A posteriori error estimators for nonconforming finite element methods of the linear elasticity problem

In this work we derive and analyze a posteriori error estimators for low-order nonconforming finite element methods of the linear elasticity problem on both triangular and quadrilateral meshes, with hanging nodes allowed for local mesh refinement. First, it is shown that equilibrated Neumann data on interelement boundaries are simply given by the local weak residuals of the numerical solution. The first error estimator is then obtained by applying the equilibrated residual method with this set of Neumann data. From this implicit estimator we also derive two explicit error estimators, one of which is similar to the one proposed by Dörfler and Ainsworth (2005) [24] for the Stokes problem. It is established that all these error estimators are reliable and efficient in a robust way with respect to the Lamé constants. The main advantage of our error estimators is that they yield guaranteed, i.e., constant-free upper bounds for the energy-like error (up to higher order terms due to data oscillation) when a good estimate for the inf-sup constant is available, which is confirmed by some numerical results.     

Block iterative solvers for higher order finite volume methods

Recently, new higher order finite volume methods (FVM) were introduced in [Z. Cai, J. Douglas, M. Park, Development and analysis of higher order finite volume methods over rectangles for elliptic equations, Adv. Comput. Math. 19 (2003) 3-33], where the linear system derived by the hybridization with Lagrange multiplier satisfying the flux consistency condition is reduced to a linear system for a pressure variable by an appropriate quadrature rule. We study the convergence of an iterative solver for this linear system. The conjugate gradient (CG) method is a natural choice to solve the system, but it seems slow, possibly due to the non-diagonal dominance of the system. In this paper, we propose block iterative methods with a reordering scheme to solve the linear system derived by the higher order FVM and prove their convergence. With a proper ordering, each block subproblem can be solved by fast methods such as the multigrid (MG) method. The numerical experiments show that these block iterative methods are much faster than CG.     

On the augmented Lagrangian approach to Signorini elastic contact problem

Summary. The Signorini problem describes the contact of a linearly elastic body with a rigid frictionless foundation. It is transformed into a saddle point problem of some augmented Lagrangian functional and then discretized by finite element methods. Optimal error estimates are obtained for general smooth domains which are not necessarily convex. The key ingredient in the analysis is a discrete inf-sup condition which guarantees the existence of the saddle point. Received January 29, 1999 / Revised version received May 2, 2000 / Published online December 19, 2000     

Small data oscillation implies the saturation assumption

Summary. The saturation assumption asserts that the best approximation error in with piecewise quadratic finite elements is strictly smaller than that of piecewise linear finite elements. We establish a link between this assumption and the oscillation of , and prove that small oscillation relative to the best error with piecewise linears implies the saturation assumption. We also show that this condition is necessary, and asymptotically valid provided . Received November 17, 2000 / Published online July 25, 2001     

Inf-sup stable non-conforming finite elements of arbitrary order on triangles

We introduce a family of scalar non-conforming finite elements of arbitrary order k≥1 with respect to the H¹-norm on triangles. Their vector-valued version generates together with a discontinuous pressure approximation of order k−1 an inf-sup stable finite element pair of order k for the Stokes problem in the energy norm. For k=1 the well-known Crouzeix-Raviart element is recovered.     

Mixed hp-FEM on anisotropic meshes II: Hanging nodes and tensor products of boundary layer meshes

Summary. The divergence stability of mixed hp Finite Element Methods for incompressible fluid flow is analyzed. A discrete inf-sup condition is proved for a general class of meshes. The meshes may be refined anisotropically, geometrically and may contain hanging nodes on geometric patches. The inf-sup constant is shown to be independent of the aspect ratio of the anisotropic elements and the dependence on the polynomial degree is analyzed. Numerical estimates of inf-sup constants confirm the theoretical results. Received October 13, 1997 / Revised version received June 8, 1998 / Published online July 28, 1999     

An iterative adaptive finite element method for elliptic eigenvalue problems

We consider the task of resolving accurately the n$n$th eigenpair of a generalized eigenproblem rooted in some elliptic partial differential equation (PDE), using an adaptive finite element method (FEM). Conventional adaptive FEM algorithms call a generalized eigensolver after each mesh refinement step. This is not practical in our situation since the generalized eigensolver needs to calculate n$n$ eigenpairs after each mesh refinement step, it can switch the order of eigenpairs, and for repeated eigenvalues it can return an arbitrary linear combination of eigenfunctions from the corresponding eigenspace. In order to circumvent these problems, we propose a novel adaptive algorithm that only calls a generalized eigensolver once at the beginning of the computation, and then employs an iterative method to pursue a selected eigenvalue–eigenfunction pair on a sequence of locally refined meshes. Both Picard’s and Newton’s variants of the iterative method are presented. The underlying partial differential equation (PDE) is discretized with higher-order finite elements (hp$hp$-FEM) but the algorithm also works for standard low-order FEM. The method is described and accompanied with theoretical analysis and numerical examples. Instructions on how to reproduce the results are provided.     

Convergence analysis of the FEM approximation of the first order projection method for incompressible flows with and without the inf-sup condition

In this paper we obtain convergence results for the fully discrete projection method for the numerical approximation of the incompressible Navier–Stokes equations using a finite element approximation for the space discretization. We consider two situations. In the first one, the analysis relies on the satisfaction of the inf-sup condition for the velocity-pressure finite element spaces. After that, we study a fully discrete fractional step method using a Poisson equation for the pressure. In this case the velocity-pressure interpolations do not need to accomplish the inf-sup condition and in fact we consider the case in which equal velocity-pressure interpolation is used. Optimal convergence results in time and space have been obtained in both cases.     

Local flux mimetic finite difference methods

We develop a local flux mimetic finite difference method for second order elliptic equations with full tensor coefficients on polyhedral meshes. To approximate the velocity (vector variable), the method uses two degrees of freedom per element edge in two dimensions and n degrees of freedom per n-gonal mesh face in three dimensions. To approximate the pressure (scalar variable), the method uses one degree of freedom per element. A specially chosen quadrature rule for the L ²-product of vector-functions allows for a local flux elimination and reduction of the method to a cell-centered finite difference scheme for the pressure unknowns. Under certain assumptions, first-order convergence is proved for both variables and second-order convergence is proved for the pressure. The assumptions are verified on simplicial meshes for a particular quadrature rule that leads to a symmetric method. For general polyhedral meshes, non-symmetric methods are constructed based on quadrature rules that are shown to satisfy some of the assumptions. Numerical results confirm the theory.     

Existence and approximation results for nonlinear mixed problems: Application to incompressible finite elasticity

Summary This paper considers the problems of minimizing Gateaux-differentiable functionals over subsets of real Banach spaces defined by a non-linear equality constraint. The existence of a Lagrange multiplier is proved, together with approximation results on the constrained subset, provided a nonlinear compatibility condition, generalizing the classical inf-sup condition, is satisfied. These ideas are applied to equilibrium problems in incompressible finite elasticity and lead to convergence results for these problems.     

A domain decomposition algorithm for elliptic problems in three dimensions

Summary Most domain decomposition algorithms have been developed for problems in two dimensions. One reason for this is the difficulty in devising a satisfactory, easy-to-implement, robust method of providing global communication of information for problems in three dimensions. Several methods that work well in two dimension do not perform satisfactorily in three dimensions.A new iterative substructuring algorithm for three dimensions is proposed. It is shown that the condition number of the resulting preconditioned problem is bounded independently of the number of subdomains and that the growth is quadratic in the logarithm of the number of degrees of freedom associated with a subdomain. The condition number is also bounded independently of the jumps in the coefficients of the differential equation between subdomains. The new algorithm also has more potential parallelism than the iterative substructuring methods previously proposed for problems in three dimensions.This work was supported in part by the National Science Foundation under grant NSF-CCR-8903003 and by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

Composite finite elements for the approximation of PDEs on domains with complicated micro-structures

Summary. Usually, the minimal dimension of a finite element space is closely related to the geometry of the physical object of interest. This means that sometimes the resolution of small micro-structures in the domain requires an inadequately fine finite element grid from the viewpoint of the desired accuracy. This fact limits also the application of multi-grid methods to practical situations because the condition that the coarsest grid should resolve the physical object often leads to a huge number of unknowns on the coarsest level. We present here a strategy for coarsening finite element spaces independently of the shape of the object. This technique can be used to resolve complicated domains with only few degrees of freedom and to apply multi-grid methods efficiently to PDEs on domains with complex boundary. In this paper we will prove the approximation property of these generalized FE spaces. Received June 9, 1995 / Revised version received February 5, 1996     

Fast parallel solvers for symmetric boundary element domain decomposition equations

Summary. The boundary element method (BEM) is of advantage in many applications including far-field computations in magnetostatics and solid mechanics as well as accurate computations of singularities. Since the numerical approximation is essentially reduced to the boundary of the domain under consideration, the mesh generation and handling is simpler than, for example, in a finite element discretization of the domain. In this paper, we discuss fast solution techniques for the linear systems of equations obtained by the BEM (BE-equations) utilizing the non-overlapping domain decomposition (DD). We study parallel algorithms for solving large scale Galerkin BE–equations approximating linear potential problems in plane, bounded domains with piecewise homogeneous material properties. We give an elementary spectral equivalence analysis of the BEM Schur complement that provides the tool for constructing and analysing appropriate preconditioners. Finally, we present numerical results obtained on a massively parallel machine using up to 128 processors, and we sketch further applications to elasticity problems and to the coupling of the finite element method (FEM) with the boundary element method. As shown theoretically and confirmed by the numerical experiments, the methods are of algebraic complexity and of high parallel efficiency, where denotes the usual discretization parameter. Received August 28, 1996 / Revised version received March 10, 1997     

Effective preconditioning of Uzawa type schemes for a generalized Stokes problem

Summary. The Schur complement of a model problem is considered as a preconditioner for the Uzawa type schemes for the generalized Stokes problem (the Stokes problem with the additional term in the motion equation). The implementation of the preconditioned method requires for each iteration only one extra solution of the Poisson equation with Neumann boundary conditions. For a wide class of 2D and 3D domains a theorem on its convergence is proved. In particular, it is established that the method converges with a rate that is bounded by some constant independent of . Some finite difference and finite element methods are discussed. Numerical results for finite difference MAC scheme are provided. Received May 2, 1997 / Revised version received May 10, 1999 / Published online May 8, 2000     

A finite element recovery approach to Green’s function approximations with applications to electrostatic potential computation

In this paper, a finite element recovery approach is proposed to improve the accuracy of finite element approximations for Green’s functions in three dimensions. This recovery approach is based on some simple postprocessing. It is proved by both theory and numerics that the recovery approach is very efficient. In particular, the approach is successfully applied to some electrostatic potential computations.     

A minimal stabilisation procedure for mixed finite element methods

Summary. Stabilisation methods are often used to circumvent the difficulties associated with the stability of mixed finite element methods. Stabilisation however also means an excessive amount of dissipation or the loss of nice conservation properties. It would thus be desirable to reduce these disadvantages to a minimum. We present a general framework, not restricted to mixed methods, that permits to introduce a minimal stabilising term and hence a minimal perturbation with respect to the original problem. To do so, we rely on the fact that some part of the problem is stable and should not be modified. Sections 2 and 3 present the method in an abstract framework. Section 4 and 5 present two classes of stabilisations for the inf-sup condition in mixed problems. We present many examples, most arising from the discretisation of flow problems. Section 6 presents examples in which the stabilising terms is introduced to cure coercivity problems. Received August 9, 1999 / Revised version received May 19, 2000 / Published online March 20, 2001     

Characterizing the inf-sup condition on product spaces

In this paper we establish characterization results for the continuous and discrete inf-sup conditions on product spaces. The inf-sup condition for each component of the bilinear form involved and suitable decompositions of the pivot space in terms of the associated null spaces are the key ingredients of our theorems. We illustrate the theory through its application to bilinear forms arising from the variational formulations of several boundary value problems. Dedicated to Professor Ivo Babuska on the occasion of his 82nd birthday. This research was partially supported by Centro de Modelamiento Matemático (CMM) of the Universidad de Chile, by Centro de Investigación en Ingenierí a Matemática (CI²MA) of the Universidad de Concepción, by FEDER/MCYT Project MTM2007-63204, and by Gobierno de Aragón (Grupo Consolidado PDIE).     

Two-scale composite finite element method for Dirichlet problems on complicated domains

In this paper, we define a new class of finite elements for the discretization of problems with Dirichlet boundary conditions. In contrast to standard finite elements, the minimal dimension of the approximation space is independent of the domain geometry and this is especially advantageous for problems on domains with complicated micro-structures. For the proposed finite element method we prove the optimal-order approximation (up to logarithmic terms) and convergence estimates valid also in the cases when the exact solution has a reduced regularity due to re-entering corners of the domain boundary. Numerical experiments confirm the theoretical results and show the potential of our proposed method.     

Spline collocation for strongly elliptic equations on the torus

Summary We prove convergence and error estimates in Sobolev spaces for the collocation method with tensor product splines for strongly elliptic pseudodifferential equations on the torus. Examples of applications include elliptic partial differential equations with periodic boundary conditions but also the classical boundary integral operators of potential theory on torus-shaped domains in three or more dimensions. For odd-degree splines, we prove convergence of nodal collocation for any strongly elliptic operator. For even-degree splines and midpoint collocation, we find an additional condition for the convergence which is satisfied for the classical boundary integral operators. Our analysis is a generalization to higher dimensions of the corresponding analysis of Arnold and Wendland [4].     

A preconditioner for the h-p version of the finite element method in two dimensions

Summary. A preconditioner, based on a two-level mesh and a two-level orthogonalization, is proposed for the - version of the finite element method for two dimensional elliptic problems in polygonal domains. Its implementation is in parallel on the subdomain level for the linear or bilinear (nodal) modes, and in parallel on the element level for the high order (side and internal) modes. The condition number of the preconditioned linear system is of order , where is the diameter of the -th subdomain, and are the diameter of elements and the maximum polynomial degree used in the subdomain. This result reduces to well-known results for the -version (i.e. ) and the -version (i.e. ) as the special cases of the - version. Received August 15, 1995 / Revised version received November 13, 1995