On variational eigenvalue approximation of semidefinite operators

Eigenvalue problems for semidefinite operators with infinite dimensional kernels appear for instance in electromagnetics. Variational discretizations with edge elements have long been analyzed in terms of a discrete compactness property. As an alternative, we show here how the abstract theory can be developed in terms of a geometric property called the vanishing gap condition. This condition is shown to be equivalent to eigenvalue convergence and intermediate between two different discrete variants of Friedrichs estimates. Next we turn to a more practical means of checking these properties. We introduce a notion of compatible operator and show how the previous conditions are equivalent to the existence of such operators with various convergence properties. In particular the vanishing gap condition is shown to be equivalent to the existence of compatible operators satisfying an Aubin-Nitsche estimate. Finally we give examples demonstrating that the implications not shown to be equivalences, indeed are not.Comment: 26 page

Nonconforming tetrahedral mixed finite elements for elasticity

This paper presents a nonconforming finite element approximation of the space of symmetric tensors with square integrable divergence, on tetrahedral meshes. Used for stress approximation together with the full space of piecewise linear vector fields for displacement, this gives a stable mixed finite element method which is shown to be linearly convergent for both the stress and displacement, and which is significantly simpler than any stable conforming mixed finite element method. The method may be viewed as the three-dimensional analogue of a previously developed element in two dimensions. As in that case, a variant of the method is proposed as well, in which the displacement approximation is reduced to piecewise rigid motions and the stress space is reduced accordingly, but the linear convergence is retained.Comment: 13 pages, 2 figure

Parameter-robust discretization and preconditioning of Biot's consolidation model

Biot's consolidation model in poroelasticity has a number of applications in science, medicine, and engineering. The model depends on various parameters, and in practical applications these parameters ranges over several orders of magnitude. A current challenge is to design discretization techniques and solution algorithms that are well behaved with respect to these variations. The purpose of this paper is to study finite element discretizations of this model and construct block diagonal preconditioners for the discrete Biot systems. The approach taken here is to consider the stability of the problem in non-standard or weighted Hilbert spaces and employ the operator preconditioning approach. We derive preconditioners that are robust with respect to both the variations of the parameters and the mesh refinement. The parameters of interest are small time-step sizes, large bulk and shear moduli, and small hydraulic conductivity.Comment: 24 page

Mixed finite element methods for linear elasticity with weakly imposed symmetry

In this paper, we construct new finite element methods for the approximation of the equations of linear elasticity in three space dimensions that produce direct approximations to both stresses and displacements. The methods are based on a modified form of the Hellinger--Reissner variational principle that only weakly imposes the symmetry condition on the stresses. Although this approach has been previously used by a number of authors, a key new ingredient here is a constructive derivation of the elasticity complex starting from the de Rham complex. By mimicking this construction in the discrete case, we derive new mixed finite elements for elasticity in a systematic manner from known discretizations of the de Rham complex. These elements appear to be simpler than the ones previously derived. For example, we construct stable discretizations which use only piecewise linear elements to approximate the stress field and piecewise constant functions to approximate the displacement field.Comment: to appear in Mathematics of Computatio

Weakly imposed symmetry and robust preconditioners for Biot's consolidation model

We discuss the construction of robust preconditioners for finite element approximations of Biot's consolidation model in poroelasticity. More precisely, we study finite element methods based on generalizations of the Hellinger-Reissner principle of linear elasticity, where the stress tensor is one of the unknowns. The Biot model has a number of applications in science, medicine, and engineering. A challenge in many of these applications is that the model parameters range over several orders of magnitude. Therefore, discretization procedures which are well behaved with respect to such variations are needed. The focus of the present paper will be on the construction of preconditioners, such that the preconditioned discrete systems are well-conditioned with respect to variations of the model parameters as well as refinements of the discretization. As a byproduct, we also obtain preconditioners for linear elasticity that are robust in the incompressible limit.Comment: 21 page

Finite element exterior calculus: from Hodge theory to numerical stability

This article reports on the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computation, the other from topology and geometry. In it we consider the numerical discretization of partial differential equations that are related to differential complexes so that de Rham cohomology and Hodge theory are key tools for the continuous problem. After a brief introduction to finite element methods, the discretization methods we consider, we develop an abstract Hilbert space framework for analyzing stability and convergence. In this framework, the differential complex is represented by a complex of Hilbert spaces and stability is obtained by transferring Hodge theoretic structures from the continuous level to the discrete. We show stable discretization is achieved if the finite element spaces satisfy two hypotheses: they form a subcomplex and there exists a bounded cochain projection from the full complex to the subcomplex. Next, we consider the most canonical example of the abstract theory, in which the Hilbert complex is the de Rham complex of a domain in Euclidean space. We use the Koszul complex to construct two families of finite element differential forms, show that these can be arranged in subcomplexes of the de Rham complex in numerous ways, and for each construct a bounded cochain projection. The abstract theory therefore applies to give the stability and convergence of finite element approximations of the Hodge Laplacian. Other applications are considered as well, especially to the equations of elasticity. Background material is included to make the presentation self-contained for a variety of readers.Comment: 74 pages, 8 figures; reorganized introductory material, added additional example and references; final version accepted by Bulletin of the AMS, added references to codes and adjusted some diagrams. Bulletin of the American Mathematical Society, to appear 201

A uniform preconditioner for a Newton algorithm for total-variation minimization and minimum-surface problems

Solution methods for the nonlinear partial differential equation of the Rudin-Osher-Fatemi (ROF) and minimum-surface models are fundamental for many modern applications. Many efficient algorithms have been proposed. First order methods are common. They are popular due to their simplicity and easy implementation. Some second order Newton-type iterative methods have been proposed like Chan-Golub-Mulet method. In this paper, we propose a new Newton-Krylov solver for primal-dual finite element discretization of the ROF model. The method is so simple that we just need to use some diagonal preconditioners during the iterations. Theoretically, the proposed preconditioners are further proved to be robust and optimal with respect to the mesh size, the penalization parameter, the regularization parameter, and the iterative step, essentially it is a parameter independent preconditioner. We first discretize the primal-dual system by using mixed finite element methods, and then linearize the discrete system by Newton\textquoteright s method. Exploiting the well-posedness of the linearized problem on appropriate Sobolev spaces equipped with proper norms, we propose block diagonal preconditioners for the corresponding system solved with the minimum residual method. Numerical results are presented to support the theoretical results.Comment: 22 pages, 24 figure