Abstract
A continuous Galerkin finite element time-stepping method for the approximation of nonlinear initial value problems is analyzed within an hp-context. We derive a priori error bounds in the L2- and H1-norm that are explicit with respect to the time steps and the approximation orders. In particular, it is shown that, for analytic solutions (with certain possible start-up singularities) exponential convergence rates can be achieved. Moreover, we prove that the scheme superconverges at the nodal points of the time partition. Numerical experiments illustrate the performance of the method.
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Wihler, T.P. An A Priori Error Analysis of the hp-Version of the Continuous Galerkin FEM for Nonlinear Initial Value Problems. J Sci Comput 25, 523–549 (2005). https://doi.org/10.1007/s10915-004-4796-2
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DOI: https://doi.org/10.1007/s10915-004-4796-2