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 Robust Adaptive Grid Method for a System of Two Singularly Perturbed Convection-Diffusion Equations with Weak Coupling

A system of singularly perturbed convection-diffusion equations with weak coupling is considered. The system is first discretized by an upwind finite difference scheme for which an a posteriori error estimate in the maximum norm is constructed. Then the a posteriori error bound is used to design an adaptive gird algorithm. Finally, a first-order rate of convergence, independent of the perturbation parameters, is established by using the theory of the discrete Green’s function. Numerical results are presented to illustrate support our theoretical results.     

Adaptive finite element heterogeneous multiscale method for homogenization problems

In this paper we present an a posteriori error analysis for elliptic homogenization problems discretized by the finite element heterogeneous multiscale method. Unlike standard finite element methods, our discretization scheme relies on macro- and microfinite elements. The desired macroscopic solution is obtained by a suitable averaging procedure based on microscopic data. As the macroscopic data (such as the macroscopic diffusion tensor) are not available beforehand, appropriate error indicators have to be defined for designing adaptive methods. We show that such indicators based only on the available macro- and microsolutions (used to compute the actual macrosolution) can be defined, allowing for a macroscopic mesh refinement strategy which is both reliable and efficient. The corresponding a posteriori estimates for the upper and lower bound are derived in the energy norm. In the case of a uniformly oscillating tensor, we recover the standard residual-based a posteriori error estimate for the finite element method applied to the homogenized problem. Numerical experiments confirm the efficiency and reliability of the adaptive multiscale method.     

An Adaptive Finite Element Method for the Wave Scattering with Transparent Boundary Condition

Consider the acoustic wave scattering by an impenetrable obstacle in two dimensions. The model is formulated as a boundary value problem for the Helmholtz equation with a transparent boundary condition. Based on a duality argument technique, an a posteriori error estimate is derived for the finite element method with the truncated Dirichlet-to-Neumann boundary operator. The a posteriori error estimate consists of the finite element approximation error and the truncation error of boundary operator which decays exponentially with respect to the truncation parameter. A new adaptive finite element algorithm is proposed for solving the acoustic obstacle scattering problem, where the truncation parameter is determined through the truncation error and the mesh elements for local refinements are marked through the finite element discretization error. Numerical experiments are presented to illustrate the competitive behavior of the proposed adaptive method.     

A posteriori error estimates for discontinuous Galerkin approximation of non-stationary convection-diffusion optimal control problems

In this paper, we investigate a discontinuous Galerkin finite element approximation of non-stationary convection dominated diffusion optimal control problems with control constraints. The state variable is approximated by piecewise linear polynomial space and the control variable is discretized by variational discretization concept. Backward Euler method is used for time discretization. With the help of elliptic reconstruction technique residual type a posteriori error estimates are derived for state variable and adjoint state variable, which can be used to guide the mesh refinement in the adaptive algorithm. Numerical experiment is presented, which indicates the good behaviour of the a posteriori error estimators.     

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.     

A robust layer adapted difference method for singularly perturbed two-parameter parabolic problems

A finite difference method for a time-dependent singularly perturbed convection–diffusion–reaction problem involving two small parameters in one space dimension is considered. We use the classical implicit Euler method for time discretization and upwind scheme on the Shishkin–Bakhvalov mesh for spatial discretization. The method is analysed for convergence and is shown to be uniform with respect to both the perturbation parameters. The use of the Shishkin–Bakhvalov mesh gives first-order convergence unlike the Shishkin mesh where convergence is deteriorated due to the presence of a logarithmic factor. Numerical results are presented to validate the theoretical estimates obtained.     

Coupling of adaptively refined dual mixed finite elements and boundary elements in linear elasticity

We investigate a coupling of mixed finite elements and Galerkin boundary elements which is stable and leads to symmetric matrices. In the FEM domain, a posteriori error estimates are employed to refine the mesh adaptively. Numerical results are given for plane strain problems.     

Some a Posteriori Error Estimators for p-Laplacian Based on Residual Estimation or Gradient Recovery

In this paper, we first derive a posteriori error estimators of residual type for the finite element approximation of the p-Laplacian, and show that they are reliable, and efficient up to higher order terms. We then construct some a posteriori error estimators based on gradient recovery. We further compare the two types of a posteriori error estimators. It is found that there exist some relationships between the two types of estimators, which are similar to those held in the case of the Laplacian. It is shown that the a posteriori error estimators based on gradient recovery are equivalent to the discretization error in a quasi-norm provided the solution is sufficiently smooth and mesh is uniform. Under stronger conditions, superconvergnece properties have been established for the used gradient recovery operator, and then some of the gradient recovery based estimates are further shown to be asymptotically exact to the discretization error in a quasi-norm. Numerical results demonstrating these a posteriori estimates are also presented.     

发展型对流占优扩散方程的FD-SD法的后验误差估计及空间网格调节技术

０．引言 流线扩散法（ｓｔｒｅａｍｌｉｎｅ　ｄｉｆｆｕｓｉｏｎ ｍｅｔｈｏｄ,简称 ＳＤ法）是由Ｈｕｇｈｅｓ和 Ｂｒｏｏｋｓ在１９８０年前后提出的一种数值求解对流占优扩散问题的新型有限元算法．随后,Ｊｏｈｎｓｏｎ和 Ｎａｖｅｒｔ把ＳＤ法推广到发展型对流扩散问题．这一方法因其兼具良好的数值稳定性和高阶精度,近年来在理论与实践方面都得到了很大发展． 对于发展型对流扩散问题的ＳＤ法均采用时空有限元,即把时间、空间同等对待,这样做虽然使关于时间、空间的精度很好地统一起来,但与传统的有限元相比,由于维数增加,计…     

Guaranteed and Fully Robust a posteriori Error Estimates for Conforming Discretizations of Diffusion Problems with Discontinuous Coefficients

We study in this paper a posteriori error estimates for H ¹-conforming numerical approximations of diffusion problems with a diffusion coefficient piecewise constant on the mesh cells but arbitrarily discontinuous across the interfaces between the cells. Our estimates give a global upper bound on the error measured either as the energy norm of the difference between the exact and approximate solutions, or as a dual norm of the residual. They are guaranteed, meaning that they feature no undetermined constants. (Local) lower bounds for the error are also derived. Herein, only generic constants independent of the diffusion coefficient appear, whence our estimates are fully robust with respect to the jumps in the diffusion coefficient. In particular, no condition on the diffusion coefficient like its monotonous increasing along paths around mesh vertices is imposed, whence the present results also include the cases with singular solutions. For the energy error setting, the key requirement turns out to be that the diffusion coefficient is piecewise constant on dual cells associated with the vertices of an original simplicial mesh and that harmonic averaging is used in the scheme. This is the usual case, e.g., for the cell-centered finite volume method, included in our analysis as well as the vertex-centered finite volume, finite difference, and continuous piecewise affine finite element ones. For the dual norm setting, no such a requirement is necessary. Our estimates are based on H(div)-conforming flux reconstruction obtained thanks to the local conservativity of all the studied methods on the dual grids, which we recall in the paper; mutual relations between the different methods are also recalled. Numerical experiments are presented in confirmation of the guaranteed upper bound, full robustness, and excellent efficiency of the derived estimators.     

A Posteriori Analysis of a Positive Streamwise Invariant Discretization of a Convection-Diffusion Equation

We consider the finite element discretization of a convection-diffusion equation, where the convection term is handled via a fluctuation splitting algorithm. We prove a posteriori error estimates which allow us to perform mesh adaptivity in order to optimize the discretization of these equations. Numerical results confirm the interest of such an approach.     

Anisotropic metrics for finite element meshes using a posteriori error estimates: Poisson and Stokes equations

In this paper, metrics derived from a posteriori error estimates for the Poisson problem and for the Stokes system solved by some finite element methods are presented. Numerical examples of mesh adaptation in two dimensions of the space are given and show that these metrics detect the singular behavior of the solution, in particular its anisotropy.     

A RT Mixed FEM/DG Scheme for Optimal Control Governed by Convection Diffusion Equations

In this paper, we provide a numerical scheme—RT mixed FEM/DG scheme for the constrained optimal control problem governed by convection dominated diffusion equations. A priori and a posteriori error estimates are obtained for both the state, the co-state and the control. The adaptive mesh refinement can be applied indicated by a posteriori error estimator provided in this paper. Numerical examples are presented to illustrate the theoretical analysis.     

A posteriori error estimates for fem–bem couplings of three-dimensional electromagnetic problems

We present residual based and p-hierarchical a posteriori error estimators for a Galerkin method coupling finite elements and boundary elements for time–harmonic interface problems in electromagnetics; special emphasis is taken for the eddy current problem. The Galerkin discretization uses lowest order Nédélec elements in the interior domain and vectorial surface rotations of continuous, piecewise linear functions on the interface boundary. Singular, weakly singular and hypersingular boundary integral operators appearing in the variational formulation show up in the terms of the error estimators as well. The estimators are derived from the defect equation using Helmholtz and Hodge decompositions. Numerical tests underline reliability and efficiency of the given error estimators yielding reasonable mesh refinements.     

An adaptive finite element scheme to solve fluid-structure vibration problems on non-matching grids

This paper deals with the computation of the vibration modes of a system consisting of a linear elastic solid interacting with an acoustic fluid. A finite element method based on meshes for each medium not matching on the fluid-solid interface is analyzed. Optimal order of convergence is proved for the approximation of the eigenfunctions, as well as a double order for the eigenvalues. Numerical tests confirming the theoretical results and showing the advantage of using non-matching grids are reported. Finally, an a posteriori error estimator for this method is introduced and combined with a mesh refinement strategy. The efficiency of this adaptive technique is tested with further numerical experiments. Received: 30 January 2001 / Accepted: 30 May 2001     

Adaptive Finite Element Approximation for an Elliptic Optimal Control Problem with Both Pointwise and Integral Control Constraints

In this paper, we study the adaptive finite element approximation for a constrained optimal control problem with both pointwise and integral control constraints. We first obtain the explicit solutions for the variational inequalities both in the continuous and discrete cases. Then a priori error estimates are established, and furthermore equivalent a posteriori error estimators are derived for both the state and the control approximation, which can be used to guide the mesh refinement for an adaptive multi-mesh finite element scheme. The a posteriori error estimators are implemented and tested with promising numerical results.     

Error control and mesh optimization for high-order finite element approximation of incompressible viscous flow

We discuss the use of a posteriori error estimates for high-order finite element methods during simulation of the flow of incompressible viscous fluids. The correlation between the error estimator and actual error is used as a criterion for the error analysis efficiency. We show how to use the error estimator for mesh optimization which improves computational efficiency for both steady-state and unsteady flows. The method is applied to two-dimensional problems with known analytical solutions (Jeffrey-Hamel flow) and more complex flows around a body, both in a channel and in an open domain.     

Adaptive mixed finite element method for Reissner–Mindlin plate

Mixed finite element methods are designed to overcome shear locking phenomena observed in the numerical treatment of Reissner–Mindlin plate models. Automatic adaptive mesh-refining algorithms are an important tool to improve the approximation behavior of the finite element discretization. In this paper, a reliable and robust residual-based a posteriori error estimate is derived, which evaluates a t-depending residual norm based on results in [D. Arnold, R. Falk, R. Winther, Math. Modell. Numer. Anal. 31 (1997) 517–557]. The localized error indicators suggest an adaptive algorithm for automatic mesh refinement. Numerical examples prove that the new scheme is efficient.     

涡流检测系统仿真分析的自适应算法

该文针对电磁场有限元计算的特点,深入研究了涡流检测系统中电磁场有限元后验误差估计的误差模选择问题,并在分析Zienkiewicz-Zhu方法在电磁场有限元后验误差估计应用中存在局限性的基础上,提出了一种适合于涡流检测系统中电有限元分析的后验误差估计新方法。在此基础上,结合James R.Stewart和Thomas J.R.Hughes所提出的简单实用的有限元算法,提出了一种适合于涡流检测系统中电磁场有限元分析的hp 自适应新算法。