The Time Discontinuous Galerkin Finite Element Methods For The Sobolev Equations

There has been a wide range of applications of Sobolev equations in fluid mechanics, thermodynamics,and many other aspects of mathematical physics,such as the percolation theory when the fluid fllows through the cracks,the transfer problem of the moisture in the soil,and the heat conduction problem in different materials and so on.This paper is divided into two chapters.Chapter 1 introduces the time discontinuous Galerkin finite element method for the quasilinear Sobolev equations.The completely discrete schemes are derived by diseretizing in the space variables by means of a Galerkin finite element method and then discretizing the time variable by a finite difference method.The time discontinuous Galerkin finite element method is to discrete the time variable also by the Galerkin method,and construct the scheme which treats the time and sapce variables similarly.The approximate solution will be sought as a piecewise polynomial of degree at most q-1 about time variable.Then,we can get high accuracy for both the space variable and the time variable.According to the above methods,this chapter built the time discontinuous Galerkin finite element scheme for the quasilinear Sobolev equations.The approximate solution was sought as a piecewise polynomial of degree at most q-1 about time variable.The error estimates were derived.The quasilinar Sobolev equations has the mixed differential term u_xxt,which increases the difficulty for the study,and it is also the significance of this paper.This chapter is divided into five sections.Sectionâ… introduces the time dis- continuous Galerkin finite element method briefly.In sectionsâ…¡andâ…¢,we give the mathematical model and the time discontinuous Galerkin finite element scheme. respectively.In sectionâ…£,the existence and uniqueness of the solution are proved by the fixed-point theorem.In sectionâ…¤,we make the L^∞(0,T;H¹(Ω))-norm error estimates by a non-standard elliptic projection and a inverse problem.Chapetr 2 introduces the time diseontinnous H¹-Galerkin mixed finite element method for the linear Sobolev equations.First,H¹-Galerkin mixed finite element method changes the model into a first-order system about unknown variable u and its fluxσ,then use H¹-Galerkin finite element method to it.The approximating finite element spaces V_h×W_h,to variable u and fluxσrespectively,were allowed to be of different polynomial degrees.Hence, estimations can be obtained which distinguished the better approximation properties of V_h and W_h.Compared to standard mixed methods,the H¹-Galerkin mixed finite element method was not subject to LBB-consistency condition.Moreover,the quasiuniformity condition was not imposed on the finite element mesh.To get high precision numerical method,this chapter combined the advantages of H¹-Galerkin mixed finite element method and the time discontinuous Galerkin finite element method to linear Sobolev equations.The schemes were bulit to approximate solution by a piecewise polynomial of degree at most q-1 about time variable.The optimal error estimates were derived.This chapter is also divided into five sections.Sectionâ… introduces the H¹-Galerkin mixed finite element method briefly.In sectionsâ…¡andâ…¢,we give the mathematical model of the linear Sobolev equations and the time discontinuous H¹-Galerkin mixed finite element scheme,respectively.In sectionâ…£,the existence and uniqueness of the solution are proved by the Gronwall inequality.In sectionâ…¤,we make the error analysis and achieve H¹-norm error estimates for variable u and its fluxσ,where the convergence orders are optimal both about the space variable and the time variable.