Adaptive Uzawa algorithm for the Stokes equation
This work provides a theoretically grounded adaptive solver for the Stokes equation, improving upon prior methods by relaxing restrictive assumptions, which is relevant for computational fluid dynamics.
The paper presents an adaptive Uzawa algorithm for the Stokes equation, proving linear convergence with optimal algebraic rates for the residual estimator, even when linear systems are solved iteratively. The method avoids discretizing given data and does not require an interior node property for refinement.
Based on the Uzawa algorithm, we consider an adaptive finite element method for the Stokes system. We prove linear convergence with optimal algebraic rates for the residual estimator (which is equivalent to the total error), if the arising linear systems are solved iteratively, e.g., by PCG. Our analysis avoids the use of discrete efficiency of the estimator. Unlike prior work, our adaptive Uzawa algorithm can thus avoid to discretize the given data and does not rely on an interior node property for the refinement.