The Space-Time Discontinuous Finite Element Methods For Parabolic Integro-Differential Equations

Discontinuous space-time finite element methods deal with the spatial and the temporal variables conformably .Thus high orders of accuracy in both dircetions are achieved . At the same time ,this kind of method is highly adaptive , and suitable for dealing with discontinuous and singular problems . The discontinuous space-time finite element methods ,that is, approximating functions is discontinuous at the nodes of partition in time, is considered for a parabolic integro-differential equations ,and the theoretic analysis of the finite element solution is given . The first part of this paper considers a semi-linear parabolic integro-differential equations with a weakly singular kernal in the memory term . Error estimates in L_∞(L₂) norm,that is maxium-norm in time ,L₂ norm in space are obtained. The second part discuss a semi-linear parabolic integro-differential equations .The approach is based on combination of finite element and finite difference techniques.In the discrete intervals of time, using properties of Lagrange interpolating polynomials at Radau nodes , eliminate the restriction to space-time meshes of conventional space-time discontinuous Galerkin methods.Error estimate in L_∞(L₂) norm are obtained.