This paper develops and analyzes two least-squares methods for the numerical so- lution of linear, stationary incompressible Newtonian fluid flow in two and three dimensions. Both approaches use the L 2 norm to define least-squares functionals. One is based on the stress-velocity formulation (see section 3.2), and it applies to general boundary conditions. The other is based on an equivalent formulation for the pseudostress and velocity (see section 4.2), and it applies to pure velocity Dirichlet boundary conditions. The velocity gradient and vorticity can be obtained alge- braically from this new tensor variable. It is shown that the homogeneous least-squares functionals are elliptic and continuous in the H(div; Ω) d × H 1 (Ω) d norm. This immediately implies optimal error estimates for conforming finite element approximations. As well, it admits optimal multigrid solution methods if Raviart-Thomas finite element spaces are used to approximate the stress or the pseudostress tensor.

Least-Squares Methods for Incompressible Newtonian Fluid Flow: Linear Stationary Problems

A computational method based on a divergence-free $H$(div) approach is presented for the Stokes equations in this article. This method is designed to find velocity approximation in an exact divergence-free subspace of the corresponding $H$(div) finite element space. That is, the continuity equation is strongly enforced a priori and the pressure is eliminated from the linear system in calculation. A strength of this approach is that the saddle-point problem for Stokes equations is reduced to a symmetric positive definite problem in a subspace for which basis functions are readily available. The resulting discrete system can then be solved by using existing sophisticated solvers. The aim of this article is to demonstrate the efficiency and robustness of $H$(div) finite element methods for Stokes equations. The results not only confirm the existing theoretical results but also reveal additional advantages of the method in dealing with discontinuous boundary conditions.

A Robust Numerical Method for Stokes Equations Based on Divergence-Free $H$(div) Finite Element Methods

Parallel computation of flow in heterogeneous media modelled by mixed finite elements

In this paper we describe an iterative method for indefinite saddle-point systems arising from mixed finite element discretizations of second-order elliptic boundary value problems subject to mixed boundary conditions and posed over polyhedral three-dimensional domains. The method is based on a decoupling of the vector of velocities in the saddle-point system from the vector of pressures, resulting in a symmetric positive definite velocity system and a triangular pressure system.
The crucial step in this approach is the construction of the divergence-free Raviart--Thomas--Nedelec elements from the curls of Nedelec's edge elements. Because of the large kernel of the curl-operator, this representation is not unique. To find a basis we consider the graph made up of the nodes and edges of the mesh and eliminate the edge elements associated with a spanning tree in this graph. To prove that this technique works in the general case considered here, we employ fundamental results from algebraic topology and graph theory.
We also include some numerical experiments, where we solve the (decoupled) velocity system by ILU-preconditioned conjugate gradients and the pressure system by simple back substitutions. We compare our method with a standard ILU-based block preconditioner for the original saddle-point system, and we find that our method is faster by a factor of at least 4.5 in all cases, with the greatest improvement occurring in the nonuniform mesh case.

/pdf/decoupling-three-dimensional-mixed-problems-using-divergence-41cnwidyq0.pdf

Decoupling Three-Dimensional Mixed Problems Using Divergence-Free Finite Elements

In this paper we consider second order scalar elliptic boundary value problems posed over three–dimensional domains and their discretization by means of mixed Raviart–Thomas finite elements [18]. This leads to saddle point problems featuring a discrete flux vector field as additional unknown. Following Ewing and Wang [26], the proposed solution procedure is based on splitting the flux into divergence free components and a remainder. It leads to a variational problem involving solenoidal Raviart–Thomas vector fields. A fast iterative solution method for this problem is presented. It exploits the representation of divergence free vector fields as 
${\bf curl}$
 s of the 
${\bf H}({\bf curl})$
 –conforming finite element functions introduced by Nedelec [43]. We show that a nodal multilevel splitting of these finite element spaces gives rise to an optimal preconditioner for the solenoidal variational problem: Duality techniques in quotient spaces and modern algebraic multigrid theory [50, 10, 31] are the main tools for the proof.

Multilevel methods for mixed finite elements in three dimensions

We consider the solution of the discrete linear system resulting from a mixed finite element discretization applied to a second-order elliptic boundary value problem in three dimensions. Based on a decomposition of the velocity space, these equations can be reduced to a discrete elliptic problem by eliminating the pressure through the use of substructures of the domain. The practicality of the reduction relies on a local basis, presented here, for the divergence-free subspace of the velocity space. We consider additive and multiplicative domain decomposition methods for solving the reduced elliptic problem, and their uniform convergence is established.

/pdf/domain-decomposition-for-a-mixed-finite-element-method-in-4i22jpztpz.pdf

Domain Decomposition for a Mixed Finite Element Method in Three Dimensions

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Domain Decomposition for a Mixed Finite Element Method in Three Dimensions