Convergence of the Discretization in Time and Space

Abstract

We study spatial approximations of Rosenbrock schemes by means of finite elements coupled with a Galerkin method. The error is evaluated in a discrete L ² _t (V)∩C ⁰ _t (H) -norm under the usual assumption that the solution is temporally smooth. Since we are mainly interested in studying Rosenbrock methods of order p≥2, we need H ^q _t (V) -regularity with q≥3. The parabolic nature of our equations often yields smooth solutions, at least after an initial transitional phase. The obtained convergence results show a natural separation of temporal and spatial error terms, which simplifies their control in an adaptive solution process. Keeping the spatial discretization error below a prescribed tolerance would nearly result in a time integration procedure similar to the unperturbed case. Variable step sizes are also allowed, but the relation between them must remain bounded (quasiuniform meshes).