Remarks around 50 lines of Matlab: short finite element implementation

A short Matlab implementation for P ₁-x ₁ finite elements on triangles and parallelograms is provided for the numerical solution of elliptic problems with mixed boundary conditions on unstructured grids. According to the shortness of the program and the given documentation, any adaptation from simple model examples to more complex problems can easily be performed. Numerical examples prove the flexibility of the Matlab tool. This revised version was published online in June 2006 with corrections to the Cover Date.     

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}     

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.     

Finite element approximation of viscoelastic fluid flow: Existence of approximate solutions and error bounds

Summary We study a finite element approximation of viscoelastic fluid flow obeying an Oldroyd B type constitutive law. The approximate stress, velocity and pressure are respectivelyP ₁ discontinuous,P ₂ continuous,P ₁ continuous. We use the method of Lesaint-Raviart for the convection of the extra stress tensor. We suppose that the continuous problem admits a sufficiently smooth and sufficiently small solution. We show by a fixed point method that the approximate problem has a solution and we give an error bound.This work has been supported in part by the GDR CNRS 901 Rhéologie der polymères fondus.     

Interpolation error estimates in W 1,p for degenerate Q 1 isoparametric elements

Optimal order error estimates in H ¹, for the Q ₁ isoparametric interpolation were obtained in Acosta and Durán (SIAM J Numer Anal37, 18–36, 1999) for a very general class of degenerate convex quadrilateral elements. In this work we show that the same conlusions are valid in W ^1,p for 1≤ p < 3 and we give a counterexample for the case p ≥ 3, showing that the result cannot be generalized for more regular functions. Despite this fact, we show that optimal order error estimates are valid for any p ≥ 1, keeping the interior angles of the element bounded away from 0 and π, independently of the aspect ratio. We also show that the restriction on the maximum angle is sharp for p ≥ 3.     

Optimal error estimate and superconvergence of the DG method for first-order hyperbolic problems

We consider the original discontinuous Galerkin method for the first-order hyperbolic problems in d-dimensional space. We show that, when the method uses polynomials of degree k, the L₂-error estimate is of order k+1 provided the triangulation is made of rectangular elements satisfying certain conditions. Further, we show the O(h^2k+1)-order superconvergence for the error on average on some suitably chosen subdomains (including the whole domain) and their outflow faces. Moreover, we also establish a derivative recovery formula for the approximation of the convection directional derivative which is superconvergent with order k+1.     

The problem of membrane locking in finite element analysis of cylindrical shells

Summary We analyse the problem of membrane locking in (h, p) finite element models of a thin hemicylindrical shell roof loaded by a smoothly varying normal pressure distribution. We show that in the standard finite element method, locking occurs especially at low values ofp and when the finite element grid is not aligned with the axis of the cylinder. A general strategy of avoiding locking by using modified bilinear forms is introduced, and a special implementation of this strategy on aligned rectangular grids is considered.     

The finite element method with nodes moving along the characteristics for convection-diffusion equations

Summary We examine the convergence properties of the finite element method with nodes moving along the characteristics for one-dimensional convection-diffusion equations. For linear elements, we demonstrate optimal rates of convergence in theL ²,H ¹ andL norms. Both linear and nonlinear problems are considered.This work forms part of the research programme of the Oxford/Reading Institute for Computational Dynamics.     

Coupling of viscous and inviscid Stokes equations via a domain decomposition method for finite elements

Summary A generalized Stokes problem is addressed in the framework of a domain decomposition method, in which the physical computational domain is partitioned into two subdomains ₁ and ₂.Three different situations are covered. In the former, the viscous terms are kept in both subdomains. Then we consider the case in which viscosity is dropped out everywhere in . Finally, a hybrid situation in which viscosity is dropped out only in ₁ is addressed. The latter is motivated by physical applications.In all cases, correct transmission conditions across the interface between ₁ and ₂ are devised, and an iterative procedure involving the successive resolution of two subproblems is proposed.The numerical discretization is based upon appropriate finite elements, and stability and convergence analysis is carried out.We also prove that the iteration-by-subdomain algorithms which are associated with the various domain decomposition approaches converge with a rate independent of the finite element mesh size.This work was partially supported by CIRA S.p.A. under the contract Coupling of Euler and Navier-Stokes equations in hypersonic flowsDeceased     

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.}     

Error estimates for a mixed finite volume method for the p-Laplacian problem

In this work we propose and analyze a mixed finite volume method for the p-Laplacian problem which is based on the lowest order Raviart–Thomas element for the vector variable and the P1 nonconforming element for the scalar variable. It is shown that this method can be reduced to a P1 nonconforming finite element method for the scalar variable only. One can then recover the vector approximation from the computed scalar approximation in a virtually cost-free manner. Optimal a priori error estimates are proved for both approximations by the quasi-norm techniques. We also derive an implicit error estimator of Bank–Weiser type which is based on the local Neumann problems.This work was supported by the Post-doctoral Fellowship Program of Korea Science & Engineering Foundation (KOSEF).     

On computingINV block preconditionings for the conjugate gradient method

The INV(k) and MINV(k) block preconditionings for the conjugate gradient method require generation of selected elements of the inverses of symmetric matrices of bandwidth 2k+1. Generalizing the previously describedk=1 (tridiagonal) case tok=2, explicit expressions for the inverse elements of a symmetric pentadiagonal matrix in terms of Green's matrix of rank two are given. These expressions are found to be seriously ill-conditioned; hence alternative computational algorithms for the inverse elements must be used. Behavior of thek=1 andk=2 preconditionings are compared for some discretized elliptic partial differential equation test problems in two dimensions.Presented by the first author at the Joint U.S.-Scandinavian Symposium on Scientific Computing and Mathematical Modeling, January 1985, in honor of Germund Dahlquist on the occasion of his 60th birthday. This work was supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy under contract DE AC03-76SF00098.     

Sur la convergence des méthodes d'éléments finis nonconformes pour des problèmes linéaires de coques minces

Résumé Dans cet article, nous étudions la convergence des méthodes d'éléments finis nonconformes pour l'approximation des problèmes de coques minces générales. Nous établissons des conditions suffisantes de convergence pour une large classe d'éléments finis, puis nous donnons des estimations de l'erreur sur les déplacements et les contraintes.

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On the convergence of nonconforming finite element method for linear thin shell problems

Summary In this paper, we study the convergence of nonconforming finite element method for the approximation of general thin shell problems. We give sufficient conditions of convergence for a large class of finite elements, then we estimate the error on the displacements and on the stresses.

     

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

Summary This work deals with theL ² condition numbers and the distribution of theL ² 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. For conforming finite elements, it was shown in the work of Goldstein, Manteuffel and Parter that if the leading part ofB is a scalar multiple (1/) of the leading part ofA, then the singular values ofB h ^–1 A h cluster and fill-in the interval [ min, max], where 0< min max are the minimum and maximum of the factor . As a generalization of these results, the current work includes nonconforming finite element methods which deal with Dirichlet boundary conditions. It will be shown that, in this more general setting, theL ² condition numbers of {B h ^–1 A h } are uniformly bounded. Moreover, the singular values also cluster and fill-in the same 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.This research was supported by the National Science Foundation under grant number DMS-8913091.}     

Projektive Newton-Verfahren bei elliptischen Randwertaufgaben

Summary In this paper we investigate projective Newton methods for nonlinear elliptic boundary value problems. These methods yield approximations by solving the linear equations of the Newton method by a projection method, e.g. the Ritz method. Using subspaces of finite elements or polynominals we obtain error estimates and optimal convergence theorems (inH ₁ andL ₂).

     

Analysis of finite element methods for second order boundary value problems using mesh dependent norms

This paper presents a new approach to the analysis of finite element methods based onC ⁰-finite elements for the approximate solution of 2nd order boundary value problems in which error estimates are derived directly in terms of two mesh dependent norms that are closely ralated to theL ₂ norm and to the 2nd order Sobolev norm, respectively, and in which there is no assumption of quasi-uniformity on the mesh family. This is in contrast to the usual analysis in which error estimates are first derived in the 1st order Sobolev norm and subsequently are derived in theL ₂ norm and in the 2nd order Sobolev norm — the 2nd order Sobolev norm estimates being obtained under the assumption that the functions in the underlying approximating subspaces lie in the 2nd order Sobolev space and that the mesh family is quasi-uniform.     

Superconvergence analysis of approximate boundary-flux calculations

Summary Certain projection post-processing techniques have been proposed for computing the boundary flux for two-dimensional problems (e.g., see Carey, et al. [5]). In a series of numerical experiments on elliptic problems they observed that these post-processing formulas for approximate fluxes were almost (O(h ²)-accurate for linear triangular elements. In this paper we prove that the computed boundary flux isO(h ² ln 1/h)-accurate in the maximum norm for the partial method of [5]. If the solutionuH ³() then the boundary flux error isO(h ^3/2) in theL ²-norm.     

Upper and lower error bounds for plate-bending finite elements

Summary. We compare the robustness of three different low-order mixed methods that have been proposed for plate-bending problems: the so-called MITC, Arnold-Falk and Arnold-Brezzi elements. We show that for free plates, the asymptotic rate of convergence in the presence of quasiuniform meshes approaches the optimal O(h) for MITC elements as the thickness approaches 0, but only approaches for the latter two. We accomplish this by establishing lower bounds for the error in the rotation. The deterioration occurs due to a consistency error associated with the boundary layer – we show how a modification of the elements at the boundary can fix the problem. Finally, we show that the Arnold-Brezzi element requires extra regularity for the convergence of the limiting (discrete Kirchhoff) case, and show that it fails to converge in the presence of point loads. Received June 9, 1998 / Published online December 6, 1999     

A posteriori error estimates for mixed finite element approximations of elliptic problems

We derive residual based a posteriori error estimates of the flux in L ²-norm for a general class of mixed methods for elliptic problems. The estimate is applicable to standard mixed methods such as the Raviart–Thomas–Nedelec and Brezzi–Douglas–Marini elements, as well as stabilized methods such as the Galerkin-Least squares method. The element residual in the estimate employs an elementwise computable postprocessed approximation of the displacement which gives optimal order.     

A p-version finite element method for nonlinear elliptic variational inequalities in 2D

This article introduces and analyzes a p-version FEM for variational inequalities resulting from obstacle problems for some quasi-linear elliptic partial differential operators. We approximate the solution by controlling the obstacle condition in images of the Gauss–Lobatto points. We show existence and uniqueness for the discrete solution u _p from the p-version for the obstacle problem. We prove the convergence of u _p towards the solution with respect to the energy norm, and assuming some additional regularity for the solution we derive an a priori error estimate. In numerical experiments the p-version turns out to be superior to the h-version concerning the convergence rate and the number of unknowns needed to achieve a certain exactness of the approximation.