Some Convergence Estimates for Semidiscrete Type Schemes for Time-Dependent Nonselfadjoint Parabolic Equations

Lj-norm error estimates are shown for semidiscrete (continuous in time) Galerkin finite element type approximations to solutions of general time-dependent nonselfadjoint second order parabolic equations under Dirichlet boundary conditions. The semidiscrete solutions are defined in terms of given methods for the corresponding elliptic problem such as the standard Galerkin method in which the boundary conditions are satisfied exactly but also methods for which this is not necessary. The results are proved by energy arguments and include estimates for the homogeneous equation with both smooth and nonsmooth initial data.