A posteriori H1 error estimation for a singularly perturbed reaction diffusion problem on anisotropic meshes

The paper deals with a singularly perturbed reaction diffusionmodel problem. The focus is on reliable a posteriori error estimatorsfor the H¹ seminorm that can be applied to anisotropic finiteelement meshes. A residual error estimator and a local problemerror estimator are proposed and rigorously analysed. They arelocally equivalent, and both bound the error reliably. Threemodifications of these estimators are introduced and discussed. Much attention is given to the performance of the error estimatorin numerical experiments. This helps to identify those estimatorsthat are suitable for practical applications.     

A Posteriori Error Bounds for Linear Elliptic Equations of Potential Type

Second-order Linear elliptic partial differential equations of potential type with Dirichlet (type 1) or Neumann (type II)boundary conditions on a simply-connected two-dimensional domainare considered. Conjugate problems, that is, a pair of one type1 and one type II problem, are introduced along with an auxiliaryeppiptic system of two equations in such a way that the energiesof the given problem, its conjugate problem, and the auxiliarysystem add to a known constant. There result two-sided boundsfor the energy of the given problem and, as a consequence, aposteriori error bounds for the norm of the difference of anapproximate solution and the exact solution of the problem.A method by which the amount of computation required to obtainthe a posteriori error bounds can be almost halved in many casesof practical interest is given. A posteriori error bounds forapproximate solutions of the auxiliary system are also given.     

An a posteriori error estimator for the Lame equation based on equilibrated fluxes

Katharina Witowski ^{We derive a new a posteriori error estimator for the Lamésystem based on H(div)-conforming elements and equilibratedfluxes. It is shown that the estimator gives rise to an upperbound where the constant is one up to higher-order terms. Thelower bound is also established using Argyris elements. Thereliability and efficiency of the proposed estimator are confirmedby some numerical tests.}     

Coupling of mixed finite elements and boundary elements

The symmetric coupling of mixed finite element and boundaryelement methods is analysed for a model interface problem withthe Laplacian. The coupling involves a further continuous ansatzfunction on the interface to link the discontinuous displacementfield to the necessarily continuous boundary ansatz function.Quasi-optimal a priori error estimates and sharp a posteriorierror estimates are established which justify adaptive mesh-refiningalgorithms. Numerical experiments prove the adaptive couplingas an efficient tool for the numerical treatment of transmissionproblems.     

The reliability of local error estimators for convection-diffusion equations

We assess the reliability of a simple a posteriori error estimatorfor steady-state convection–diffusion equations in caseswhere convection dominates. Our estimator is computed by solvinga local Poisson problem with Neumann boundary conditions. Itgives global upper and local lower bounds on the error measuredin the H¹ semi-norm. However, the error may be overestimatedlocally within boundary layers if these are not resolved bythe mesh, that is, when the local mesh Péclet numberis significantly greater than unity. We discuss the implicationsof this overestimation in a practical context where the estimatoris used as a local error indicator within a self-adaptive meshrefinement process. Received 18 June 1999. Accepted 7 March 2000.     

A posteriori error analysis of a cell-centered finite volume method for semilinear elliptic problems

In this paper, we conduct a goal-oriented a posteriori analysis for the error in a quantity of interest computed from a cell-centered finite volume scheme for a semilinear elliptic problem. The a posteriori error analysis is based on variational analysis, residual errors and the adjoint problem. To carry out the analysis, we use an equivalence between the cell-centered finite volume scheme and a mixed finite element method with special choice of quadrature.     

Reduced-basis output bound methods for parabolic problems

^** Email: rovas{at}uiuc.edu^*** Email: luc_machiels{at}mckinsey.com^**** Corresponding author. Email: maday{at}ann.jussieu.fr In this paper, we extend reduced-basis output bound methodsdeveloped earlier for elliptic problems, to problems describedby ‘parameterized parabolic’ partial differentialequations. The essential new ingredient and the novelty of thispaper consist in the presence of time in the formulation andsolution of the problem. First, without assuming a time discretization,a reduced-basis procedure is presented to ‘efficiently’compute accurate approximations to the solution of the parabolicproblem and ‘relevant’ outputs of interest. In addition,we develop an error estimation procedure to ‘a posteriorivalidate’ the accuracy of our output predictions. Second,using the discontinuous Galerkin method for the temporal discretization,the reduced-basis method and the output bound procedure areanalysed for the semi-discrete case. In both cases the reduced-basisis constructed by taking ‘snapshots’ of the solutionboth in time and in the parameters: in that sense the methodis close to Proper Orthogonal Decomposition (POD).     

A posteriori L2 error estimation on anisotropic tetrahedral finite element meshes

A new a posteriori L₂ norm error estimator is proposed for thePoisson equation. The error estimator can be applied to anisotropictetrahedral or triangular finite element meshes. The estimatoris rigorously analysed for Dirichlet and Neumann boundary conditions. The lower error bound relies on specifically designed anisotropicbubble functions and the corresponding inverse inequalities.The upper error bound utilizes non-standard anisotropic interpolationestimates. Its proof requires H² regularity of the Poisson problem,and its quality depends on how good the anisotropic mesh resolvesthe anisotropy of the problem. This is measured by a so-called‘matching function’. A numerical example supports the anisotropic error analysis.     

A Fast Method for the Solution of Fredholm Integral Equations

Methods described to date for the solution of linear Fredholmintegral equations have a computing time requirement of O(N³),where N is the number of expansion functions or discretizationpoints used. We describe here a Tchebychev expansion method,based on the FFT, which reduces this time to O(N² ln N), andreport some comparative timings obtained with it. We give alsoboth a priori and a posteriori error estimates which are cheapto compute, and which appear more reliable than those used previously.     

A posteriori error control for finite element approximations of elliptic eigenvalue problems

We develop a new approach to a posteriori error estimation for Galerkin finite element approximations of symmetric and nonsymmetric elliptic eigenvalue problems. The idea is to embed the eigenvalue approximation into the general framework of Galerkin methods for nonlinear variational equations. In this context residual-based a posteriori error representations are available with explicitly given remainder terms. The careful evaluation of these error representations for the concrete situation of an eigenvalue problem results in a posteriori error estimates for the approximations of eigenvalues as well as eigenfunctions. These suggest local error indicators that are used in the mesh refinement process.     

Error Analysis of the Enthalpy Method for the Stefan Problem

In this paper an error bound is derived for a practical piecewiselinear finite-element approximation of an enthalpy formulationof the multidimensional Stefan problem with an implicit timediscretization. It is shown that if the time step t is O(h),then the error in the temperature measured in the L² norm isO(h).     

Numerical differentiation by radial basis functions approximation

Based on radial basis functions approximation, we develop in this paper a new com-putational algorithm for numerical differentiation. Under an a priori and an a posteriori choice rules for the regularization parameter, we also give a proof on the convergence error estimate in reconstructing the unknown partial derivatives from scattered noisy data in multi-dimension. Numerical examples verify that the proposed regularization strategy with the a posteriori choice rule is effective and stable to solve the numerical differential problem. *The work described in this paper was partially supported by a grant from CityU (Project No. 7001646) and partially supported by the National Natural Science Foundation of China (No. 10571079).     

The Kačanov method for some nonlinear problems

The Kačanov method is an iteration method for solving some nonlinear partial differential equation problems. In each iteration, a linear problem is solved. In this paper, we discuss the use of the Kačanov method in the context of two model problems. We show the convergence of the Kačanov iteration sequences, and derive a posteriori error estimates for the Kačanov iterates. Numerical examples are given showing the convergence of the method and the effectiveness of the a posteriori error estimates.     

A posteriori error estimation for convection dominated problems on anisotropic meshes

A singularly perturbed convection–diffusion problem in two and three space dimensions is discretized using the streamline upwind Petrov Galerkin (SUPG) variant of the finite element method. The dominant convection frequently gives rise to solutions with layers; hence anisotropic finite elements can be applied advantageously. The main focus is on a posteriori energy norm error estimation that is robust in the perturbation parameter and with respect to the mesh anisotropy. A residual error estimator and a local problem error estimator are proposed and investigated. The analysis reveals that the upper error bound depends on the alignment of the anisotropies of the mesh and of the solution. Hence reliable error estimation is possible for suitable anisotropic meshes. The lower error bound depends on the problem data via a local mesh Peclet number. Thus efficient error estimation is achieved for small mesh Peclet numbers. Altogether, error estimation approaches for isotropic meshes are successfully extended to anisotropic elements. Several numerical experiments support the analysis. Copyright © 2003 John Wiley & Sons, Ltd.     

On Mixed Error Estimates for Elliptic Obstacle Problems

We establish in this paper sharp error estimates of residual type for finite element approximation to elliptic obstacle problems. The estimates are of mixed nature, which are neither of a pure a priori form nor of a pure a posteriori form but instead they are combined by an a priori part and an a posteriori part. The key ingredient in our derivation for the mixed error estimates is the use of a new interpolator which enables us to eliminate inactive data from the error estimators. One application of our mixed error estimates is to construct a posteriori error indicators reliable and efficient up to higher order terms, and these indicators are useful in mesh-refinements and adaptive grid generations. In particular, by approximating the a priori part with some a posteriori quantities we can successfully track the free boundary for elliptic obstacle problems.     

Probabilistic Analysis of Simpson′s Quadrature

We study probabilistic properties of Simpson′s quadrature, assuming that the class of integrands is equipped with a variant of the r-fold Wiener measure. In the average case setting, we show that the error of Simpson′s quadrature is minimal (modulo a multiplicative constant) when equally spaced points are used. Furthermore, composite Simpson′s quadrature with equally spaced points is almost optimal among all algorithms iff the regularity degree r does not exceed 3. We are also interested in computing a posteriori bounds on the error of Simpson′s quadrature. The error bounds as well as the approximation to ∫¹₀ ƒ(x) dx are computed based on a (fixed) finite number of function values. We derive a new a posteriori error bound for Simpson′s quadrature and show that, from a probabilistic point of view, it is significantly better than a bound that is commonly used in practice.     

A posteriori error estimates for a time discrete scheme for a phase-field system of Penrose-Fife type

A time discrete scheme is used to approximate the solution toa phase field system of Penrose–Fife type with a non-conservedorder parameter. An a posteriori error estimate is presentedthat allows the estimation of the difference between continuousand semidiscrete solutions by quantities that can be calculatedfrom the approximation and given data.     

A C0 interior penalty method for a fourth‐order variational inequality of the second kind

In this article, we propose a C⁰ interior penalty method for the frictional plate contact problem and derive both a priori and a posteriori error estimates. We derive an abstract error estimate in the energy norm without additional regularity assumption on the exact solution. The a priori error estimate is of optimal order whenever the solution is regular. Further, we derive a reliable and efficient a posteriori error estimator. Numerical experiments are presented to illustrate the theoretical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 36–59, 2016