Error Estimates For The Discontinuous Space-Time Finite Element Solution Of Singular Parabolic Problems

The parabolic equations with singular coefficients are very important ones in practical problems e.g. problems in nuclear physics, gas dynamics, mechanics, boundary layer theory, nonlinear field and optics. In recent years, many researchers have studied such kind of problems by difference method, projection method, etc.. And they have achieved excellent results. As to the finite element methods (FEM) for singular differential equations, the main achievements were gained by traditional FEM. In recent years, the FEM developed quickly. Many nonstandard finite element methods were proposed, such as space-time FEM, moving FEM and discontinuous FEM, etc.. However, the papers which are on the discussion of singular differential equations solved by nonstandard FEM are not appeared as so for.In this paper, we give the theoretical error estimates on linear and semi-linear singular parabolic equations by discontinuous space-time nonstandard FEM which is discontinuous in time but continuous in space. For linear problem, we give the superconvergent results at the nodes of time and the optimal order estimate in the interior of the time intervals, respectively. These results are gained through the stability of dual problems and Galerkin orthogonality with the limitation of the space-time meshes; For the semi- linear equation, the technique of combination of the FEM and finite difference method is adopted to prove the existence and uniqueness of the weak solution without the limitation of the space-time meshes. And the error estimate for maximum norm in time, weighted L₂ -norm in space is obtained.