Instationary Stokes Equations Research Articles

A Well-Posed First Order System Least Squares Formulation of the Instationary Stokes Equations

In this paper, a well-posed simultaneous space-time First Order System Least Squares formulation is constructed of the instationary incompressible Stokes equations with slip boundary conditions. As a consequence of this well-posedness, the minimization over any conforming triple of finite element spaces for velocities, pressure and stress tensor gives a quasi-best approximation from that triple. The formulation is practical in the sense that all norms in the least squares functional can be efficiently evaluated. Being of least squares type, the formulation comes with an efficient and reliable a posteriori error estimator. In addition, a priori error estimates are derived, and numerical results are presented.

Optimal Sobolev regularity for the Stokes equations on a 2D wedge domain

In this note we prove that the solution of the stationary and the instationary Stokes equations subject to perfect slip boundary conditions on a 2D wedge domain admits optimal regularity in the $$L^p$$ -setting, in particular it is $$W^{2,p}$$ in space. This improves known results in the literature to a large extend. For instance, in Maier and Saal (Discrete Contain Dyn Syst Ser S 7(5):1045–1063, 2014) [Theorem 1.1 and Corollary 3] it is proved that the Laplace and the Stokes operator in the underlying setting have maximal regularity. In that result the range of p admitting $$W^{2,p}$$ regularity, however, is restricted to the interval $$1<p<1+\delta $$ for small $$\delta >0$$ , depending on the opening angle of the wedge. This note gives a detailed answer to the question, whether the optimal Sobolev regularity extends to the full range $$1<p<\infty $$ . We will show that for the Laplacian this does only hold on a suitable subspace, but, depending on the opening angle of the wedge domain, not for every $$p\in (1,\infty )$$ on the entire $$L^p$$ -space. On the other hand, for the Stokes operator in the space of solenoidal fields $$L^p_\sigma $$ we obtain optimal Sobolev regularity for the full range $$1<p<\infty $$ and for all opening angles less than $$\pi $$ . Roughly speaking, this relies on the fact that an existing “bad” part of $$L^p$$ for the Laplacian is complemented to the space of solenoidal vector fields.

Parameter-dependent Stokes problems in vector-valued spaces and applications

The stationary and instationary Stokes equations with operator coefficients in abstract function spaces are studied. The problems are considered in the whole space, and equations include small parameters. The uniform separability of these problems is established.MSC:35Q30, 76D05, 34G10, 35J25.

Efficient Solution of Large-Scale Saddle Point Systems Arising in Riccati-Based Boundary Feedback Stabilization of Incompressible Stokes Flow

We investigate numerical methods for solving large-scale saddle point systems which arise during the feedback control of flow problems. We focus on the instationary Stokes equations that describe instationary, incompressible flows for moderate viscosities. After a mixed finite element discretization we get a differential-algebraic system of differential index two [J. Weickert, Navier-Stokes Equations as a Differential-Algebraic System, Preprint SFB393/96-08, Department of Mathematics, Chemnitz University of Technology, Chemnitz, Germany, 1996]. To reduce this index, we follow the analytic ideas of [J.-P. Raymond, SIAM J. Control Optim., 45 (2006), pp. 790--828] coupled with the projection idea of [M. Heinkenschloss, D. C. Sorensen, and K. Sun, SIAM J. Sci. Comput., 30 (2008), pp. 1038--1063]. Avoiding this explicit projection leads to solving a series of large-scale saddle point systems. In this paper we construct iterative methods to solve such saddle point systems by deriving efficient preconditioners b...

Reduced Basis A Posteriori Error Bounds for the Instationary Stokes Equations: A Penalty Approach

AbstractWe present reduced basis approximations and associated rigorous a posteriori error bounds for the instationary Stokes equations. The proposed method is an extension of the penalty approach introduced in Gerner and Veroy (2011b) to the time-dependent setting: The introduction of a penalty term enables us to develop a posteriori error bounds that do not rely on the expensive calculation of inf-sup stability constants and that are thus computationally very efficient. Numerical results show the rapid convergence of reduced basis approximations and demonstrate the effects of the penalty parameter.

Divergence-free wavelet bases on the hypercube: Free-slip boundary conditions, and applications for solving the instationary Stokes equations

We construct wavelet Riesz bases for the usual Sobolev spaces of divergence free functions on that have vanishing normals at the boundary. We give a simultaneous space-time variational formulation of the instationary Stokes equations that defines a boundedly invertible mapping between a Bochner space and the dual of another Bochner space. By equipping these Bochner spaces by tensor products of temporal and divergence-free spatial wavelets, the Stokes problem is rewritten as an equivalent well-posed bi-infinite matrix vector equation. This equation can be solved with an adaptive wavelet method in linear complexity with best possible rate, that, under some mild Besov smoothness conditions, is nearly independent of the space dimension. For proving one of the intermediate results, we construct an eigenfunction basis of the stationary Stokes operator.

The instationary Stokes equations in weighted Bessel-potential spaces

We investigate the solvability of the instationary Navier–Stokes equations with fully inhomogeneous data in a bounded domain \(\Omega \subset {{\mathbb{R}}^{n}} \). The class of solutions is contained in \(L^{r}(0, T; H^{\beta, q}_{w} (\Omega))\), where \({H^{\beta, q}_{w}} (\Omega)\) is a Bessel-Potential space with a Muckenhoupt weight w. In this context we derive solvability for small data, where this smallness can be realized by the restriction to a short time interval. Depending on the order of this Bessel-Potential space we are dealing with strong solutions, weak solutions, or with very weak solutions.

The stationary Navier-Stokes equations in weighted Bessel-potential spaces

We investigate the solvability of the instationary Stokes equations with fully inhomogeneous data in \({L^r(0,T;H^{\beta,q}_w(\Omega))}\) , where \({H^{\beta,q}_w(\Omega)}\) is a Bessel-potential space with a Muckenhoupt weight w. Depending on the order of this Bessel-potential space we are dealing with strong solutions or with very weak solutions. Whereas in the context of lowest regularity one obtains solvability with respect to inhomogeneous data by dualization, this is more delicate in the case of higher regularity, where one has to introduce some additional time regularity. As a preparation, we introduce a generalization of the Stokes operator that is appropriate to the context of very weak solutions in weighted Bessel-potential spaces.

The penalty method applied to the instationary stokes equations

A mathematical analysis of the penalty method applied to the initial boundary value problem for the Stokes equations is presented. We prove a regularity result for the penalty equations by means of a spectral Galerkin method. Based upon this result the discretization in time (by a multistep integration formula) and in space (by a finite element method) is shown to give an optimal L 2-error estimate.