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.     

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.     

A priori and a posteriori error analysis of an augmented mixed finite element method for incompressible fluid flows

In this paper we extend recent results on the a priori and a posteriori error analysis of an augmented mixed finite element method for the linear elasticity problem, to the case of incompressible fluid flows with symmetric stress tensor. Similarly as before, the present approach is based on the introduction of the Galerkin least-squares type terms arising from the constitutive and equilibrium equations, and from the relations defining the pressure in terms of the stress tensor and the rotation in terms of the displacement, all of them multiplied by stabilization parameters. We show that these parameters can be suitably chosen so that the resulting augmented variational formulation is defined by a strongly coercive bilinear form, whence the associated Galerkin scheme becomes well-posed for any choice of finite element subspaces. Next, we present a reliable and efficient residual-based a posteriori error estimator for the augmented mixed finite element scheme. Finally, several numerical results confirming the theoretical properties of this estimator, and illustrating the capability of the corresponding adaptive algorithm to localize the singularities and the large stress regions of the solution, are reported.     

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 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.     

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.     

Viscous incompressible flow by a penalty function finite element method

A quantitative evaluation for the penalty function finite element method for two-dimensional viscous incompressible flow using primitive variables is made, using bilinear and biquadratic elements. We conclude that this procedure, more efficient than full velocity pressure formulation, can achieve excellent accuracy using rather coarse meshes, that biquadratic elements produce improved accuracy over the more commonly used bilinears, and that the penalty method effectively imposes the continuity constraint. We point out the dangers of modelling problems containing singularities, as is the case of the driven cavity flow, with finite element formulations using primitive variables and how to overcome these problems. The driven cavity flow for Reynolds number up to 400 is solved and the answers compared with the best available solutions, the Jeffery-Hamel flow in a convergent control channel is solved for Reynolds number up to 61 and the results compared with the analytical solution. Finally extensions of the method are indicated via an example of natural convection in a square cavity for Rayleigh number up to 10⁶, and the advantage of the method discussed.     

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.     

二维不可压缩Navier-Stokes方程的并行谱有限元法求解

针对不可压缩Navier-Stokes （N-S）方程求解过程中的有限元法存在计算网格量大、收敛速度慢的缺点,提出了基于面积坐标的三角网格剖分谱有限元法（TSFEM）并进一步给出了利用OpenMP对其并行化的方法。该算法结合谱方法和有限元法思想,选取具有无限光滑特性的指数函数取代传统有限元法中的多项式函数作为基函数,能够有效减少计算网格数量,提高算法的精度和收敛速度;利用面积坐标便于三角形单元计算的特点,选取三角单元作为计算单元,增强了适用性;在顶盖方腔驱动流问题上对该算法进行验证。实验结果表明,TSFEM较传统有限元法（FEM）无论是收敛速度还是计算效率都有了显著提高。     

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 anisotropic pressure-stabilized finite element method for incompressible flow problems

We consider a pressure-stabilized, finite element approximation of incompressible flow problems in primitive velocity–pressure variables, which is based on a projection of the gradient of the discrete pressure onto the space of discrete functions. Equal order interpolation for the velocity and the pressure can be employed with this formulation. The method introduced here is specially developed to be used on anisotropic finite element meshes with large element aspect ratios.     

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.