Star-Based a Posteriori Error Estimates for Elliptic Problems

We give an a posteriori error estimator for low order nonconforming finite element approximations of diffusion-reaction and Stokes problems, which relies on the solution of local problems on stars. It is proved to be equivalent to the energy error up to a data oscillation, without requiring Helmholtz decomposition of the error nor saturation assumption. Numerical experiments illustrate the good behavior and efficiency of this estimator for generic elliptic problems.     

A flux-free a posteriori error estimator for the incompressible Stokes problem using a mixed FE formulation

In this contribution, we present an a posteriori error estimator for the incompressible Stokes problem valid for a conventional mixed FE formulation. Due to the saddle-point property of the problem, conventional error estimators developed for pure minimization problems cannot be utilized straight-forwardly. The new estimator is built up by two key ingredients. At first, a computed error approximation, exactly fulfilling the continuity equation for the error, is obtained via local Dirichlet problems. Secondly, we adopt the approach of solving local equilibrated flux-free problems in order to bound the remaining, incompressible, error. In this manner, guaranteed upper and lower bounds, of the velocity “energy norm” of the error as well as goal-oriented (linear) output functionals, with respect to a reference (overkill) mesh are obtained. In particular, it should be noted that this approach requires no computation of hybrid fluxes. Furthermore, the estimator is applicable to mixed FE formulations using continuous pressure approximations, such as the Mini and Taylor–Hood class of elements. In conclusion, a few simple numerical examples are presented, illustrating the accuracy of the error bounds.     

Lower and upper bounds of Stokes eigenvalue problem based on stabilized finite element methods

Two stabilized finite element methods for the Stokes eigenvalue problem based on the lowest equal-order finite element pair are given. They are stabilized conforming element and nonconforming element with local Gauss integration. By using the stabilized nonconforming finite element method, the lower bound of the Stokes eigenvalue is obtained; by using the stabilized conforming finite element method, the upper bound of the Stokes eigenvalue is given. Moreover, numerical tests confirm the theoretical results of the presented methods.     

On boundary-hybrid finite element methods for the Laplace equation

About two decades ago, I. Babu ka, J.T. Oden and J.K. Lee introduced finite element methods that calculate the normal derivative of the solution along the mesh interfaces and recover the solution via local Neumann problems. These methods for the treatment of the homogeneous Laplace equation were called ‘boundary-hybrid methods’. The approach was revisited in [12] for general symmetric and positive definite elliptic equations with homogeneous boundary conditions. The new approximation is nonconforming and lends itself well for an a posteriori error estimator for conforming finite element approximations. Numerical tests presented in [12] corroborated that the error estimates are accurate and cheap for conforming approximations. This paper provides the iterative solution methods and Galerkin discretization methods on which the numerical approximations in [12] were based.     

On a posteriori error estimates for the linear triangular finite element

Based on equilibration of side fluxes, an a posteriori error estimator is obtained for the linear triangular element for the Poisson equation, which can be computed locally. We present a procedure for constructing the estimator in which we use the Lagrange multiplier similar to the usual equilibrated residual method introduced by Ainsworth and Oden. The estimator is shown to provide guaranteed upper bound, and local lower bounds on the error up to a multiplicative constant depending only on the geometry. Based on this, we give another error estimator which can be directly constructed without solving local Neumann problems and also provide the two-sided bounds on the error. Finally, numerical tests show our error estimators are very efficient.     

Coupled model- and solution-adaptivity in the finite-element method

Modeling of elastic thin-walled beams, plates and shells as ID- and 2D-boundary value problems is valid in undisturbed subdomains. Disturbances near supports and free edges, in the vicinity of concentrated loads and at thickness jumps cannot be described in a sufficient way by 1D- and 2D-BVPs. In these disturbed subdomains dimensional (d)-adaptivity and model (m)-adaptivity have to be performed coupled with h- and/or p-adaptivity using hierarchically expanded test spaces in order to guarantee reliable and efficient overall results. The expansion strategy is applied for enhancing the spatial dimension and the model which is more efficient and evident for engineers than the reduction method.

Using local residual error estimators of the primal problem in the energy norm by solving Dirichlet-problems on element patches, an efficient integrated adaptive calculation of the discretization—and the dimensional error is possible and reasonable, demonstrated by examples.

We also present an error estimator of the dual problem, namely a posterior equilibrium method (PEM) for calculation of the interface tractions on local patches with Neumann boundary conditions, using orthogonality conditions. These tractions are equilibrated with respect to the global equilibrium condition of the stress resultants. An upper bound error estimator based on differences between the new tractions and the discontinuous tractions calculated from the stresses of the current finite element solution. The introduction of new element boundary tractions yields a method which can be regarded as a stepwise hybrid displacement method or as Trefftz method for local Neumann problems of element patches.

An important advantage of PEM is the coupled computation of local discretization, dimensional- and model errors by an additive split.     

A general construction of local error estimators for conforming finite elements

A local a posteriori error estimation for conforming finite elements in linear elasticity is presented. The error is measured approximately by error estimators in the form of a local energy norm per element. These estimators can be derived in a systematic way for any type of conforming model. An essential feature of the construction is a polynomial interpolation of an integral type to discretize the defects inherent in a finite element approximation.     

Two guaranteed equilibrated error estimators for Harmonic formulations in eddy current problems

In this paper a guaranteed equilibrated error estimator is developed for the 3D harmonic magnetodynamic problem of Maxwell’s system. This system is recasted in the classical $\mathbf{A}?\phi$ potential formulation and solved by the Finite Element method. The error estimator is built starting from the $\mathbf{A}?\phi$ numerical solution by a local flux reconstruction technique. Its equivalence with the error in the energy norm is established. A comparison of this estimator with an equilibrated error estimator already developed through a complementary problem points out the advantages and drawbacks of these two estimators. In particular, an analytical benchmark test illustrates the obtained theoretical results and a physical benchmark test shows the efficiency of these two estimators.     

A posteriori error analysis of time-dependent Stokes problem by Chorin-Temam scheme

We present a posteriori-residual analysis for the approximate time-dependent Stokes model Chorin-Temam projection scheme (Chorin in Math. Comput. 23:341–353, 1969; Temam in Arch. Ration. Mech. Appl. 33:377–385, 1969). Based on the multi-step approach introduced in Bergam et al. (Math. Comput. 74(251):1117–1138, 2004), we derive error estimators, with respect to both time and space approximations, related to diffusive and incompressible parts of Stokes equations. Using a conforming finite element discretization, we prove the equivalence between error and estimators under specific conditions.     

A posteriori error estimates for the primary and dual variables for the div first-order least-squares finite element method

We propose a posteriori error estimators for first-order div least-squares (LS) finite element method for linear elasticity, Stokes equations and general second-order scalar elliptic problems. Our main interest is obtaining a posteriori error estimators for the dual variables (fluxes, strains, stress, etc.) which are main quantity of interest in many applications. We also provide a posteriori error estimators for the primary variable. These estimators are obtained from the local least-squares functional by assigning weight coefficients scaling the respective residuals. The weight coefficients are given in terms of local meshsize h_K. We establish the global upper bounds and local lower bounds for the estimators. The estimators can be easily computed from the finite element solution together with the given problem data and provide basis for mesh refinement criteria for efficient computation of finite element solution (the indicators and estimators are identical). Numerical experiments show a superior performance of our a posteriori estimators for user-specific norm over the standard LS functional.     

Robust equilibrated a posteriori error estimators for the Reissner-Mindlin system

We consider a conforming finite element approximation of the Reissner-Mindlin system. We propose a new robust a posteriori error estimator based on H(div) conforming finite elements and equilibrated fluxes. It is shown that this estimator gives rise to an upper bound where the constant is one up to higher order terms. Lower bounds can also be established with constants depending on the shape regularity of the mesh. The reliability and efficiency of the proposed estimator are confirmed by some numerical tests.     

Local residual error estimator and adaptive finite element analysis of Poisson’s problems

A local residual error estimator based on the least-square residuals of the balance equation, constitutive relation and irrotationality condition is presented to enable adaptive finite element analyses of Poisson’s problems. The efficiency of the residual error indicator and the quality of the L² error indicator are investigated in problems with sharp and relatively sharp concentration of gradients. Both error estimators are highly dependent on the accuracy of the recovered or post-processed derivatives. Two types of post-processing are adopted: the L² projection and a local macroelement technique. In the proposed macroelement post-processing technique, the derivatives are recovered by solving local variational problems involving the residuals of the balance equation irrotationality condition and constitutive relation at special superconvergence points.     

Nonconforming Least-Squares Method for Elliptic Partial Differential Equations with Smooth Interfaces

In this paper a least-squares based method is proposed for elliptic interface problems in two dimensions, where the interface is smooth. The underlying method is spectral element method. The least-squares formulation is based on the minimization of a functional as defined in (4.1). The jump in the solution and its normal derivative across the interface are enforced?(in an appropriate Sobolev norm) in the functional. The solution is obtained by solving the normal equations using preconditioned conjugate gradient method. Essentially the method is nonconforming, so a block diagonal matrix is constructed as a preconditioner based on the stability estimate where each diagonal block is decoupled. A conforming solution is obtained by making a set of corrections to the nonconforming solution as in Schwab (p and h?Cp Finite Element Methods, Clarendon Press, Oxford, 1998) and an error estimate in H ¹-norm is given which shows the exponential convergence of the proposed method.     

A posteriori error estimates for nonconforming approximations of Steklov eigenvalue problems

This paper deals with a posteriori error estimators for the non conforming Crouzeix-Raviart finite element approximations of the Steklov eigenvalue problem. First, we define an error estimator of the residual type which can be computed locally from the approximate eigenpair and we prove the equivalence between this estimator and the broken energy norm of the error with constants independent of the corresponding eigenvalue. Next, we prove that edge residuals dominate the volumetric part of the residual and that the volumetric part of the residual terms dominate the normal component of the jumps of the discrete fluxes across interior edges. Finally, based on these results, we introduce two simpler equivalent error estimators. The analysis shows that these a posteriori error estimates are optimal up to higher order terms and that may be used for the design of adaptive algorithms.     

Convergence and Optimality of the Adaptive Nonconforming Linear Element Method for the Stokes Problem

In this paper, we analyze the convergence and optimality of a standard adaptive nonconforming linear element method for the Stokes problem. After establishing a special quasi-orthogonality property for both the velocity and the pressure in this saddle point problem, we introduce a new prolongation operator to carry through the discrete reliability analysis for the error estimator. We then use a specially defined interpolation operator to prove that, up to oscillation, the error can be bounded by the approximation error within a properly defined nonlinear approximate class. Finally, by introducing a new parameter-dependent error estimator, we prove the convergence and optimality estimates.     

A posteriori error analysis of nonconforming finite-element discretization for the Stokes equations

In this paper, we extend the unifying theory for a posteriori error analysis of the nonconforming finite-element methods to the Stokes problems. We present explicit residual-based computable error indicators, we prove its reliability and efficiency based on two assumptions concerning both the weak continuity and the weak orthogonality of the nonconforming finite-element spaces, respectively, and we apply the unified framework to various nonconforming finite elements from the literature.     

A new local stabilized nonconforming finite element method for the Stokes equations

In this paper, we propose and study a new local stabilized nonconforming finite method based on two local Gauss integrations for the two-dimensional Stokes equations. The nonconforming method uses the lowest equal-order pair of mixed finite elements (i.e., NCP ₁–P ₁). After a stability condition is shown for this stabilized method, its optimal-order error estimates are obtained. In addition, numerical experiments to confirm the theoretical results are presented. Compared with some classical, closely related mixed finite element pairs, the results of the present NCP ₁–P ₁ mixed finite element pair show its better performance than others.     

An improved mean estimator for judgment post-stratification

We prove that the standard nonparametric mean estimator for judgment post-stratification is inadmissible under squared error loss within a certain class of linear estimators. We derive alternate estimators that are admissible in this class, and we show that one of them is always better than the standard estimator. The reduction in mean squared error from using this alternate estimator can be as large as 10% for small set sizes and small sample sizes.     

Optimal convex error estimators for classification

A cross-validation error estimator is obtained by repeatedly leaving out some data points, deriving classifiers on the remaining points, computing errors for these classifiers on the left-out points, and then averaging these errors. The 0.632 bootstrap estimator is obtained by averaging the errors of classifiers designed from points drawn with replacement and then taking a convex combination of this “zero bootstrap” error with the resubstitution error for the designed classifier. This gives a convex combination of the low-biased resubstitution and the high-biased zero bootstrap. Another convex error estimator suggested in the literature is the unweighted average of resubstitution and cross-validation. This paper treats the following question: Given a feature-label distribution and classification rule, what is the optimal convex combination of two error estimators, i.e. what are the optimal weights for the convex combination. This problem is considered by finding the weights to minimize the MSE of a convex estimator. It also considers optimality under the constraint that the resulting estimator be unbiased. Owing to the large amount of results coming from the various feature-label models and error estimators, a portion of the results are presented herein and the main body of results appears on a companion website. In the tabulated results, each table treats the classification rules considered for the model, various Bayes errors, and various sample sizes. Each table includes the optimal weights, mean errors and standard deviations for the relevant error measures, and the MSE and MAE for the optimal convex estimator. Many observations can be made by considering the full set of experiments. Some general trends are outlined in the paper. The general conclusion is that optimizing the weights of a convex estimator can provide substantial improvement, depending on the classification rule, data model, sample size and component estimators. Optimal convex bootstrap estimators are applied to feature-set ranking to illustrate their potential advantage over non-optimized convex estimators.