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First-order system least squares (FOSLS) is a commonly used technique in a wide range of physical applications. FOSLS discretizations are straightforward to implement and offer many advantages over traditional Galerkin or saddle-point formulations. Often, these problems are formulated in H(div) spaces and implemented using H(div)-conforming elements. These elements have fewer regularity assumptions than the more commonly used H¹-conforming elements and are therefore often believed to be more suited for less regular problems. This paper compares the approximation properties of the H(div)-conforming Raviart-Thomas and Brezzi-Douglas-Marini elements to H¹-conforming piecewise polynomials for H(div) problem formulations. Furthermore, for each problem, a corresponding H¹-formulation is derived and compared. The traditional Poisson and Stokes problems are examined with less regular solutions addressed by adaptive refinement strategies. The results imply that for smooth problems or problems exhibiting a point singularity H¹-conforming piecewise polynomials are a viable, and sometimes more efficient, choice even if an H(div)-formulation is used. Smooth problems do additionally benefit by using an H¹-formulation instead of an H(div)-formulation. [ABSTRACT FROM AUTHOR]

In previous work, the first-order system LL* (FOSLL*) method was developed for linear partial differential equations. This approach seeks to minimize the residual of the equations in a dual norm induced by the differential operator, yielding approximations accurate in L²(Ω) rather than H¹(Ω) or H(Div). In this paper, the general framework of FOSLL* is extended to a wide range of nonlinear problems. Four approaches to propagating an inexact Newton iteration based on a FOSLL* approximation are presented, and theory for robust convergence in L²(O) is established. Numerical results are presented for two formulations of the steady incompressible Navier-Stokes equations and for a diffusion equation with reduced regularity due to a discontinuous diffusion coefficient. [ABSTRACT FROM AUTHOR]

In the case that the domain has reentrant edges, the standard finite element method loses its global accuracy because of singularities on the boundary. To overcome this difficulty, FOSLL* is applied in this paper. FOSLL* is a methodology for solving PDEs using the dual operator. Here, a modified FOSLL* method is developed that employs a partially weighted functional and allows the use of a standard finite element scheme without losing global accuracy. [ABSTRACT FROM AUTHOR]

Following earlier work for Stokes equations, a least squares functional is developed for two- and three-dimensional Oseen equations. By introducing a velocity flux variable and associated curl and trace equations, ellipticity is established in an appropriate product norm. The form of Oseen equations examined here is obtained by linearizing the incompressible Navier–Stokes equations. An algorithm is presented for approximately solving steady-state, incompressible Navier–Stokes equations with a nested iteration-Newton-FOSLS-AMG iterative scheme, which involves solving a sequence of Oseen equations. Some numerical results for Kovasznay flow are provided. Copyright © 2006 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Least-squares finite element methods (LSFEMs) for the inviscid Burgers equation are studied. The scalar nonlinear hyperbolic conservation law is reformulated by introducing the flux vector, or the associated flux potential, explicitly as additional dependent variables. This reformulation highlights the smoothness of the flux vector for weak solutions, namely, $\bff(u) \in H({\rm div},\Omega)$. The standard least-squares (LS) finite element (FE) procedure is applied to the reformulated equations using H (div)-conforming FE spaces and a Gauss--Newton nonlinear solution technique. Numerical results are presented for the one-dimensional Burgers equation on adaptively refined space-time domains, indicating that the H (div)-conforming FE methods converge to the entropy weak solution of the conservation law. The H (div)-conforming LSFEMs do not satisfy a discrete exact conservation property in the sense of Lax and Wendroff. However, weak conservation theorems that are analogous to the Lax--Wendroff theorem for conservative finite difference methods are proved for the H (div)-conforming LSFEMs. These results illustrate that discrete exact conservation in the sense of Lax and Wendroff is not a necessary condition for numerical conservation but can be replaced by minimization in a suitable continuous norm. [ABSTRACT FROM AUTHOR]

The standard multigrid algorithm is widely known to yield optimal convergence whenever all high-frequency error components correspond to large relative eigenvalues. This property guarantees that smoothers like Gauss–Seidel and Jacobi will significantly dampen all the high-frequency error components, and thus, produce a smooth error. This has been established for matrices generated from standard discretizations of most elliptic equations. In this paper, we address a system of equations that is generated from a perturbation of the non-elliptic operator I-grad div by a negative ℇ Δ. For ℇnear to one, this operator is elliptic, but as ℇapproaches zero, the operator becomes non-elliptic as it is dominated by its non-elliptic part. Previous research on the non-elliptic part has revealed that discretizing I-grad div with the proper finite element space allows one to define a robust geometric multigrid algorithm. The robustness of the multigrid algorithm depends on a relaxation operator that yields a smooth error. We use this research to assist in developing a robust discretization and solution method for the perturbed problem. To this end, we introduce a new finite element space for tensor product meshes that is used in the discretization, and a relaxation operator that succeeds in dampening all high-frequency error components. The success of the corresponding multigrid algorithm is first demonstrated by numerical results that quantitatively imply convergence for any ℇis bounded by the convergence for ℇequal to zero. Then we prove that convergence of this multigrid algorithm for the case of ℇ equal to zero is independent of mesh size. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper develops first-order system least-squares (FOSLS) functionals for solving the pure traction problem in three-dimensional linear elasticity. It is a direct extension of an earlier paper on planar elasticity [Z. Cai, T. A. Manteuffel, S. F. McCormick, and S. V. Parter, SIAM J. Numer. Anal., 35 (1998), pp. 320–335]. Two functionals are developed, one involving L2 norms of the first-order system and the other involving dual norms. These functionals are shown to be equivalent to appropriate product Sobolev norms, uniformly in the Poisson ratio. These results imply that standard finite element discretization and iterative solver techniques can be applied to obtain performance that is optimal—even as the material nears the incompressible limit. [ABSTRACT FROM AUTHOR]

This paper develops two first-order system least-squares (FOSLS) approaches for the solution of the pure traction problem in planar linear elasticity. Both are two-stage algorithms that first solve for the gradients of displacement (which immediately yield deformation and stress), then for the displacement itself (if desired). One approach, which uses L² norms to define the FOSLS functional, is shown under certain H² regularity assumptions to admit optimal H¹-like performance for standard finite element discretization and standard multigrid solution methods that is uniform in the Poisson ratio for all variables. The second approach, which is based on H-1 norms, is shown under general assumptions to admit optimal uniform performance for displacement flux in an L² norm and for displacement in an HSup1; norm. These methods do not degrade as other methods generally do when the material properties approach the incompressible limit. [ABSTRACT FROM AUTHOR]

Mathematical modeling of compliant blood vessels generally involves the Navier-Stokes equations on the evolving fluid domain and constitutive structural equations on the tissue domain. Coupling these systems while accounting for the changing shape of the fluid domain is a major challenge in numerical simulation. Many techniques have been developed to model compliant vessels, but all suffer from disproportionate increase in computational cost as problem complexity increases (i.e., larger domains, more dimensions, and more variables.). Even the best standard methods result in computational cost that typically grows quadratically with the degrees of freedom. Using a least-squares formulation of the problem, elliptic grid generation for the changing fluid domain, and an algebraic multigrid solver for the linear system can overcome many shortcomings of standard techniques. Most notably, the computational cost of solving the problem increases linearly with the degrees of freedom and the associated functional provides an a posteriori error measurement. Least squares represents a systematic approach for formulating the original problem so that the numerical process becomes straightforward and optimal, and it avoids restrictions that often limit other methods. Results are presented for a two-dimensional test problem consisting of a Newtonian fluid with properties of blood in a linear-elastic vessel with properties of smooth muscle.