A convergent adaptive finite element method for an optimal design problem

The optimal design problem for maximal torsion stiffness of an infinite bar of given geometry and unknown distribution of two materials of prescribed amounts is one model example in topology optimisation. It eventually leads to a degenerate convex minimisation problem. The numerical analysis is therefore delicate for possibly multiple primal variables u but unique derivatives σ : = DW(D u). Even fine a posteriori error estimates still suffer from the reliability-efficiency gap. However, it motivates a simple edge-based adaptive mesh-refining algorithm (AFEM) that is not a priori guaranteed to refine everywhere. Its convergence proof is therefore based on energy estimates and some refined convexity control. Numerical experiments illustrate even nearly optimal convergence rates of the proposed AFEM. Supported by the DFG Research Center MATHEON “Mathematics for key technologies” in Berlin.     

A posteriori error estimates for elliptic problems with Dirac delta source terms

The aim of this paper is to introduce residual type a posteriori error estimators for a Poisson problem with a Dirac delta source term, in L ^p norm and W^1,p seminorm. The estimators are proved to yield global upper and local lower bounds for the corresponding norms of the error. They are used to guide adaptive procedures, which are experimentally shown to lead to optimal orders of convergence.     

Adaptive finite element for semi-linear convection–diffusion problems

In this article a strategy of adaptive finite element for semi-linear problems, based on minimizing a residual-type estimator, is reported. We get an a posteriori error estimate which is asymptotically exact when the mesh size h tends to zero. By considering a model problem, the quality of this estimator is checked. It is numerically shown that without constraint on the mesh size h, the efficiency of the a posteriori error estimate can fail dramatically. This phenomenon is analysed and an algorithm which equidistributes the local estimators under the constraint h ⩽ h _max is proposed. This algorithm allows to improve the computed solution for semi-linear convection–diffusion problems, and can be used for stabilizing the Lagrange finite element method for linear convection–diffusion problems. This revised version was published online in June 2006 with corrections to the Cover Date.     

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.     

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).     

Error estimates for the numerical solution of a time-periodic linear parabolic problem

In this paper we consider the numerical solution of a time-periodic linear parabolic problem. We derive optimal order error estimates inL ₂() for approximate solutions obtained by discretizing in space by a Galerkin finite-element method and in time by single-step finite difference methods, using known estimates for the associated initial value problem. We generalize this approach and obtain error estimates for more general discretization methods in the norm of a Banach spaceB L ₂(), e.g.,B=H ₀ ¹ () orL (). Finally, we consider some computational aspects and give a numerical example.     

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)].     

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.     

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}     

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.}     

Convergent adaptive finite elements for the nonlinear Laplacian

Summary. The numerical solution of the homogeneous Dirichlet problem for the p-Laplacian, , is considered. We propose an adaptive algorithm with continuous piecewise affine finite elements and prove that the approximate solutions converge to the exact one. While the algorithm is a rather straight-forward generalization of those for the linear case p=2, the proof of its convergence is different. In particular, it does not rely on a strict error reduction. Received December 29, 2000 / Revised version received August 30, 2001 / Published online December 18, 2001 RID="*" ID="*" Current address: Dipartimento di Matematica, Università degli Studi di Milano, Via C. Saldini 50, 20133 Milano, Italy; e-mail: veeser@mat.unimi.it     

A posteriori error estimator based on gradient recovery by averaging for discontinuous Galerkin methods

We consider some (anisotropic and piecewise constant) diffusion problems in domains of R², approximated by a discontinuous Galerkin method with polynomials of any fixed degree. We propose an a posteriori error estimator based on gradient recovery by averaging. It is shown that this estimator gives rise to an upper bound where the constant is one up to some additional terms that guarantee reliability. The lower bound is also established. Moreover these additional terms are negligible when the recovered gradient is superconvergent. The reliability and efficiency of the proposed estimator is confirmed by some numerical tests.     

Fourth order eigenvalue approximation by extrapolation on domains with reentrant corners

Summary The eigenvalue problem of the Laplace operator is considered on a non-convex domain composed of rectangles. This model problem may be solved by the finite element method with bilinear elements on a rectangular mesh. It is known thatO(h) ²(<1) convergence can be obtained for the eigenvalues, if the mesh hasO(h) ^–2 points. A simple extrapolation scheme is presented which, on appropriately graded meshes, increases the rate of convergence toO(h) ⁴ This work was supported by the Deutsche Forschungsgemeinschaft (DFG), SFB 123 Stochatistische Mathematische Modelle, Universität Heidelberg     

Convergent numerical approximations of the thermomechanical phase transitions in shape memory alloys

Summary Discrete approximations are constructed to a nonlinear evolutionary system of partial differential equations arising from modelling the dynamics of solid-state phase transitions of thermomechenical nature in the case of one space dimension. The class of problems considered includes the so-called shape memory alloys, in particular. It is shown that the obtained discrete solutions converge to the solution of the original problem, and numerical simulations for the shape memory alloy Au₂₃Cu₃₀Zn₄₇ demonstrate the quality of the discrete model.Partially supported by Research Program RP.1.02Supported by DFG, SPP Anwendungsbezogene Optimierung und Steuerung     

A finite difference analysis of a streamline diffusion method on a Shishkin mesh

We consider streamline diffusion finite element methods applied to a singularly perturbed convection–diffusion two‐point boundary value problem whose solution has a single boundary layer. To analyse the convergence of these methods, we rewrite them as finite difference schemes. We first consider arbitrary meshes, then, in analysing the scheme on a Shishkin mesh, we consider two formulations on the fine part of the mesh: the usual streamline diffusion upwinding and the standard Galerkin method. The error estimates are given in the discrete L _∞ norm; in particular we give the first analysis that shows precisely how the error depends on the user-chosen parameter _τ0 specifying the mesh. When _τ0 is too small, the error becomes O(1), but for _τ0 above a certain threshold value, the error is small and increases either linearly or quadratically as a function of . Numerical tests support our theoretical results. This revised version was published online in August 2006 with corrections to the Cover Date.     

Quadratic spline collocation methods for elliptic partial differential equations

We consider Quadratic Spline Collocation (QSC) methods for linear second order elliptic Partial Differential Equations (PDEs). The standard formulation of these methods leads to non-optimal approximations. In order to derive optimal QSC approximations, high order perturbations of the PDE problem are generated. These perturbations can be applied either to the PDE problem operators or to the right sides, thus leading to two different formulations of optimal QSC methods. The convergence properties of the QSC methods are studied. OptimalO(h ^3–j ) global error estimates for thejth partial derivative are obtained for a certain class of problems. Moreover,O(h ^4–j ) error bounds for thejth partial derivative are obtained at certain sets of points. Results from numerical experiments verify the theoretical behaviour of the QSC methods. Performance results also show that the QSC methods are very effective from the computational point of view. They have been implemented efficiently on parallel machines.This research was supported in part by David Ross Foundation (U.S.A) and NSERC (Natural Sciences and Engineering Research Council of Canada).     

The finite element method with non-uniform mesh sizes applied to the exterior Helmholtz problem

Summary The finite element method with non-uniform mesh sizes is employed to approximately solve Helmholtz type equations in unbounded domains. The given problem on an unbounded domain is replaced by an approximate problem on a bounded domain with the radiation condition replaced by an approximate radiation boundary condition on the artificial boundary. This approximate problem is then solved using the finite element method with the mesh graded systematically in such a way that the element mesh sizes are increased as the distance from the origin increases. This results in a great reduction in the number of equations to be solved. It is proved that optimal error estimates hold inL ²,H ¹ andL , provided that certain relationships hold between the frequency, mesh size and outer radius.     

Fourier analysis of schwarz alternating methods for piecewise hermite bicubic orthogonal spline collocation

The rates of convergence of two Schwarz alternating methods are analyzed for the iterative solution of a discrete problem which arises when orthogonal spline collocation with piecewise Hermite bicubics is applied to the Dirichlet problem for Poisson's equation on a rectangle. In the first method, the rectangle is divided into two overlapping subrectangles, while three overlapping subrectangles are used in the second method. Fourier analysis is used to obtain explicit formulas for the convergence factors by which theH ¹-norm of the errors is reduced in one iteration of the Schwarz methods. It is shown numerically that while these factors depend on the size of overlap, they are independent of the partition stepsize. Results of numerical experiments are presented which confirm the established rates of convergence of the Schwarz methods.This research was supported in part by funds from the National Science Foundation grant CCR-9103451.     

Boundary element preconditioners for a hypersingular integral equation on an interval

We propose an almost optimal preconditioner for the iterative solution of the Galerkin equations arising from a hypersingular integral equation on an interval. This preconditioning technique, which is based on the single layer potential, was already studied for closed curves [11,14]. For a boundary element trial space, we show that the condition number is of order (1 + | log h _min|)², where h _min is the length of the smallest element. The proof requires only a mild assumption on the mesh, easily satisfied by adaptive refinement algorithms.     

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.