A posteriori error estimation in sensitivity analysis

We present a posteriori error estimators suitable for automatic mesh refinement in the numerical evaluation of sensitivity by means of the finite element method. Both diffusion (Poisson-type) and elasticity problems are considered, and the equivalence between the true error and the proposed error estimator is proved. Application to shape sensitivity is briefly addressed.     

Remarks on a posteriori error estimation for inaccurate finite element solutions

Usually, error estimators for adaptive refinement require exact discrete solutions. In this paper, we show how inaccurate solutions (e.g., iterative approximations) can be used, too. As a side remark we characterise iterative solution schemes that are particularly suited to producing good approximations for error estimators. This work was supported by Deutsche Forschungsgemeinschaft (Project Ha 1324/9).     

A posteriori error estimation for acoustic wave propagation problems

Summary The main purpose of this paper is to review a posteriori error estimators for the simulation of acoustic wave propagation problems by computational methods. Residual-type (explicit and implicit) and recovery-type estimators are presented in detail in the case of the Helmholtz problem. Recent work on goal-oriented error estimation techniques with respect to so-called quantities of interest or output functionals are also accounted for. Fundamental results from a priori error estimation are presented and issues dealing with pollution error at large wave numbers are extensively discussed.     

Uniform convergence and a posteriori error estimation for assumed stress hybrid finite element methods

Assumed stress hybrid methods are known to improve the performance of standard displacement-based finite elements and are widely used in computational mechanics. The methods are based on the Hellinger–Reissner variational principle for the displacement and stress variables. This work analyzes two existing 4-node hybrid stress quadrilateral elements due to Pian and Sumihara [T.H.H. Pian, K. Sumihara, Rational approach for assumed stress finite elements, Int. J. Numer. Methods Engrg. 20 (9) (1984) 1685–1695] and due to Xie and Zhou [X.P. Xie, T.X. Zhou, Optimization of stress modes by energy compatibility for 4-node hybrid quadrilaterals, Int. J. Numer. Methods Engrg. 59 (2004) 293–313], which behave robustly in numerical benchmark tests. For the finite elements, the isoparametric bilinear interpolation is used for the displacement approximation, while different piecewise-independent 5-parameter modes are employed for the stress approximation. We show that the two schemes are free from Poisson-locking, in the sense that the error bound in the a priori estimate is independent of the relevant Lamé constant λ. We also establish the equivalence of the methods to two assumed enhanced strain schemes. Finally, we derive reliable and efficient residual-based a posteriori error estimators for the stress in L²-norm and the displacement in H¹-norm, and verify the theoretical results by some numerical experiments.     

Error-tolerant sign retrieval using visual features and maximum a posteriori estimation

This paper proposes an efficient error-tolerant approach to retrieving sign words from a Taiwanese Sign Language (TSL) database. This database is tagged with visual gesture features and organized as a multilist code tree. These features are defined in terms of the visual characteristics of sign gestures by which they are indexed for sign retrieval and displayed using an anthropomorphic interface. The maximum a posteriori estimation is exploited to retrieve the most likely sign word given the input feature sequence. An error-tolerant mechanism based on mutual information criterion is proposed to retrieve a sign word of interest efficiently and robustly. A user-friendly anthropomorphic interface is also developed to assist learning TSL. Several experiments were performed in an educational environment to investigate the system's retrieval accuracy. Our proposed approach outperformed a dynamic programming algorithm in its task and shows tolerance to user input errors.     

A multiscale a posteriori error estimate

We introduce a hierarchic a posteriori error estimate for singularly perturbed reaction–diffusion problems. The estimator is based on a Petrov–Galerkin method in which the trial space is enriched with nonpolynomial functions or multiscale functions. We study the equivalence between the a posteriori estimate and the exact error in the energy norm. Moreover, we prove a relationship between the hierarchic estimator and an explicit residual estimator. The approach provides accurate estimates for the boundary layer regions which is confirmed by numerical experiments.     

A posteriori discontinuous finite element error estimation for two-dimensional hyperbolic problems

We analyze the discontinuous finite element errors associated with p-degree solutions for two-dimensional first-order hyperbolic problems. We show that the error on each element can be split into a dominant and less dominant component and that the leading part is O(h^p+1) and is spanned by two (p+1)-degree Radau polynomials in the x and y directions, respectively. We show that the p-degree discontinuous finite element solution is superconvergent at Radau points obtained as a tensor product of the roots of (p+1)-degree Radau polynomial. For a linear model problem, the p-degree discontinuous Galerkin solution flux exhibits a strong O(h^2p+2) local superconvergence on average at the element outflow boundary. We further establish an O(h^2p+1) global superconvergence for the solution flux at the outflow boundary of the domain. These results are used to construct simple, efficient and asymptotically correct a posteriori finite element error estimates for multi-dimensional first-order hyperbolic problems in regions where solutions are smooth.     

A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems

We analyze the spatial discretization errors associated with solutions of one-dimensional hyperbolic conservation laws by discontinuous Galerkin methods (DGMs) in space. We show that the leading term of the spatial discretization error with piecewise polynomial approximations of degree p is proportional to a Radau polynomial of degree p+1 on each element. We also prove that the local and global discretization errors are O(Δx^2(p+1)) and O(Δx^2p+1) at the downwind point of each element. This strong superconvergence enables us to show that local and global discretization errors converge as O(Δx^p+2) at the remaining roots of Radau polynomial of degree p+1 on each element. Convergence of local and global discretization errors to the Radau polynomial of degree p+1 also holds for smooth solutions as p→∞. These results are used to construct asymptotically correct a posteriori estimates of spatial discretization errors that are effective for linear and nonlinear conservation laws in regions where solutions are smooth.     

Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers’ equation

In this paper we present rigorous a posteriori L ² error bounds for reduced basis approximations of the unsteady viscous Burgers’ equation in one space dimension. The a posteriori error estimator, derived from standard analysis of the error-residual equation, comprises two key ingredients—both of which admit efficient Offline-Online treatment: the first is a sum over timesteps of the square of the dual norm of the residual; the second is an accurate upper bound (computed by the Successive Constraint Method) for the exponential-in-time stability factor. These error bounds serve both Offline for construction of the reduced basis space by a new POD-Greedy procedure and Online for verification of fidelity. The a posteriori error bounds are practicable for final times (measured in convective units) T≈O(1) and Reynolds numbers ν ⁻¹≫1; we present numerical results for a (stationary) steepening front for T=2 and 1≤ν ⁻¹≤200. This work was supported by AFOSR Grants FA9550-05-1-0114 and FA-9550-07-1-0425 and the Singapore-MIT Alliance. We acknowledge many helpful discussions with Professor Yvon Maday of University of Paris VI and Dr. Paul Fischer of Argonne National Laboratory and University of Chicago.     

A feedback finite element method with a posteriori error estimation: Part I. The finite element method and some basic properties of the a posteriori error estimator

This paper is the first in a series of two in which we discuss some theoretical and practical aspects of a feedback finite element method for solving systems of linear second-order elliptic partial differential equations (with particular interest in classical linear elasticity). In this first part we introduce some nonstandard finite element spaces, which, though based on the usual square bilinear elements, permit local mesh refinement. The algebraic structure of these spaces and their approximation properties are analyzed. An “equivalent estimator” for the H¹ finite element error is developed. In the second paper we shall discuss the asymptotic properties of the estimator and computational experience.     

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.     

Application of a posteriori error estimation to finite element simulation of incompressible Navier-Stokes flow

The main goal of this paper is to study adaptive mesh techniques, using a posteriori error estimates, for the finite element solution of the Navier-Stokes equations modeling steady and unsteady flows of an incompressible viscous fluid. Among existing operator splitting techniques, the θ-scheme is used for time integration of the Navier-Stokes equations. Then, a posteriori error estimates, based on the solution of a local system for each triangular element, are presented in the framework of the generalized incompressible Stokes problem, followed by its practical application to the case of incompressible Navier-Stokes problem. Hierarchical mesh adaptive techniques are developed in response to the a posteriori error estimation. Numerical simulations of viscous flows associated with selected geometries are performed and discussed to demonstrate the accuracy and efficiency of our methodology.     

Application of a posteriori error estimation to finite element simulation of compressible Navier-Stokes flow

In this paper, we discuss how to improve the adaptive finite element simulation of compressible Navier-Stokes flow via a posteriori error estimate analysis. We use the moving space-time finite element method to globally discretize the time-dependent Navier-Stokes equations on a series of adapted meshes. The generalized compressible Stokes problem, which is the Stokes problem in its most generalized form, is presented and discussed. On the basis of the a posteriori error estimator for the generalized compressible Stokes problem, a numerical framework of a posteriori error estimation is established corresponding to the case of compressible Navier-Stokes equations. Guided by the a posteriori errors estimation, a combination of different mesh adaptive schemes involving simultaneous refinement/unrefinement and point-moving are applied to control the finite element mesh quality. Finally, a series of numerical experiments will be performed involving the compressible Stokes and Navier-Stokes flows around different aerodynamic shapes to prove the validity of our mesh adaptive algorithms.     

A posteriori error estimation by postprocessor independent of method of flowfield calculation

We consider a postprocessor that is able to analyze the flow-field generated by an external (unknown) code so as to determine the error of useful functionals. The residuals engendered by the action of a high-order finite-difference stencil on a numerically computed flow-field are used for adjoint based a posteriori error estimation. The method requires information on the physical model (PDE system), flowfield parameters and corresponding grid and may be constructed without availability of detailed information on the numerical method used for the flow computation.     

A variational multiscale a posteriori error estimation method for mixed form of nearly incompressible elasticity

This paper presents an error estimation framework for a mixed displacement–pressure finite element method for nearly incompressible elasticity. The proposed method is based on Variational Multiscale (VMS) concepts, wherein the displacement field is decomposed into coarse scales that can be resolved by a given finite element mesh and fine scales that are beyond the resolution capacity of the mesh. Variational projection of fine scales onto the coarse-scale space via variational embedding of the fine-scale solution into the coarse-scale formulation leads to the stabilized method with two major attributes: first, it is free of volumetric locking and, second, it accommodates arbitrary combinations of interpolation functions for the displacement and pressure fields. This VMS-based stabilized method is equipped with naturally derived error estimators and offers various options for numerical computation of the error. Specifically, two error estimators are explored. The first method employs an element-based strategy and a representation of error via a fine-scale error equation defined over element interiors which is evaluated by a direct post-solution evaluation. This quantity when combined with the global pollution error results in a simple explicit error estimator. The second method involves solving the fine-scale error equation through localization to overlapping patches spread across the domain, thereby leading to an implicit calculation of the local error. This implicit calculation when combined with the global pollution error results in an implicit error estimator. The performance of the stabilized method and the error estimators is investigated through numerical convergence tests conducted for two model problems on uniform and distorted meshes. The sharpness and robustness of the estimators is shown to be consistent across the test cases employed.     

Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the Maxwell operator

In this paper we develop the a posteriori error estimation of mixed discontinuous Galerkin finite element approximations of the Maxwell operator. In particular, by employing suitable Helmholtz decompositions of the error, together with the conservation properties of the underlying method, computable upper bounds on the error, measured in terms of a natural (mesh-dependent) energy norm, are derived. Numerical experiments testing the performance of our a posteriori error bounds for problems with both smooth and singular analytical solutions are presented.     

A posteriori error estimation for the heterogeneous Maxwell equations on isotropic and anisotropic meshes

We consider residual-based a posteriori error estimators for the heterogeneous Maxwell equations using isotropic as well as anisotropic meshes. The continuous problem is approximated by using conforming approximated spaces with minimal assumptions. Lower and upper bounds are obtained under standard assumptions on the meshes. The lower bound holds unconditionally, while the upper bound depends on alignment properties of the meshes with respect to the solution. In particular for isotropic meshes the upper bound also holds unconditionally. A numerical test is presented which confirms our theoretical results.     

Robust topology optimization using a posteriori error estimator for the finite element method

In our work, we consider the classical density-based approach to the topology optimization. We propose to modify the discretized cost functional using a posteriori error estimator for the finite element method. It can be regarded as a new technique to prevent checkerboards. It also provides higher regularity of solutions and robustness of results.     

On estimation error bounds for receding-horizon filters using quadratic boundedness

Quadratic boundedness is used to deal with stability and design of receding-horizon estimators. Upper bounds on the norm of the estimation error have been found by means of invariant sets that can be constructed by using quadratic boundedness. Moreover, these bounds are expressed in terms of linear matrix inequalities and are well-suited to being minimized for the purpose of design.