Goal-Oriented Error Estimation for the Fractional Step Theta Scheme

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 timestepping 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.2010 Mathematical subject classification: 65M60, 65L60, 65L70, 65M15.

On the Adjoint Equation in Fluid-Structure Interaction

“…For several benchmark problems, excellent error reduction and effectivity indices could be measured. Similar to Section 4, the adjoint equation was derived from a space-time formulation and discretized with the help of a Galerkin representation of the Fractional-Step-θ scheme [34,35].…”

Error estimation and adaptivity for differential equations with multiple scales in time

“…However, the consistency error between the Galerkin approach and the trapezoidal rule is of the same order such that both approaches must be considered as separate discretization schemes. In [14,15] we have demonstrated how the more efficient trapezoidal rule can be used for solving the problem while including the consistency error within the estimator. It shows that the quadrature error is indeed of the same (or even higher) order than further contributions to the error estimator.…”

Goal Oriented Time Adaptivity Using Local Error Estimates

“…The standard approach for goal oriented adaptivity is the dual weighted residual (DWR) method [4,5]. It is most commonly used in the context of space-adaptivity for PDEs, but there is also work specifically on time-adaptivity [6][7][8]. DWR uses the adjoint (dual) problem to perform a posteriori adaptivity and get error bounds for the QoI.…”