Finite element methods for the Stokes problem on complicated domains

Abstract

Abstract. It is a standard assumption in the error analysis of finite element methods that the underlying finite element mesh has to resolve the physical domain of the modeled process. In case of complicated domains appearing in many applications such as ground water flows this requirement sometimes becomes a bottleneck. The resolution condition links the computational complexity a priorily to the number (and size) of geometric details. Therefore even the coarsest available discretization can lead to a huge number of unknowns. In this paper, we will relax the resolution condition and introduce coarse (optimal order) approximation spaces for Stokes problems on complex domains. The described method will be efficient in the sense that the number of unknowns is only linked to the properties of the solution and not to the problem data. The presentation picks up the concept of composite finite elements for the Stokes problem presented in a previous paper of the authors. Here, the a priori error and stability analysis of the proposed mixed method is generalized to quite general, i.e. slip and leak boundary conditions that are of great importance in practical applications. Keywords Problem settingWe consider the stationary Stokes equationsdescribing the motion of a viscous incompressible fluid in a bounded Lipschitz domain Ω under the general mixed boundary conditions proposed by Navier [25] Thereby we use the following notationBoth, the equations (1.1) and the boundary conditions (1.2), can be generalized by replacing the zeros on the right hand sides by some given functions. In this paper, we are especially interested in the limit cases of the boundary conditions, i.e., Dirichlet (λ ν = λ τ = 1), Neumann (λ ν = λ τ = 0), slip (λ ν = 1, λ τ = 0) and leak (λ ν = 0, λ τ = 1) boundary conditions. In particular, we assume the boundary Γ := ∂ Ω to consist of four relatively closed disjoint partsThe leak and slip parts Γ s and Γ l are supposed to be of class C 1 . We define the coefficient functions from (1.2) in the following way:This choice leads to the following set of boundary conditionsWhile the mathematical literature on Dirichlet and Neumann boundary conditions is vast, leak and slip boundary conditions have been studied less extensively. However, they are of great practical interest. For theoretical studies of these boundary conditions we refer to [38; 13; 34] and for some applications to [19; 20].The Sobolev space that contains those velocity fields which fulfill the essential parts (conditions on the left in (1.6)) of the boundary conditions is denoted byThe (mixed) weak formulation of problem (1.1) together with the boundary conditions (1.6) reads:The bilinear forms a :In general, problem (1.8) is not uniquely solvable. The bilinear form a has a nontrivial kernel given by the set of rigid body motionsMoreover, every v ∈ R is divergence-free, i.e. the pairs (v, q), q ∈ R, is a solution of the homogeneous Stokes problem. Thus, a solution of (1.8) can only be unique up to elements of H 1 ess ∩ R. Remark 1.To assure ...

Composite finite elements for elliptic interface problems

“…To keep notation and technicalities at a minimum, the simplest possible setting has been chosen. Generalizations, not only to general linear elliptic problems but also saddle point problems such as Stokes' problem, are straightforward with regard to the previous work [18,19].…”

“…In previous CFEs [18,19,22], for the treatment of essential boundary conditions on unfitted meshes (with respect to the boundary of the domain), the adaptation of shape was done in such a way that the prescribed boundary condition was fulfilled in an approximate way. Now, in the context of interface problems, finite element shape functions are adapted on a submesh such that the continuity across the interface is preserved in an approximate way.…”

Reconstruction of missing boundary conditions from partially overspecified data : the Stokes system

“…Dirichlet-Dirichlet case : In this case first condition follows from the assumption of the same unknown Dirichlet condition on Γ i . Second condition is a simple result of condition (22). Neumann-Neumann case : Equivalently, first condition follows from the assumption of the same unknown Neumann condition on Γ i .…”

The sub-Cauchy–Stokes problem: Solvability issues and Lagrange multiplier methods with artificial boundary conditions

“…The upper bound of a η (·, ·) is straightforward with C 2 = max(2, η). The lower bound follows from Korn's inequality applied for H [34], that of (3.21)-(3.22) can be ensured using the same arguments together with the analysis done in [35]. Hence the solution operators D i , i = D, N , are well-defined and in order to ensure that (u D , p D ) = (u N , p N ) in all of Ω, the resulting solutions are enforced to verify the equality of the velocity trace on the inaccessible boundary Γ I (see Theorem 9).…”