Interior a posteriori error estimates for time discrete approximations of parabolic problems

a posteriori error estimates for time discrete approximations o

Numerical integrators for motion under a strong constraining force

This paper deals with the numerical integration of Hamiltonian systems in which a stiff anharmonic potential causes highly oscillatory solution behavior with solution-dependent frequencies. The impulse method, which uses micro- and macro-steps for the integration of fast and slow parts, respectively, does not work satisfactorily on such problems. Here it is shown that variants of the impulse method with suitable projection preserve the actions as adiabatic invariants and yield accurate approximations, with macro-stepsizes that are not restricted by the stiffness parameter

Backward difference time discretization of parabolic differential equations on evolving surfaces

A linear parabolic differential equation on a moving surface is discretized in space by evolving surface finite elements and in time by backward difference formulas (BDF). Using results from Dahlquist's G-stability theory and Nevanlinna & Odeh's multiplier technique together with properties of the spatial semi-discretization, stability of the full discretization is proven for the BDF methods up to order 5 and optimal-order convergence is shown. Numerical experiments illustrate the behaviour of the fully discrete method