Space-Time Finite Element Methods For Singular Parabolic Problems

Singular differential equations are widely used in modeling problems rising from the fields of engineering and physics. So it's of practical importance to solve these equations numerically. Nowadays, many researchers in the field of computational mathematics are dedicating themselves to finite difference methods, standard finite element methods (FEMs) and spectral methods. However, these methods are confronted with some difficulties in solving problems with variable domains or discontinuity. This paper focuses on the space-time FEM, whose main feature is that the spatial and the temporal variables are considered equivalently. Thus high orders of accuracy in both directions are achieved. At the meantime, this kind of method is highly adaptive, and suitable for dealing with discontinuous and singular problems. In this paper, linear and semi-linear singular parabolic problems are discussed. The existence and uniqueness of the finite element solutions and their theoretical error estimates are obtained. Numericalsimulation results are presented.The first part of this paper discusses a linear singular parabolic problem. In chapter one, the Lions Theorem is used in proving the existence of the generalized solution. Then a continuous space-time FEM is analyzed and the optimal error estimates in weighted H~1 norm and weighted L~2 norm are presented in chapter two. Part two focuses on a semi-linear problem. The existence of the generalized solution is proved first. And in the fourth chapter, a space-time FEM continuous in space and discontinuous in time is discussed. A weighted H~1 semi-norm error estimate is obtained as well as a weighted L~2 norm estimate for the jumping terms. At last,numerical simulation results in (weighted) Hl semi-norm and (weighted) L2 norm of two experiments are presented in chapter five. The examples include a semi-linear singular parabolic problem and a forward-backward heat equation.