The combined effect of numerical integration and approximation of the boundary in the finite element method for eigenvalue problems

Summary. The paper deals with the finite element analysis of second order elliptic eigenvalue problems when the approximate domains are not subdomains of the original domain and when at the same time numerical integration is used for computing the involved bilinear forms. The considerations are restricted to piecewise linear approximations. The optimum rate of convergence for approximate eigenvalues is obtained provided that a quadrature formula of first degree of precision is used. In the case of a simple exact eigenvalue the optimum rate of convergence for approximate eigenfunctions in the -norm is proved while in the -norm an almost optimum rate of convergence (i.e. near to is achieved. In both cases a quadrature formula of first degree of precision is used. Quadrature formulas with degree of precision equal to zero are also analyzed and in the case when the exact eigenfunctions belong only to the convergence without the rate of convergence is proved. In the case of a multiple exact eigenvalue the approximate eigenfunctions are compard (in contrast to standard considerations) with linear combinations of exact eigenfunctions with coefficients not depending on the mesh parameter . Received September 18, 1993 / Revised version received September 26, 1994     

A note on the effect of numerical quadrature in finite element eigenvalue approximation

Summary In a recent work by the author and J.E. Osborn, it was shown that the finite element approximation of the eigenpairs of differential operators, when the elements of the underlying matrices are approximated by numerical quadrature, yield optimal order of convergence when the numerical quadrature satisfies a certain precision requirement. In this note we show that this requirement is indeed sharp for eigenvalue approximation. We also show that the optimal order of convergence for approximate eigenvectors can be obtained, using numerical quadrature with less precision.The author would like to thank Prof. I. Babuka for several helpful discussions. This work was done during the author's visit to the Institute of Physical Sciences and Technology and the Department of Mathematics of University of Maryland, College Park, MD 20742, USA, and was supported in part by the Office of Naval Research under Naval Research Grant N0001490-J-1030     

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.     

On the influence of numerical integration on mixed finite element approximations of a Maxwell eigenvalue problem

The behaviour of electromagnetic resonances in cavities is modelled by a Maxwell eigenvalue problem (EVP). In the present work, we rewrite the corresponding variational problem, as it arises with a view to the application of a finite element method, in a mixed formulation. For the modelling of realistic problems the integrals occurring in this mixed formulation often cannot be evaluated exactly. We take into account the error arising from numerical quadrature and show convergence to the approximations using exact integration. Finally, some numerical results are presented.     

Numerische Quadratur bei Projektionsverfahren

Summary In this paper we investigate the influence of the numerical quadrature in projection methods. In particular we derive conditions for the order of the quadrature formulas in finite element methods under which the order of convergence is not perturbed. It seems that this question has been discussed only for the Ritz method. There is an essential difference between this method on one side and the Galerkin and least squares methods on the other side. The methods using numerical integration are only in the latter case still projection methods. The resulting conditions for the quadrature formulas are often much weaker than those for the Ritz method. Numerical examples using cubic splines and polynomials show that the conditions derived are realistic. These examples also allow the comparison of some projection methods.

     

The finite element approximation of a coupled reaction-diffusion problem with non-Lipschitz nonlinearities

Summary. A coupled semilinear elliptic problem modelling an irreversible, isothermal chemical reaction is introduced, and discretised using the usual piecewise linear Galerkin finite element approximation. An interesting feature of the problem is that a reaction order of less than one gives rise to a "dead core" region. Initially, one reactant is assumed to be acting as a catalyst and is kept constant. It is shown that error bounds previously obtained for a scheme involving numerical integration can be improved upon by considering a quadratic regularisation of the nonlinear term. This technique is then applied to the full coupled problem, and optimal and error bounds are proved in the absence of quadrature. For a scheme involving numerical integration, bounds similar to those obtained for the catalyst problem are shown to hold. Received May 25, 1993 / Revised version received July 5, 1994     

Numerical verification of solutions for variational inequalities

In this paper, we consider a numerical technique that enables us to verify the existence of solutions for variational inequalities. This technique is based on the infinite dimensional fixed point theorems and explicit error estimates for finite element approximations. Using the finite element approximations and explicit a priori error estimates for obstacle problems, we present an effective verification procedure that through numerical computation generates a set which includes the exact solution. Further, a numerical example for an obstacle problem is presented. Received October 28,1996 / Revised version received December 29,1997     

Integration of singular Galerkin-type boundary element integrals for 3D elasticity problems

Summary. A Galerkin approximation of both strongly and hypersingular boundary integral equation (BIE) is considered for the solution of a mixed boundary value problem in 3D elasticity leading to a symmetric system of linear equations. The evaluation of Cauchy principal values (v. p.) and finite parts (p. f.) of double integrals is one of the most difficult parts within the implementation of such boundary element methods (BEMs). A new integration method, which is strictly derived for the cases of coincident elements as well as edge-adjacent and vertex-adjacent elements, leads to explicitly given regular integrand functions which can be integrated by the standard Gauss-Legendre and Gauss-Jacobi quadrature rules. Problems of a wide range of integral kernels on curved surfaces can be treated by this integration method. We give estimates of the quadrature errors of the singular four-dimensional integrals. Received June 25, 1995 / Revised version received January 29, 1996     

Pointwise a posteriori error estimates for monotone semi-linear equations

We derive upper and lower a posteriori estimates for the maximum norm error in finite element solutions of monotone semi-linear equations. The estimates hold for Lagrange elements of any fixed order, non-smooth nonlinearities, and take numerical integration into account. The proof hinges on constructing continuous barrier functions by correcting the discrete solution appropriately, and then applying the continuous maximum principle; no geometric mesh constraints are thus required. Numerical experiments illustrate reliability and efficiency properties of the corresponding estimators and investigate the performance of the resulting adaptive algorithms in terms of the polynomial order and quadrature.     

Numerical integration with Taylor truncations for the quadrilateral and hexahedral finite elements

For general quadrilateral or hexahedral meshes, the finite-element methods require evaluation of integrals of rational functions, instead of traditional polynomials. It remains as a challenge in mathematics to show the traditional Gauss quadratures would ensure the correct order of approximation for the numerical integration in general. However, in the case of nested refinement, the refined quadrilaterals and hexahedra converge to parallelograms and parallelepipeds, respectively. Based on this observation, the rational functions of inverse Jacobians can be approximated by the Taylor expansion with truncation. Then the Gauss quadrature of exact order can be adopted for the resulting integrals of polynomials, retaining the optimal order approximation of the finite-element methods. A theoretic justification and some numerical verification are provided in the paper.     

Asymptotic expansions and extrapolations of eigenvalues for the stokes problem by mixed finite element methods

This paper derives a general procedure to produce an asymptotic expansion for eigenvalues of the Stokes problem by mixed finite elements. By means of integral expansion technique, the asymptotic error expansions for the approximations of the Stokes eigenvalue problem by Bernadi–Raugel element and _Q2-_P1$Q_2-P_1$ element are given. Based on such expansions, the extrapolation technique is applied to improve the accuracy of the approximations.     

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.     

A finite element lumped mass scheme for solving eigenvalue problems of circular arches

Summary In this paper, we present a finite element lumped mass scheme for eigenvalue problems of circular arch structures, and give error estimates for the approximation. They assert that approximate eigenvalues and eigenfuctions converge to the exact ones. Some numerical examples are also given to illustrate our results.     

Nonconforming streamline-diffusion-finite-element-methods for convection-diffusion problems

Summary. We analyze nonconforming finite element approximations of streamline-diffusion type for solving convection-diffusion problems. Both the theoretical and numerical investigations show that additional jump terms have to be added in the nonconforming case in order to get the same order of convergence in L as in the conforming case for convection dominated problems. A rigorous error analysis supported by numerical experiments is given. Received June 26, 1996 / Revised version received November 20, 1996     

A Zienkiewicz-type finite element applied to fourth-order problems

This paper deals with convergence analysis and applications of a Zienkiewicz-type (Z-type) triangular element, applied to fourth-order partial differential equations. For the biharmonic problem we prove the order of convergence by comparison to a suitable modified Hermite triangular finite element. This method is more natural and it could be applied to the corresponding fourth-order eigenvalue problem. We also propose a simple postprocessing method which improves the order of convergence of finite element eigenpairs. Thus, an a posteriori analysis is presented by means of different triangular elements. Some computational aspects are discussed and numerical examples are given.     

Effect of numerical integration for elliptic obstacle problems

Summary. An elliptic obstacle problem is approximated by piecewise linear finite elements with numerical integration on the penalty and forcing terms. This leads to diagonal nonlinearities and thereby to a practical scheme. Optimal error estimates in the maximum norm are derived. The proof is based on constructing suitable super and subsolutions that exploit the special structure of the penalization, and using quite precise pointwise error estimates for an associated linear elliptic problem with quadrature via the discrete maximum principle. Received March 19, 1993     

Numerical analysis of the equations of small strains quasistatic elastoviscoplasticity

Summary The present paper deals with the mathematical and the numerical analysis of small strains elastoviscoplasticity. By considering the problem as an evolution equation whose only unknown is the stress field, the quasistatic elastoviscoplastic evolution problem is proved to be well-posed, consistent mixed finite element approximations are introduced, and classical numerical algorithms are interpreted. In particular, augmented Lagrangian methods operating on the velocity appear as standard alternating-directions time-integrations of this stress evolution problem.     

A unified approach to a posteriori error estimation using element residual methods

Summary This paper deals with the problem of obtaining numerical estimates of the accuracy of approximations to solutions of elliptic partial differential equations. It is shown that, by solving appropriate local residual type problems, one can obtain upper bounds on the error in the energy norm. Moreover, in the special case of adaptiveh-p finite element analysis, the estimator will also give a realistic estimate of the error. A key feature of this is the development of a systematic approach to the determination of boundary conditions for the local problems. The work extends and combines several existing methods to the case of fullh-p finite element approximation on possibly irregular meshes with, elements of non-uniform degree. As a special case, the analysis proves a conjecture made by Bank and Weiser [Some A Posteriori Error Estimators for Elliptic Partial Differential Equations, Math. Comput.44, 283–301 (1985)].     

On the locking phenomenon for a class of elliptic problems

Summary. In this paper we study the numerical behaviour of elliptic problems in which a small parameter is involved and an example concerning the computation of elastic arches is analyzed using this mathematical framework. At first, the statements of the problem and its Galerkin approximations are defined and an asymptotic analysis is performed. Then we give general conditions ensuring that a numerical scheme will converge uniformly with respect to the small parameter. Finally we study an example in computation of arches working in linear elasticity conditions. We build one finite element scheme giving a locking behaviour, and another one which does not. Revised version received October 25, 1993     

On the connection between some linear triangular Reissner-Mindlin plate bending elements

Summary. We consider three triangular plate bending elements for the Reissner-Mindlin model. The elements are the MIN3 element of Tessler and Hughes [19], the stabilized MITC3 element of Brezzi, Fortin and Stenberg [5] and the T3BL element of Xu, Auricchio and Taylor [2, 17, 20]. We show that the bilinear forms of the stabilized MITC3 and MIN3 elements are equivalent and that their implementation may be simplified by using numerical integration of reduced order. The T3BL element is shown to be essentially the same as the MIN3 and stabilized MITC3 elements with reduced integration. We finally introduce a general stabilized finite element formulation which covers all three methods. For this class of methods we prove the stability and optimal convergence properties. Received November 4, 1996 / Revised version received May 29, 1997 / Published online January 27, 2000