlinear unsteady RLW equation,a secondorder（Backward Differentiation Formula 2（BDF2））fully-discrete finite element approximation scheme is constructed,and its stability,the existence and uniqueness of the numerical solution are proved.By use of the boundedness of the numerical solution in H¹-norm and the properties of the bilinear elements,the superclose and superconvergence results are deduced without the restriction of step ratio,which improves the conditional results in the previous literature.In addition,the above results are also proved by combining the two-grid algorithm with the proposed discrete scheme.Further,numerical experiments are done not only to the proposed scheme in this paper and the other literatures,but also to the nonconforming EQ₁^rot,and the numerical results show that the two-grid method can save about 2/3 CPU time compared with the traditional method.Secondly,the MFEMs for the nonlinear fourth-order Rosenau equation is discussed.This part includes two aspects:firstly,a conservative implicit Crank Nicolson（CN）fully discrete scheme is constructed,and the existence and uniqueness of its solution are proved.With the help of the properties and estimation techniques of the conforming bilinear element,the unconditional superclose and superconvergence results of the original and intermediate variables in H¹-norm are obtained.Secondly,the problem is analyzed from the perspective of the nonconforming finite element.By use of the properties of the nonconforming QuasiWilson element,whose conforming and nonconforming part are orthogonal,a modified fully discrete approximation scheme is proposed and its conservation is deduced.The existence and uniqueness of its solution is proved through the fixed point Brouwer theorem,and the unconditional superclose and superconvergence results in the broken norm are obtained by means of the constructed scheme and the special characters of the selected element.Thirdly,the Galerkin finite element method for nonlinear unsteady fourthorder Rosenau equation is discussed.Firstly,from the perspective of the conforming Galerkin FEM for the parabolic Rosenau equation,a fully implicit backward leapfrog scheme is constructed,and its energy conservation,existence and uniqueness of solution are proved.By using the high-precision properties of the conforming bicubic Hermite element,the superclose result in H²-norm is deduced.Secondly,for the hyperbolic Rosenau equation,the existence and uniqueness of solution and convergence analysis for the semi-discrete scheme are given.Moreover,a fully discrete approximation scheme is constructed and its stability is proved.The convergent result in H²-norm is obtained with the lower smoothness of the solution by use of the projection operator.Finally,the nonlinear unsteady fourth-order Rosenau-RLW equation is analysed by MFEMs,which combines of RLW and Rosenau equation.The backward Euler（BE）and CN fully discrete schemes are given,and their stability,the existence and uniqueness of numerical solutions are proved,respectively.By use of the the high-accuracy properties of bilinear element and low-order Nedelec element,the superclose and superconvergence results of the original variable in H¹-norm and the related variable（L²）²-norm are achieved.Moreover,for the constant φ（X,t）studied in the previous literature,we prove that the constructed schemes still maintain conservation and give the numerical examples to verify our analysis.It is worth mentioning that we provide corresponding numerical examples for each parts to verify the correctness of the theoretical analysis and the effectiveness of the proposed algorithm.