Adaptive finite element methods for the Stokes problem with discontinuous viscosity

Abstract:

Discontinuity in viscosities is of interest in many applications. Classical adaptive numerical methods perform under the restricting assumption that the discontinuities of the viscosity are captured by the initial partition. This excludes applications where the jump of the viscosity takes place across curves, manifolds or at a priori unknown positions. We present a novel estimate measuring the distortion of the viscosity in $L^{q}$ for a $q<\infty$, thereby allowing for any type of discontinuities. This estimate requires the velocity $\mathbf {u}$ of the Stokes system to satisfy the extra regularity assumption $\nabla \mathbf {u} \in L^{r}(\Omega )^{d\times d}$ for some $r>2$. We show that the latter holds on any bounded Lipschitz domain provided the data belongs to a smaller class than those required to obtain well-posedness. Based on this theory, we introduce adaptive finite element methods which approximate the solution of Stokes equations with possible discontinuous viscosities. We prove that these algorithms are quasi-optimal in terms of error compared to the number of cells. Finally, the performance of the adaptive algorithm is numerically illustrated on insightful examples.