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.     

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 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 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.     

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.     

Collocation finite element solution of a compressible flow

In this paper, the subcritical compressible small disturbance equation is employed to simulate the nonlifting flow over a 6% thick circular arc. The governing nonlinear elliptic boundary value problem is solved numerically with the collocation finite element technique using the Hermite bicubic shape functions. The lack of quadrature calculations makes the overall simulation fast and accurate. The calculation results are compared with experimental measurements supporting their validity.     

A posteriori estimation of the error in the recovered derivatives of the finite element solution

This work addresses the accuracy of the solution derivatives which are recovered by local averaging of the finite element solution. The main results of the study are: (1) The error in the locally averaged derivatives (e.g. the derivatives which are recovered by the Zienkiewicz-Zhu Superconvergent Patch Recovery (ZZ-SPR) or other similar local recoveries) can be more than the error in the derivatives computed directly from the finite element solution, especially in the case of unsmooth solutions and/or coarse meshes. (2) In order to determine which solution derivatives should be relied upon, the locally averaged ones or the ones computed directly from the finite element solution, one must be able to estimate their errors. It is shown that one can obtain indicators of the error in the derivatives recovered by the ZZ-SPR by employing an additional local averaging of the recovered derivatives (recycling of the ZZ-SPR) or by comparing the derivatives computed by the ZZ-SPR with the derivatives obtained using a different local averaging which takes into account the character of the exact solution (harmonic averaging).     

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.     

A posteriori error estimates¶for nonconforming finite element schemes

We derive a posteriori error estimates for nonconforming discretizations of Poisson's and Stokes' equations. The estimates are residual based and make use of weight factors obtained by a duality argument. Crouzeix-Raviart elements on triangles and rotated bilinear elements are considered. The quadrilateral case involves the introduction of additional local trial functions. We show that their influence is of higher order and that they can be neglected. The validity of the estimate is demonstrated by computations for the Laplacian and for Stokes' equations. Received: November 1998 / Accepted: January 1999     

A review of some a posteriori error estimates for adaptive finite element methods

Recently, the adaptive finite element methods have gained a very important position among numerical procedures for solving ordinary as well as partial differential equations arising from various technical applications. While the classical a posteriori error estimates are oriented to the use in h-methods the contemporary higher order hp-methods usually require new approaches in a posteriori error estimation.     

Strict error estimation for a spectral method of compressible fluid flow

In this paper, a Fourier Spectral method for compressible flow in n-dimensional space with periodic boundary conditions is constructed. We give a strict error estimation, from which the convergence follows with some assumptions.     

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.     

A posteriori error estimates for finite element discretizations of the heat equation

We consider discretizations of the heat equation by A-stable -schemes in time and conforming finite elements in space. For these discretizations we derive residual a posteriori error indicators. The indicators yield upper bounds on the error which are global in space and time and yield lower bounds that are global in space and local in time. The ratio between upper and lower bounds is uniformly bounded in time and does not depend on any step-size in space or time. Moreover, there is no restriction on the relation between the step-sizes in space and time.     

A priori and a posteriori error analysis for a large-scale ocean circulation finite element model

We consider the finite element solution of the stream function–vorticity formulation for a large-scale ocean circulation model. First, we study existence and uniqueness of solution for the continuous and discrete problems. Under appropriate regularity assumptions we prove that the stream function can be computed with an error of order h in H¹-seminorm. Second, we introduce and analyze an h-adaptive mesh refinement strategy to reduce the spurious oscillations and poor resolution which arise when convective terms are dominant. We propose an a posteriori anisotropic error indicator based on the recovery of the Hessian from the finite element solution, which allows us to obtain well adapted meshes. The numerical experiments show an optimal order of convergence of the adaptive scheme. Furthermore, this strategy is efficient to eliminate the oscillations around the boundary layer.     

Reliable and efficient a posteriori error estimates for finite element approximations of the parabolic p-Laplacian

We generalize the a posteriori techniques for the linear heat equation in Verfürth (Calcolo 40(3):195–212, 2003) to the case of the nonlinear parabolic $p$ -Laplace problem thereby proving reliable and efficient a posteriori error estimates for a fully discrete implicite Euler Galerkin finite element scheme. The error is analyzed using the so-called quasi-norm and a related dual error expression. This leads to equivalence of the error and the residual, which is the key property for proving the error bounds.     

An error estimation method in finite element structural analysis

In the paper a new interpretation of the finite element approach is described and illustrated numerically. The conventional shape functions of the displacement—and stress—type finite element models are treated as constraints imposed on the continuous medium considered. This enables a consistent error estimation analysis based on a concept of so-called reaction forces and deformation incompatibilities.     

An adaptive mesh refinement of quadrilateral finite element meshes based upon a posteriori error estimation of quantities of interest: linear static response

A posteriori error estimation in finite element analysis serves as an important guide to the meshing tool in an adaptive refinement process. However, the traditional posteriori error estimates, which are often defined in the energy or energy-type norms over the entire domain, provide users insufficient information regarding the accuracy of specific quantities in the solution. This paper describes an adaptive quadrilateral refinement process with a goal-oriented error estimation, in which a posteriori error is estimated with respect to the specified quantity of interest. A highlight of this paper is the demonstration of tools described in the paper used in a practical industrial environment. The performance of this process is demonstrated on several practical problems where the comparison is with the adaptive process based on the traditional error estimation.     

An adaptive mesh refinement of quadrilateral finite element meshes based upon an a posteriori error estimation of quantities of interest: modal response

Modal analysis is commonly performed in a vehicle development process to assess dynamic responses of structure designs. This paper presents an adaptive quadrilateral refinement process for modal analysis of elastic shells based upon a posteriori error estimation in natural frequencies. The process provides engineers with an estimation of their modal analysis quality and an effective adaptive refinement tool for quadrilateral meshes. The effectiveness of the process is demonstrated on the eigenvalue analyses of two numerical examples, a shock tower cap and a roof structure. It shows that the solution error in the frequency of interest is effectively reduced through the adaptive refinement process, and the resulting frequency of interest converges to the solution of a very fine model.     

A posteriori error analysis of space-time finite element discretizations of the time-dependent Stokes equations

We present a novel a posteriori error analysis of space-time finite element discretizations of the time-dependent Stokes equations. Our analysis is based on the equivalence of error and residual and a suitable decomposition of the residual into spatial and temporal contributions. In contrast to existing results we directly bound the error of the full space-time discretization and do not resort to auxiliary semi-discretizations. We thus obtain sharper bounds. Moreover the present analysis covers a wider range of discretizations both with respect to time and to space.