Mesh dependent stability and condition number estimates for finite element approximations of parabolic problems

In this paper, we discuss the effects of spatial simplicial meshes on the stability and the conditioning of fully discrete approximations of a parabolic equation using a general finite element discretization in space with explicit or implicit marching in time. Based on the new mesh dependent bounds on extreme eigenvalues of general finite element systems defined for simplicial meshes, we derive a new time step size condition for the explicit time integration schemes presented, which provides more precise dependence not only on mesh size but also on mesh shape. For the implicit time integration schemes, some explicit mesh-dependent estimates of the spectral condition number of the resulting linear systems are also established. Our results provide guidance to the studies of numerical stability for parabolic problems when using spatially unstructured adaptive and/or possibly anisotropic meshes.