The Weak Galerkin Finite Element Method For Nonlinear Klein-gordon Equation

As a relativistic form of schrodinger equation,Klein-Gordon equation has many ap-plications in mathematics and physics.In this paper,the numerical solution of nonlinear Klein-Gordon equation by weak Galerkin finite element method is studied.Compared with the traditional Galerkin finite element method,the weak Galerkin finite element method uses discrete weak gradient to replace the classical gradient,allows discontinuous function between the interior domain and the edges of element,and the element parti-tion shape can be changeable,which has the advantage of flexibility.This paper can be divided into six parts.In chapter one,we introduce the historical background of the Klein-Gordon equation first,which plays an important role in the study of nonlinear dynamics.Furthermore,we summarize the basic definition of weak Galerkin finite element method and its application in various partial differential equations.In chapter two,the concepts of discrete weak finite element approximation space and discrete weak gradient are proposed,which allows the use of completely discontinuous functions in finite element processes.In chapter three,for the one-dimensional nonlinear Klein-Gordon problem,we assume that the nonlinear term satisfies the Lipschitz continuity condition.Besides,a weak Galerkin finite element scheme with backward Euler discretization for time is proposed,and the results show that the errors of numerical schemes converge optimally in the energy norm and suboptimally in the L~2norm.Chapter four and five are aimed at two special nonlinear Klein-Gordon equations.They have the following general form:u_tt-Δu+αu_t+δg（u）=f in （?）×(0,T],（1）u（t）=0 on（?）（?）×[0,T],（2）u（0）=φ₀,u_t（x,0）=φ₁ in （?）.（3）We analyze the weak Galerkin finite element scheme when the nonlinear term g（u）=sinu or|u|^γu,0≤γ<∞respectively.And the Lipschitz continuity of the nonlinear term g（u）is also proved.At the same time,we propose a weak Galerkin finite element method with implicitθ-scheme discretization for time,and obtain that the numerical scheme error of the grid size h converges to the optimal order under the energy norm.Numerical results are in agreement with the theoretical analysis.The last chapter is the summary of this paper and the prospect of the future research direction.