Convergence of the Discretization in Time and Space
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 t 2 (V)∩C t 0 (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 t q (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).
KeywordsInterpolation Operator Variable Step Size Rosenbrock Method Quasi Uniform Mesh Parabolic Nature
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