Galerkin Time-stepping Methods for Nonlinear Parabolic Equations

The interest for time Galerkin and corresponding space-time finite element methods has been linked during the last decade to the development of adaptivity of mesh selection for evolution PDE’s. Certain issues as, e.g., a posteriori estimates, estimates of optimal order and regularity, fully discrete schemes with mesh modification, etc., have been extensively considered in the framework of Continuous and Discontinuous Galerkin methods, cf., e.g., [2,7–16]. This is probably partly due to the fact that in principle space-time Galerkin methods provide freedom for (almost) arbitrary selection of the space time mesh, [11], and partly due merely to the fact that, as in the elliptic case, the properties of variational methods can be studied in an easier, more systematic and clearer way than properties of their pointwise counterparts, i.e., finite difference methods. Still many issues related to the above problems have to be investigated, mainly for nonlinear evolution PDE’s. The purpose of this paper is to provide a rather comprehensive a priori analysis of a class of variational in time methods, and in particular of the Discontinuous and Continuous Galerkin methods, for nonlinear parabolic equations. It turns out that the limitations of the approach of [7, 8] that was further developed (although from a different perspective) for nonlinear problems in [2] can be overcome by adopting a direct approach based on energy type variational arguments.