Abstract.
In this work, we derive a goal-oriented a posteriori error estimator for the error due to
time-discretization of nonlinear parabolic partial differential equations by the fractional step
theta method. This time-stepping scheme is assembled by three steps of the general theta method,
that also unifies simple schemes like forward and backward Euler as well as the Crank–Nicolson
method. Further, by combining three substeps of the theta time-stepping scheme, the fractional
step theta time-stepping scheme is derived. It possesses highly desired stability and numerical
dissipation properties and is second order accurate.
The derived error estimator is based on a Petrov–Galerkin formulation that is up to a numerical
quadrature error equivalent to the theta time-stepping scheme. The error estimator is assembled
as one weighted residual term given by the dual weighted residual method and one additional
residual estimating the Galerkin error between time-stepping scheme and Petrov–Galerkin
formulation.
Goal-Oriented Error Estimation for the Fractional Step Theta Scheme
