Mixed Finite Element Method For Several Partial Differential Equations

This paper discusses the numerical solutions to several partial differential equations. By using two splitting least squares mixed finite element method to study the second order parabolic equations and Burgers equation, using the adaptive least squares mixed finite element method to study high order parabolic equation and two dimensional viscoelastic problems. Two splitting least squares mixed finite element method is first discretizing the equation in time layer, then introducing an unknown variable to get a coupled equations. And then decompose the domain, the coupled system can be split into two independent sub-systems, thus reduce the difficulty and scale of primal problems. Theoretical analysis shows that the method yield the approximate solutions for the primal problems with optimal accuracy in ( )L2Ωnorm. Adaptive least-squares mixed finite element method transformate the original problem into pairs of low-order equations by introduction of an unknown variable, and use this method to each of equation, we can obtain the optimal approximation of the primal problem and its flux. We composed a posteriori error estimators with the least-squares functional used in adaptive refinement algorithm, and it estimated the problems effectively. Since the method can reduce the smoothness requirements of the finite element space, so that it can more easily construct a mixed finite element space and improve the accuracy of calculation.The main results of this paper are outlined as follows:In chapter one, the paper briefly introduces the development process of the finite element method and some basic knowledge used in the latter part of this paper.In chapter two and three, this paper discuss the second order parabolic equations and Burgers equation by using two splitting least squares mixed finite element method, respectively. In chapter four and five, this paper discuss the high order parabolic equation and two dimensional viscoelastic problems by using adaptive least squares mixed finite element method, respectively.