A second order finite difference error indicator for adaptive transonic flow computations

Summary. An adaptive finite element method for the calculation of transonic potential flows was developed. An error indicator based on first order finite differences of gradients is introduced as a local error estimator. It measures second order distributional derivatives. Estimates involving this error estimator, a residual and the error are given. The error estimator can be used as a criterion for mesh refinement. We also give some computational results. Received September 16, 1993 / Revised version received June 7, 1994     

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.     

A residual based error estimator for mortar finite element discretizations

Summary. A residual based error estimator for the approximation of linear elliptic boundary value problems by nonconforming finite element methods is introduced and analyzed. In particular, we consider mortar finite element techniques restricting ourselves to geometrically conforming domain decomposition methods using P1 approximations in each subdomain. Additionally, a residual based error estimator for Crouzeix-Raviart elements of lowest order is presented and compared with the error estimator obtained in the more general mortar situation. It is shown that the computational effort of the error estimator can be considerably reduced if the special structure of the Lagrange multiplier is taken into account. Received July 18, 1997 / Revised version received July 27, 1998 / Published online September 7, 1999     

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).     

A posteriori error estimators for the Stokes equations II non-conforming discretizations

Summary We present an a posteriori error estimator for the non-conforming Crouzeix-Raviart discretization of the Stokes equations which is based on the local evaluation of residuals with respect to the strong form of the differential equation. The error estimator yields global upper and local lower bounds for the error of the finite element solution. It can easily be generalized to the stationary, incompressible Navier-Stokes equations and to other non-conforming finite element methods. Numerical examples show the efficiency of the proposed error estimator.     

A posteriori error estimations of residual type for Signorini's problem

This paper presents an a posteriori error analysis for the linear finite element approximation of the Signorini problem in two space dimensions. A posteriori estimations of residual type are defined and upper and lower bounds of the discretization error are obtained. We perform several numerical experiments in order to compare the convergence of the terms in the error estimator with the discretization error.     

A posteriori error estimation of residual type for anisotropic diffusion-convection-reaction problems

This paper presents a robust a posteriori residual error estimator for diffusion-convection-reaction problems with anisotropic diffusion, approximated by a SUPG finite element method on isotropic or anisotropic meshes in R^d, d=2 or 3. The equivalence between the energy norm of the error and the residual error estimator is proved. Numerical tests confirm the theoretical results.     

An a posteriori error estimator for anisotropic refinement

Summary. Besides an algorithm for local refinement, an a posteriori error estimator is the basic tool of every adaptive finite element method. Using information generated by such an error estimator the refinement of the grid is controlled. For 2nd order elliptic problems we present an error estimator for anisotropically refined grids, like -d cuboidal and 3-d prismatic grids, that gives correct information about the size of the error; additionally it generates information about the direction into which some element has to be refined to reduce the error in a proper way. Numerical examples are presented for 2-d rectangular and 3-d prismatic grids. Received March 15, 1994 / Revised version received June 3, 1994     

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.     

Robustness in a posteriori error analysis for FEM flow models

Summary. We derive a residual-based a posteriori error estimator for a stabilized finite element discretization of certain incompressible Oseen-like equations. We focus our attention on the behaviour of the effectivity index and we carry on a numerical study of its sensitiveness to the problem and mesh parameters. We also consider a scalar reaction-convection-diffusion problem and a divergence-free projection problem in order to investigate the effects on the robustness of our a posteriori error estimator of the reaction-convection-diffusion phenomena and, separately, of the incompressibility constraint. Received March 21, 2001 / Revised version received July 30, 2001 / Published online October 17, 2001     

A mixed finite element method for solving the nonstationary stokes equation

Summary This paper deals with a mixed finite element method for approximating a fourth order initial value problem arising from the nonstationary Stokes problem. For piecewise linear shape functions error estimates are given with convergence rates similar to the elliptic case. Some numerical computations will illustrate the theoretical results.     

A posteriori error estimates for mixed FEM in elasticity

A residue based reliable and efficient error estimator is established for finite element solutions of mixed boundary value problems in linear, planar elasticity. The proof of the reliability of the estimator is based on Helmholtz type decompositions of the error in the stress variable and a duality argument for the error in the displacements. The efficiency follows from inverse estimates. The constants in both estimates are independent of the Lamé constant , and so locking phenomena for are properly indicated. The analysis justifies a new adaptive algorithm for automatic mesh–refinement. Received July 17, 1997     

Adaptive finite elements for exterior domain problems

Summary. We present an adaptive finite element method for solving elliptic problems in exterior domains, that is for problems in the exterior of a bounded closed domain in , . We describe a procedure to generate a sequence of bounded computational domains , , more precisely, a sequence of successively finer and larger grids, until the desired accuracy of the solution is reached. To this end we prove an a posteriori error estimate for the error on the unbounded domain in the energy norm by means of a residual based error estimator. Furthermore we prove convergence of the adaptive algorithm. Numerical examples show the optimal order of convergence. Received July 8, 1997 /Revised version received October 23, 1997     

On the asymptotic exactness of error estimators for linear triangular finite elements

Summary This paper deals with a-posteriori error estimates for piecewise linear finite element approximations of elliptic problems. We analyze two estimators based on recovery operators for the gradient of the approximate solution. By using superconvergence results we prove their asymptotic exactness under regularity assumptions on the mesh and the solution.One of the estimators can be easily computed in terms of the jumps of the gradient of the finite element approximation. This estimator is equivalent to the error in the energy norm under rather general conditions. However, we show that for the asymptotic exactness, the regularity assumption on the mesh is not merely technical. While doing this, we analyze the relation between superconvergence and asymptotic exactness for some particular examples.     

Finite element analysis of a holonomic elastic-plastic problem

Summary A thorough analysis of the finite element method is given for a holonomic elastic-plastic problem. An inequality of the Cea's lemma type is proved, which is the basis of error estimates for various finite element solutions. Difficulty caused by a non-differentiable term in the problem can be overcome by using two convergent procedures, an iterative procedure and a regularization procedure. An a-posteriori quantitative error estimate is derived for the regularized solution.The work was done while the author was at the Department of Mathematics, University of Maryland, College Park. The research was partially supported by the National Science Foundation grant CCR-88-20279     

A framework for obtaining guaranteed error bounds for finite element approximations

We give an overview of our recent progress in developing a framework for the derivation of fully computable guaranteed posteriori error bounds for finite element approximation including conforming, non-conforming, mixed and discontinuous finite element schemes. Whilst the details of the actual estimator are rather different for each particular scheme, there is nonetheless a common underlying structure at work in all cases. We aim to illustrate this structure by treating conforming, non-conforming and discontinuous finite element schemes in a single framework. In taking a rather general viewpoint, some of the finer details of the analysis that rely on the specific properties of each particular scheme are obscured but, in return, we hope to allow the reader to ‘see the wood despite the trees’.     

Dual-mixed finite element approximation of Stokes and nonlinear Stokes problems using trace-free velocity gradients

In this work a finite element method for a dual-mixed approximation of Stokes and nonlinear Stokes problems is studied. The dual-mixed structure, which yields a twofold saddle point problem, arises in a formulation of this problem through the introduction of unknown variables with relevant physical meaning. The method approximates the velocity, its gradient, and the total stress tensor, but avoids the explicit computation of the pressure, which can be recovered through a simple postprocessing technique. This method improves an existing approach for these problems and uses Raviart-Thomas elements and discontinuous piecewise polynomials for approximating the unknowns. Existence, uniqueness, and error results for the method are given, and numerical experiments that exhibit the reduced computational cost of this approach are presented.     

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     

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)].     

AN ADAPTIVE INVERSE ITERATION FEM FOR THE INHOMOGENEOUS DIELECTRIC WAVEGUIDES

We introduce an adaptive finite element method for computing electromagnetic guided waves in a closed, inhomogeneous, pillared three-dimensional waveguide at a given frequency based on the inverse iteration method. The problem is formulated as a generalized eigenvalue problems. By modifying the exact inverse iteration algorithm for the eigenvalue problem, we design a new adaptive inverse iteration finite element algorithm. Adaptive finite element methods based on a posteriori error estimate are known to be successful in resolving singularities of eigenfunctions which deteriorate the finite element convergence. We construct a posteriori error estimator for the electromagnetic guided waves problem. Numerical results are reported to illustrate the quasi-optimal performance of our adaptive inverse iteration finite element method.