Finite element methods on piecewise equidistant meshes for interior turning point problems

We consider linear second order singularly perturbed two-point boundary value problems with interior turning points. Piecewise linear Galerkin finite element methods are constructed on various piecewise equidistant meshes designed for such problems. These methods are proved to be convergent, uniformly in the singular perturbation parameter, in a weighted energy norm and the usualL ² norm. Supporting numerical results are presented.     

Preconditioning nonconforming finite element methods for treating Dirichlet boundary conditions. II

Summary This work deals with theH ¹ condition numbers and the distribution of theB _h singular values of the preconditioned operators {B _h ^–1 A _h }_{0, whereA h andB h are finite element discretizations of second order elliptic operators,A andB respectively.B is also assumed to be self-adjoint and positive definite. For conforming finite elements, Parter and Wong have shown that the singular values cluster in a positive finite interval. Goldstein also has derived results on the spectral distribution ofB h ^–1 A h using a different approach. As a generalization of the results of Parter and Wong, the current work includes nonconforming finite element methods which deal with Dirichlet boundary conditions. It will be shown that, in this more general setting, the singular values also cluster in a positive finite interval. In particular, if the leading part ofB is the same as the leading part ofA, then the singular values cluster about the point {1}. Two specific methods are given as applications of this theory. They are the penalty method of Babuka and the method of nearly zero boundary conditions of Nitsche. Finally, it will be shown that the same results can be proven by an approach generalized from the work of Goldstein.This research was supported by the National Science Foundation under grant number DMS-8913091.}     

Eigenvalue estimates for a preconditioned Galerkin matrix arising from mixed finite element discretizations of viscous incompressible flows

The article deals with Galerkin matrices arising with finite element discretizations of the Navier–Stokes system. Usually these matrices are indefinite and nonsymmetric. They have to be preconditioned if a related linear system is to be solved efficiently by an iterative method. We consider preconditioning by a pressure mass matrix. It is shown how upper and lower bounds of the eigenvalues of a preconditioned Galerkin matrix may be found by variational arguments.     

Circulant preconditioners for second order hyperbolic equations

Linear systems arising from implicit time discretizations and finite difference space discretizations of second-order hyperbolic equations in two dimensions are considered. We propose and analyze the use of circulant preconditioners for the solution of linear systems via preconditioned iterative methods such as the conjugate gradient method. Our motivation is to exploit the fast inversion of circulant systems with the Fast Fourier Transform (FFT). For second-order hyperbolic equations with initial and Dirichlet boundary conditions, we prove that the condition number of the preconditioned system is ofO() orO(m), where is the quotient between the time and space steps andm is the number of interior gridpoints in each direction. The results are extended to parabolic equations. Numerical experiments also indicate that the preconditioned systems exhibit favorable clustering of eigenvalues that leads to a fast convergence rate.     

An asymptotic finite element method for improvement of solutions of boundary layer problems

Summary A scheme that uses singular perturbation theory to improve the performance of existing finite element methods is presented. The proposed scheme improves the error bounds of the standard Galerkin finite element scheme by a factor of O(ⁿ⁺¹) (where is the small parameter andn is the order of the asymptotic approximation). Numerical results for linear second order O.D.E.'s are given and are compared with several other schemes.     

On the eigenvalue distribution of a class of preconditioning methods

Summary A class of preconditioning methods depending on a relaxation parameter is presented for the solution of large linear systems of equationAx=b, whereA is a symmetric positive definite matrix. The methods are based on an incomplete factorization of the matrixA and include both pointwise and blockwise factorization. We study the dependence of the rate of convergence of the preconditioned conjugate gradient method on the distribution of eigenvalues ofC ^–1 A, whereC is the preconditioning matrix. We also show graphic representations of the eigenvalues and present numerical tests of the methods.     

Zur numerischen Lösung des ersten biharmonischen Randwertproblems

Summary The system of equations resulting from a mixed finite element approximation of the first biharmonic boundary value problem is solved by various preconditioned Uzawa-type iterative methods. The preconditioning matrices are based on simple finite element approximations of the Laplace operator and some factorizations of the corresponding matrices. The most efficient variants of these iterative methods require asymptoticallyO(h ^–0,5In ^–1) iterations andO(h ^–p–0,5In ^–1) arithmetic operations only, where denotes the relative accuracy andh is a mesh-size parameter such that the number of unknowns grows asO(h ^–p ),h0.

     

A note on conjugate gradient convergence – Part III

Summary. In this paper we again consider the rate of convergence of the conjugate gradient method. We start with a general analysis of the conjugate gradient method for uniformly bounded solutions vectors and matrices whose eigenvalues are uniformly bounded and positive. We show that in such cases a fixed finite number of iterations of the method gives some fixed amount of improvement as the the size of the matrix tends to infinity. Then we specialize to the finite element (or finite difference) scheme for the problem . We show that for some classes of function we see this same effect. For other functions we show that the gain made by performing a fixed number of iterations of the method tends to zero as the size of the matrix tends to infinity. Received July 9, 1998 / Published online March 16, 2000     

PreconditioningP 1 nonconforming finite elements: Condition numbers and singular value distributions

Summary This work deals with the condition numbers and the distribution of theB _h singular values of the preconditioned operators {B _h ^–1 A _h }_{0, whereA h andB h are discretizations of second order elliptic operatorsA andB usingP 1 nonconforming finite elements of Crouzeix and Raviart.B is also assumed to be self-adjoint and positive definite. For conforming finite elements, Parter and Wong have shown that the singular values cluster in a positive finite interval. These reults are being extended to the aforementioned nonconforming finite elements. It will be shown that, for quasiuniform grids, theB h singular values are bounded above and below by positive constants which are independent of the grid sizeh. Moreover, they also cluster in a smaller but usually estimable interval. Issues of implementation are also discussed.This research was supported by the National Science Foundation under grant number DMS-8913091}     

Additive Schwarz algorithms for the p version of the Galerkin boundary element method

Summary. We study some additive Schwarz algorithms for the version Galerkin boundary element method applied to some weakly singular and hypersingular integral equations of the first kind. Both non-overlapping and overlapping methods are considered. We prove that the condition numbers of the additive Schwarz operators grow at most as independently of h, where p is the degree of the polynomials used in the Galerkin boundary element schemes and h is the mesh size. Thus we show that additive Schwarz methods, which were originally designed for finite element discretisation of differential equations, are also efficient preconditioners for some boundary integral operators, which are non-local operators. Received June 15, 1997 / Revised version received July 7, 1998 / Published online February 17, 2000     

A multilevel preconditioner for the C-R FEM for elliptic problems with discontinuous coefficients

In this paper, we propose a multilevel preconditioner for the Crouzeix-Raviart finite element approximation of second-order elliptic partial differential equations with discontinuous coefficients. Since the finite element spaces are nonnested, weighted intergrid transfer operators, which are stable under the weighted L2 norm, are introduced to exchange information between different meshes. By analyzing the eigenvalue distribution of the preconditioned system, we prove that except a few small eigenvalues, all the other eigenvalues are bounded below and above nearly uniformly with respect to the jump and the mesh size. As a result, we get that the convergence rate of the preconditioned conjugate gradient method is quasi-uniform with respect to the jump and the mesh size. Numerical experiments are presented to confirm our theoretical analysis.     

Superlinearly convergent CG methods via equivalent preconditioning for nonsymmetric elliptic operators

Summary. The convergence of the conjugate gradient method is studied for preconditioned linear operator equations with nonsymmetric normal operators, with focus on elliptic convection-diffusion operators in Sobolev space. Superlinear convergence is proved first for equations whose preconditioned form is a compact perturbation of the identity in a Hilbert space. Then the same convergence result is verified for elliptic convection-diffusion equations using different preconditioning operators. The convergence factor involves the eigenvalues of the corresponding operators, for which an estimate is also given. The above results enable us to verify the mesh independence of the superlinear convergence estimates for suitable finite element discretizations of the convection-diffusion problems.Mathematics Subject Classification (2000): 65J10, 65F10, 65N15The second author was supported by the Hungarian Research Grant OTKA No. T. 043765.Dedicated to David M. Young on the occasion of his 80th birthday.     

Multilevel diagonal scaling preconditioners for boundary element equations on locally refined meshes

Summary. We study a multilevel preconditioner for the Galerkin boundary element matrix arising from a symmetric positive-definite bilinear form. The associated energy norm is assumed to be equivalent to a Sobolev norm of positive, possibly fractional, order m on a bounded (open or closed) surface of dimension d, with . We consider piecewise linear approximation on triangular elements. Successive levels of the mesh are created by selectively subdividing elements within local refinement zones. Hanging nodes may be created and the global mesh ratio can grow exponentially with the number of levels. The coarse-grid correction consists of an exact solve, and the correction on each finer grid amounts to a simple diagonal scaling involving only those degrees of freedom whose associated nodal basis functions overlap the refinement zone. Under appropriate assumptions on the choice of refinement zones, the condition number of the preconditioned system is shown to be bounded by a constant independent of the number of degrees of freedom, the number of levels and the global mesh ratio. In addition to applying to Galerkin discretisation of hypersingular boundary integral equations, the theory covers finite element methods for positive-definite, self-adjoint elliptic problems with Dirichlet boundary conditions. Received October 5, 2001 / Revised version received December 5, 2001 / Published online April 17, 2002 The support of this work through Visiting Fellowship grant GR/N21970 from the Engineering and Physical Sciences Research Council of Great Britain is gratefully acknowledged. The second author was also supported by the Australian Research Council     

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.     

On conjugate gradient type methods and polynomial preconditioners for a class of complex non-hermitian matrices

Summary We consider conjugate gradient type methods for the solution of large linear systemsA x=b with complex coefficient matrices of the typeA=T+iI whereT is Hermitian and a real scalar. Three different conjugate gradient type approaches with iterates defined by a minimal residual property, a Galerkin type condition, and an Euclidean error minimization, respectively, are investigated. In particular, we propose numerically stable implementations based on the ideas behind Paige and Saunder's SYMMLQ and MINRES for real symmetric matrices and derive error bounds for all three methods. It is shown how the special shift structure ofA can be preserved by using polynomial preconditioning, and results on the optimal choice of the polynomial preconditioner are given. Also, we report on some numerical experiments for matrices arising from finite difference approximations to the complex Helmholtz equation.This work was supported in part by Cooperatives Agreement NCC 2-387 between the National Aeronautics and Space Administration (NASA) and the Universities Space Research Association (USRA) and by National Science Foundation Grant DCR-8412314     

Additive Schwarz Method for Mortar Discretization of Elliptic Problems with <Emphasis Type="Italic">P</Emphasis><Subscript>1</Subscript> Nonconforming Finite Elements

An additive Schwarz preconditioner for nonconforming mortar finite element discretization of a second order elliptic problem in two dimensions with arbitrary large jumps of the discontinuous coefficients in subdomains is described. An almost optimal estimate of the condition number of the preconditioned problem is proved. The number of preconditioned conjugate gradient iterations is independent of jumps of the coefficients and is proportional to (1+log(H/h)), where H,h are mesh sizes. AMS subject classification (2000) 65N55, 65N30, 65N22     

Convergence and superconvergence analysis of an anisotropic nonconforming finite element methods for singularly perturbed reaction-diffusion problems

The numerical approximation by a lower order anisotropic nonconforming finite element on appropriately graded meshes are considered for solving singular perturbation problems. The quasi-optimal order error estimates are proved in the ε-weighted H¹-norm valid uniformly, up to a logarithmic factor, in the singular perturbation parameter. By using the interpolation postprocessing technique, the global superconvergent error estimates in ε-weighted H¹-norm are obtained. Numerical experiments are given to demonstrate validity of our theoretical analysis.     

Backward perturbation analysis of certain characteristic subspaces

Summary This paper gives optimal backward perturbation bounds and the accuracy of approximate solutions for subspaces associated with certain eigenvalue problems such as the eigenvalue problemAx=x, the generalized eigenvalue problem Ax=Bx, and the singular value decomposition of a matrixA. This paper also gives residual bounds for certain eigenvalues, generalized eigenvalues and singular values.This subject was supported by the Swedish Natural Science Research Council and the Institute of Information Processing of the University of Umeå.     

Convergent nonnegative matrices and iterative methods for consistent linear systems

Summary In this paper we study linear stationary iterative methods with nonnegative iteration matrices for solving singular and consistent systems of linear equationsAx=b. The iteration matrices for the schemes are obtained via regular and weak regular splittings of the coefficients matrixA. In certain cases when only some necessary, but not sufficient, conditions for the convergence of the iterations schemes exist, we consider a transformation on the iteration matrices and obtain new iterative schemes which ensure convergence to a solution toAx=b. This transformation is parameter-dependent, and in the case where all the eigenvalues of the iteration matrix are real, we show how to choose this parameter so that the asymptotic convergence rate of the new schemes is optimal. Finally, some applications to the problem of computing the stationary distribution vector for a finite homogeneous ergodic Markov chain are discussed.Research sponsored in part by US Army Research Office     

Nonconforming finite element methods for eigenvalue problems in linear plate theory

The paper deals with nonconforming finite element methods for approximating fourth order eigenvalue problems of type ² w=w. The methods are handled within an abstract Hilbert space framework which is a special case of the discrete approximation schemes introduced by Stummel and Grigorieff. This leads to qualitative spectral convergence under rather weak conditions guaranteeing the basic properties of consistency and discrete compactness for the nonconforming methods. Further asymptotic error estimates for eigenvalues and eigenfunctions are derived in terms of the given orders of approximability and nonconformity. These results can be applied to various nonconforming finite elements used by Adini, Morley, Zienkiewicz, de Veubeke e.a. This is carried out for the simple elements of Adini and Morley and is illustrated by some numerical results at the end.