Numerical Theory And Calculation Of Two Kinds Of 1d Linear Equations By The Weak Galerkin Mixed Finite Element Methods

Sobolev equation,which contains a third-order spatiotemporal mixed derivatives,is a partial differential equation.It can describe wave motion in media and steep non-linear waves with dispersion and diffusion equilibrium.It plays an essential role in fluid mechanics and other disciplines.Kd V equation is a class of unsteady-state partial differential equations with(2n+1)th-order spatial derivatives.It is used to describe nonlinear long wave problems,which contains soliton wave phenomena in the analyt-ical solution,Is a significant odd order differential equation.The main task content of this dissertation is to innovate the stabilizer free weak Galerkin mixed finite ele-ment methods and use them to solve Sobolev equation and Kd V equation.Stabilizer free weak Galerkin mixed finite element methods are extensions of the weak Galerkin mixed finite element method.Firstly,the primary problems are transformed into a sys-tem of first-order equations by introducing one or more intermediate variables to obtain the Galerkin variational form.Then,the weak functions are defined to approximate the continuous functions,and the discrete weak derivatives in P^k+1(Ih)are defined to approximate the primary derivatives,thus to achieve the stability of numerical solution and the optimal convergence rate of error without stabilizer in a suitable piecewise finite element space.These methods can simplify the structure of the existing weak Galerkin finite element scheme to a certain extent,and make the numerical analysis and practical calculation of Sobolev Equation and Kd V equation easier.The first part of this dissertation is to solve the one-dimensional variable coeffi-cient linear Sobolev Equation by using the new stabilizer free H~1-weak Galerkin mixed finite element method.Firstly,we combine the traditional H~1-Galerkin mixed finite element method with the stabilizer free weak Galerkin finite element method,and con-struct a semi discrete variational scheme.Moreover,without the need to prove the inf-sup condition(also known as the ladyzhenskaya-Babu?ka-Brezzi condition,generally shortened to the LBB condition),we prove the well posedness of the semi discrete vari-ational scheme,as well as the optimal order prior error estimates of the weak derivative and weak function under the L~2norm,by the bounded relationship between the weak derivative L~2norm and the weak function L~2norm.Then we use the implicit?scheme to discretize the time to obtain the fully discrete stabilizer free H~1-weak Galerkin mixed fi-nite element scheme,and prove the well posedness of the problem.Moreover we obtain the spatiotemporal optimal order error estimates for weak derivative and weak function in L~2norm.Finally,the theoretical analysis results are verified by a numerical example of constant coefficients and a numerical examples of variable coefficients.The second part of this dissertation is to solve the one dimensional linear Korteweg-de Vries equation(abbreviated as Kd V equation)by using the new modified weak Galerkin mixed finite element method.We use the definition of weak function at the el-ement boundary in modified weak finite element method,that is,the boundary in weak function is replaced by the mean value of the interior,which contribute to reduce the de-gree of freedom in the numerical calculation operation.In this work,the stabilizer free semi discrete scheme is obtained by using new modified weak Galerkin mixed finite el-ement to approximate the continuous mixed variational form.Then,three conservation laws of the scheme are proved based on special equation for weak derivative,and the optimal order error estimates under uniform gird is proved when the polynomial order k is even and the mesh number N is odd by using special global projection.Finally,numerical examples are given to verify the feasibility of the method.