Least-Squares Galerkin Procedures For Some Evolution Equations

The development of modern science and technology is greatly dependent on achievement and progress of physics, chemistry and biology, while accuracy of those subjects is an important guarantee for making progress. Mathematical models are the basis of an accuracy of those subjects. A large number of phenomena are modelled by evolutionary partial differential equations. It is frequently the case that a good approximation of some function of the gradient of the solution is at least as important as the approximation of the solution itself. So the mixed Galerkin finite element method is widely used [1, 2. 3, 4]. However, as is well known, for stability the finite element spaces are required to satisfy the Ladyzhenskaya [5]-Babuska [6]-Brezzi [2] (LBB) consistency condition. It restricts the choice of approximation subspaces, with some of the best known and widely used finite element spaces excluded. In the least-squares mixed (LSM) approach a least-squares residual minimization is introduced for the mixed system. This method has some advantages as follows: Firstly, the LSM method is not subject to the the LBB condition, so we can select the finite element spaces more flexibly. It has been proved in [7]. Secondly, the discrete system is symmetric positive definite.There has been much research on least-squares finite element schemes and their applications to various boundary value problems of elliptic equations. Some systematic theory on the ellipticity of schemes and convergence of approximate solutions have been established, see [7-15]. Moreover, the least-squares finite element methods have been extended to time-dependent problems, see [16, 17, 18, 19, 20], in which some very effective numerical results were shown.In this dissertation, we analyze the least-squares finite element schemes for the models of some evolution equations: parabolic integro-differential equations. Sobolev...