Error estimates and automatic time step control for nonlinear parabolic problems, I
SIAM Journal on Numerical Analysis
Continuous finite element methods which preserve energy properties for nonlinear problems
Applied Mathematics and Computation
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Galerkin Finite Element Methods for Parabolic Problems (Springer Series in Computational Mathematics)
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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.