Superconvergence in the generalized finite element method

In this paper, we address the problem of the existence of superconvergence points of approximate solutions, obtained from the Generalized Finite Element Method (GFEM), of a Neumann elliptic boundary value problem. GFEM is a Galerkin method that uses non-polynomial shape functions, and was developed in (Babuška et al. in SIAM J Numer Anal 31, 945–981, 1994; Babuška et al. in Int J Numer Meth Eng 40, 727–758, 1997; Melenk and Babuška in Comput Methods Appl Mech Eng 139, 289–314, 1996). In particular, we show that the superconvergence points for the gradient of the approximate solution are the zeros of a system of non-linear equations; this system does not depend on the solution of the boundary value problem. For approximate solutions with second derivatives, we have also characterized the superconvergence points of the second derivatives of the approximate solution as the roots of a system of non-linear equations. We note that smooth generalized finite element approximation is easy to construct. I. Babuška’s research was partially supported by NSF Grant # DMS-0341982 and ONR Grant # N00014-99-1-0724. U. Banerjee’s research was partially supported by NSF Grant # DMS-0341899. J. E. Osborn’s research was supported by NSF Grant # DMS-0341982.     

Weighted quadrature rules for finite element methods

We discuss the numerical integration of polynomials times non-polynomial weighting functions in two dimensions arising from multiscale finite element computations. The proposed quadrature rules are significantly more accurate than standard quadratures and are better suited to existing finite element codes than formulas computed by symbolic integration. We validate this approach by introducing the new quadrature formulas into a multiscale finite element method for the two-dimensional reaction–diffusion equation.     

Mixed finite element methods for elastic rods of arbitrary geometry

Summary The boundary-value problem for rods having arbitrary geometry, and subjected to arbitrary loading, is studied within the context of the small-strain theory. The basic assumptions underlying the rod kinematics are those corresponding to the Timoshenko hypotheses in the plane rectilinear case: that is, plane sections normal to the line of centroids in the undeformed state remain plane, but not necessarily normal. The problem is formulated in both the standard and mixed variational forms, and after establishing the existence and uniqueness of solutions to these equivalent problems, the corresponding discrete problems are studied. Finite element approximations of the mixed problem are shown to be stable and convergent. It is shown that the equivalence between the mixed problem and the standard problem with selective reduced integration holds only for the case of rods having constant curvature and torsion, though. The results of numerical experiments are presented; these confirm the convergent behaviour of the mixed problem.     

Analysis and finite element approximation of a Ladyzhenskaya model for viscous flow in streamfunction form

In this paper we consider a model for the motion of incompressible viscous flows proposed by Ladyzhenskaya. The Ladyzhenskaya model is written in terms of the velocity and pressure while the studied model is written in terms of the streamfunction only. We derived the streamfunction equation of the Ladyzhenskaya model and present a weak formulation and show that this formulation is equivalent to the velocity–pressure formulation. We also present some existence and uniqueness results for the model. Finite element approximation procedures are presented. The discrete problem is proposed to be well posed and stable. Some error estimates are derived. We consider the 2D driven cavity flow problem and provide graphs which illustrate differences between the approximation procedure presented here and the approximation for the streamfunction form of the Navier–Stokes equations. Streamfunction contours are also displayed showing the main features of the flow.     

Modified edge finite elements for photonic crystals

We consider Maxwell’s equations with periodic coefficients as it is usually done for the modeling of photonic crystals. Using Bloch/Floquet theory, the problem reduces in a standard way to a modification of the Maxwell cavity eigenproblem with periodic boundary conditions. Following [8], a modification of edge finite elements is considered for the approximation of the band gap. The method can be used with meshes of tetrahedrons or parallelepipeds. A rigorous analysis of convergence is presented, together with some preliminary numerical results in 2D, which fully confirm the robustness of the method. The analysis uses well established results on the discrete compactness for edge elements, together with new sharper interpolation estimates.     

Superconvergence of new mixed finite element spaces

In this paper we prove some superconvergence of a new family of mixed finite element spaces of higher order which we introduced in [ETNA, Vol. 37, pp. 189-201, 2010]. Among all the mixed finite element spaces having an optimal order of convergence on quadrilateral grids, this space has the smallest unknowns. However, the scalar variable is only suboptimal in general; thus we have employed a post-processing technique for the scalar variable. As a byproduct, we have obtained a superconvergence on a rectangular grid. The superconvergence of a velocity variable naturally holds and can be shown by a minor modification of existing theory, but that of a scalar variable requires a new technique, especially for k=1. Numerical experiments are provided to support the theory.     

Numerical analysis of electric field formulations of the eddy current model

This paper deals with finite element methods for the numerical solution of the eddy current problem in a bounded conducting domain crossed by an electric current, subjected to boundary conditions involving only data easily available in applications. Two different cases are considered depending on the boundary data: input current intensities or differences of potential. Weak formulations in terms of the electric field are given in both cases. In the first one, the input current intensities are imposed by means of integrals on curves lying on the boundary of the domain and joining current entrances and exit. In the second one, the electric potentials are imposed by means of Lagrange multipliers, which are proved to represent the input current intensities. Optimal error estimates are proved in both cases and implementation issues are discussed. Finally, numerical tests confirming the theoretical results are reported. Partially supported by Xunta de Galicia research grant PGIDIT02PXIC20701PN (Spain). Partially supported by FONDAP in Applied Mathematics (Chile). Partially supported by MCYT (Spain) project DPI2003-01316.     

On the multi-level splitting of finite element spaces

Summary In this paper we analyze the condition number of the stiffness matrices arising in the discretization of selfadjoint and positive definite plane elliptic boundary value problems of second order by finite element methods when using hierarchical bases of the finite element spaces instead of the usual nodal bases. We show that the condition number of such a stiffness matrix behaves like O((log )²) where is the condition number of the stiffness matrix with respect to a nodal basis. In the case of a triangulation with uniform mesh sizeh this means that the stiffness matrix with respect to a hierarchical basis of the finite element space has a condition number behaving like instead of for a nodal basis. The proofs of our theorems do not need any regularity properties of neither the continuous problem nor its discretization. Especially we do not need the quasiuniformity of the employed triangulations. As the representation of a finite element function with respect to a hierarchical basis can be converted very easily and quickly to its representation with respect to a nodal basis, our results mean that the method of conjugate gradients needs onlyO(log n) steps andO(n log n) computer operations to reduce the energy norm of the error by a given factor if one uses hierarchical bases or related preconditioning procedures. Heren denotes the dimension of the finite element space and of the discrete linear problem to be solved.     

Duality and finite spaces

Using that finite topological spaces are just finite orders, we develop a duality theory for sheaves of Abelian groups over finite spaces following closely Grothendieck's duality theory for coherent sheaves over proper schemes. Since the geometric realization of a finite space is a polyhedron, we relate this duality with the duality theory for Abelian sheaves over polyhedra.     

Local and parallel finite element algorithms for the stokes problem

Based on two-grid discretizations, some local and parallel finite element algorithms for the Stokes problem are proposed and analyzed in this paper. These algorithms are motivated by the observation that for a solution to the Stokes problem, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel procedure. One technical tool for the analysis is some local a priori estimates that are also obtained in this paper for the finite element solutions on general shape-regular grids. Y. He was partially subsidized by the NSF of China 10671154 and the National Basic Research Program under the grant 2005CB321703; A. Zhou was partially supported by the National Science Foundation of China under the grant 10425105 and the National Basic Research Program under the grant 2005CB321704; J. Li was partially supported by the NSF of China under the grant 10701001. J. Xu was partially supported by Alexander von Humboldt Research Award for Senior US Scientists, NSF DMS-0609727 and NSFC-10528102.     

Two preconditioners based on the multi-level splitting of finite element spaces

Summary The hierarchical basis preconditioner and the recent preconditioner of Bramble, Pasciak and Xu are derived and analyzed within a joint framework. This discussion elucidates the close relationship between both methods. Special care is devoted to highly nonuniform meshes; exclusively local properties like the shape regularity of the finite elements are utilized.The author was supported by the Konrad-Zuse-Zentrum für Informationstechnik Berlin, Federal Republic of Germany     

Multilevel iterative methods for mixed finite element discretizations of elliptic problems

Summary For solving second order elliptic problems discretized on a sequence of nested mixed finite element spaces nearly optimal iterative methods are proposed. The methods are within the general framework of the product (multiplicative) scheme for operators in a Hilbert space, proposed recently by Bramble, Pasciak, Wang, and Xu [5,6,26,27] and make use of certain multilevel decomposition of the corresponding spaces for the flux variable.     

Discrete Hodge operators

Summary. Many linear boundary value problems arising in computational physics can be formulated in the calculus of differential forms. Discrete differential forms provide a natural and canonical approach to their discretization. However, much freedom remains concerning the choice of discrete Hodge operators, that is, discrete analogues of constitutive laws. A generic discrete Hodge operator is introduced and it turns out that most finite element and finite volume schemes emerge as its specializations. We reap the possibility of a unified convergence analysis in the framework of discrete exterior calculus. Received November 26, 1999 / Revised version received November 2, 2000 / Published online May 30, 2001     

High-order finite element methods for time-fractional partial differential equations

The aim of this paper is to develop high-order methods for solving time-fractional partial differential equations. The proposed high-order method is based on high-order finite element method for space and finite difference method for time. Optimal convergence rate O((Δt)^2−α+N^−r) is proved for the (r−1)th-order finite element method (r≥2).     

Bubble finite elements for the primitive equations of the ocean

We introduce in this paper two original Mixed methods for the numerical resolution of the (stationary) Primitive Equations (PE) of the Ocean. The PE govern the behavior of oceanic flows in shallow domains for large time scales. We use a reduced formulation (Lions et al. [28]) involving horizontal velocities and surface pressures. By using bubble functions constructed ad-hoc, we are able to define two stable Mixed Methods requiring a low number of degrees of freedom. The first one is based on the addition of bubbles of reduced support to velocities elementwise. The second one makes use of conic bubbles of extended support along the vertical coordinate. The latter constitutes a genuine mini-element for the PE, e.g., it requires the least number of extra degrees of freedom to stabilize piecewise linear hydrostatic pressures. Both methods verify a specific inf-sup condition and provide stability and convergence. Finally, we compare several numerical features of the proposed pairs in the context of other FE methods found in the literature.     

Additive Schwarz methods for the Crouzeix-Raviart mortar finite element for elliptic problems with discontinuous coefficients

In this paper, we propose two variants of the additive Schwarz method for the approximation of second order elliptic boundary value problems with discontinuous coefficients, on nonmatching grids using the lowest order Crouzeix-Raviart element for the discretization in each subdomain. The overall discretization is based on the mortar technique for coupling nonmatching grids. The convergence behavior of the proposed methods is similar to that of their closely related methods for conforming elements. The condition number bound for the preconditioned systems is independent of the jumps of the coefficient, and depend linearly on the ratio between the subdomain size and the mesh size. The performance of the methods is illustrated by some numerical results. This work has been supported by the Alexander von Humboldt Foundation and the special funds for major state basic research projects (973) under 2005CB321701 and the National Science Foundation (NSF) of China (No.10471144) This work has been supported in part by the Bergen Center for Computational Science, University of Bergen     

Convergence in the incompressible limit of finite element approximations based on the Hu-Washizu formulation

The classical Hu–Washizu mixed formulation for plane problems in elasticity is examined afresh, with the emphasis on behavior in the incompressible limit. The classical continuous problem is embedded in a family of Hu–Washizu problems parametrized by a scalar α for which corresponds to the classical formulation, with λ and μ being the Lamé parameters. Uniform well- posedness in the incompressible limit of the continuous problem is established for α ≠ − 1. Finite element approximations are based on the choice of piecewise bilinear approximations for the displacements on quadrilateral meshes. Conditions for uniform convergence are made explicit. These conditions are shown to be met by particular choices of bases for stresses and strains, and include bases that are well known, as well as newly constructed bases. Though a discrete version of the spherical part of the stress exhibits checkerboard modes, it is shown that a λ-independent a priori error estimate for the displacement can be established. Furthermore, a λ-independent estimate is established for the post-processed stress. The theoretical results are explored further through selected numerical examples.     

A general finite element preconditioning for the conjugate gradient method

Discretizing a symmetric elliptic boundary value problem by a finite element method results in a system of linear equations with a symmetric positive definite coefficient matrix. This system can be solved iteratively by a preconditioned conjugate gradient method. In this paper a preconditioning matrix is proposed that can be constructed for all finite element methods if a mild condition for the node numbering is fulfilled. Such a numbering can be constructed using a variant of the reverse Cuthill-McKee algorithm.     

Mixed finite elements on sparse grids

Summary. This paper generalizes the idea of approximation on sparse grids to discrete differential forms that include )- and -conforming mixed finite element spaces as special cases. We elaborate on the construction of the spaces, introduce suitable nodal interpolation operators on sparse grids and establish their approximation properties. We discuss how nodal interpolation operators can be approximated. The stability of -conforming finite elements on sparse grids, when used to approximate second order elliptic problems in mixed formulation, is investigated both theoretically and in numerical experiments. Received November 2, 2000 / Revised version received October 23, 2001 / Published online January 30, 2002 This work was supported by DFG. This paper is dedicated to Ch. Zenger on the occasion of his 60th birthday.     

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.