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Higher order variational time discretizations for nonlinear systems of ordinary differential equations

by Matthies, G.; Schieweck, F..

Series: 2011-23, Preprints

65M12 Stability and convergence of numerical methods
65M60 Finite elements, ~Rayleigh-Ritz and Galerkin methods, finite methods

Starting from the known continuous Galerkin-Petrov (cGP) and discontinuous Galerkin (dG) time discretization method of some polynomial order $k$, we propose two new variational time discretizations for the system of ordinary differential equations (ODE) associated with the semi-discrete finite element solution of a parabolic partial differential equation. Both methods are based on a time polynomial ansatz of higher order $k+1$ where the global smoothness of the discrete solution is also one level higher than that of the original method cGP or dG, respectively. Therefore, the total number of unknowns and the computational costs are not increasing but the accuracy is improved by one order. For the new methods, we prove optimal order a priori error estimates in the maximum norm for a general nonlinear ODE-system with a Lipschitz-continuous right-hand side. We show that the $C^1$-continuous Galerkin-Petrov methods are A-stable and that the $C^0$-continuous dG-methods are strongly A-stable (L-stable). Moreover, we prove that, in each case, the new higher order method and the original method coincide at the endpoints of the time intervals and that their difference can be computed by a simple post-processing step with low computational costs. With this relationship we have proven in the nonlinear case that, in particular, the cGP(2)- and dG(1)-method are superconvergent of order 4 and 3, respectively, at the endpoints of the time intervals. Finally, we present numerical results for the Burgers equation which confirm the theoretical results.

discontinuous Galerkin method, continuous Galerkin--Petrov method, A-stability, L-stability, error estimates