A new a posteriori error estimator in adaptive direct boundary element methods: the Dirichlet problem

In this paper we propose a new a posteriori error estimator for a boundary element solution related to a Dirichlet problem with a second order elliptic partial differential operator. The method is based on an approximate solution of a boundary integral equation of the second kind by a Neumann series to estimate the error of a previously computed boundary element solution. For this one may use an arbitrary boundary element method, for example, a Galerkin, collocation or qualocation scheme, to solve an appropriate boundary integral equation. Due to the approximate solution of the error equation the proposed estimator provides high accuracy. A numerical example supports the theoretical results. Received: June 1999 / Accepted: September 1999     

A posteriori error estimate for the symmetric coupling of finite elements and boundary elements

In this note we study a posteriori error estimates for a model problem in the symmetric coupling of boundary element and finite elements methods. Emphasis is on the use of the Poincaré-Steklov operator and its discretization which are analyzed in general for both a priori and a posteriori error estimates. Combining arguments from [6] and [9, 10] we refine the a posteriori error estimate obtained in [9, 10]. For quasi-uniform meshes on the boundary, we prove some inequality of a reverse type using techniques from [5] and [36]. This indicates efficiency of the new estimate as illustrated in a numerical example.     

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.     

Error Estimates of the Finite Element Method for Interior Transmission Problems

The interior transmission problem (ITP) plays an important role in the investigation of the inverse scattering problem. In this paper we propose the finite element method for solving the ITP. Based on the $\mathbb T $ -coercivity, we derive both priori error estimate and a posteriori error estimate of the finite element approximation. Numerical experiments are also included to illustrate the accuracy of the finite element method.     

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.     

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…     

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 FE/BE coupling for the 3D time-dependent eddy current problem. Part II: a posteriori error estimates and adaptive computations

The discontinuous Galerkin method in time for the coupling of conforming finite element and boundary element methods was established in Part I of this paper, where quasi-optimal a priori error estimates are provided. In the second part, we establish a posteriori error estimates and so justify an adaptive space/time-mesh refinement algorithm for the efficient numerical treatment of the time-dependent eddy current problem.     

An a posteriori error estimate for a linear-nonlinear transmission problem in plane elastostatics

We consider the coupling of dual-mixed finite element and boundary element methods to solve a linear-nonlinear transmission problem in plane hyperelasticity with mixed boundary conditions. Besides the displacement and the stress tensor, we introduce the strain tensor as an additional unknown, which yields a two-fold saddle point operator equation as the corresponding variational formulation. We derive a reliable a posteriori error estimate that depends on the solution of local Dirichlet problems and on residual terms on the transmission and Neumann boundaries, which are given in a negative order Sobolev norm. Our approach does not need the exact Galerkin solution, but any reasonable approximation of it. In addition, the analysis does not depend on special finite element or boundary element subspaces. However, for certain specific subspaces we are able to provide two fully local a posteriori error estimates, in which the residual terms are bounded by weighted local L ²-norms. Further, one of the error estimates does not require the explicit solution of the local problems. Received: November 2000 / Revised version: December 2001 This research was partially supported by Fondecyt-Chile through research projects 1980122 and 2000124, and the FONDAP Program in Applied Mathematics, and by the Dirección de Investigación of the Universidad de Concepción through the Advanced Research Groups Program.     

A posteriori error estimates for a mixed-FEM formulation of a non-linear elliptic problem

We consider the numerical solution, via the mixed finite element method, of a non-linear elliptic partial differential equation in divergence form with Dirichlet boundary conditions. Besides the temperature u and the flux $\text{σ}$, we introduce ∇u as a further unknown, which yields a variational formulation with a twofold saddle point structure. We derive a reliable a posteriori error estimate that depends on the solution of a local linear boundary value problem, which does not need any equilibrium property for its solvability. In addition, for specific finite element subspaces of Raviart–Thomas type we are able to provide a fully explicit a posteriori error estimate that does not require the solution of the local problems. Our approach does not need the exact finite element solution, but any reasonable approximation of it, such as, for instance, the one obtained with a fully discrete Galerkin scheme. In particular, we suggest a scheme that uses quadrature formulas to evaluate all the linear and semi-linear forms involved. Finally, several numerical results illustrate the suitability of the explicit error estimator for the adaptive computation of the corresponding discrete solutions.     

Finite element approximations of nonlinear eigenvalue problems in quantum physics

In this paper, we study finite element approximations of a class of nonlinear eigenvalue problems arising from quantum physics. We derive both a priori and a posteriori finite element error estimates and obtain optimal convergence rates for both linear and quadratic finite element approximations. In particular, we analyze the convergence and complexity of an adaptive finite element method. In our analysis, we utilize certain relationship between the finite element eigenvalue problem and the associated finite element boundary value approximations. We also present several numerical examples in quantum physics that support our theory.     

A FE/BE coupling for the 3D time-dependent eddy current problem. Part I: a priori error estimates

In this paper the discontinuous Galerkin method in time for the coupling of conforming finite element and boundary element methods is established. We derive quasi-optimal a priori error estimates. Numerical examples prove the new scheme to be useful in practice. A posteriori error control and an adaptive algorithm are studied in Part II of this paper.     

Adaptive methods for frictionless contact problems

In this paper a posteriori error indicators for frictionless contact problems are presented. In detail, error indicators relying on superconvergence properties and error estimators based on duality principles are investigated. Applications are to 3D solids under the hypothesis of non-linear elastic material behaviour associated with finite deformations. A penalization technique is applied to enforce multilateral boundary conditions due to contact. The approximate solution of the problem is obtained by using the finite element method. Several numerical results are reported to show the applicability of the adaptive algorithm to the considered problems.     

Adaptive error control for multigrid finite element

We consider the problem of adaptive error control in the finite element method including the error resulting from, inexact solution of the discrete equations. We prove a posteriori error estimates for a prototype elliptic model problem discretized by the finite element with a canomical multigrid algorithm. The proofs are based on a combination of so-called strong stability and, the orthogonality inherent in both the finite element method can the multigrid algorithm.     

Adaptive Local Postprocessing Finite Element Method for the Navier-Stokes Equations

An adaptive local postprocessing finite element method for the Navier-Stokes equations is presented in this paper. We firstly solve the problem on a relative coarse grid to get a rough approximation. Then, we correct the rough approximation by solving a series of approximate local residual equations defined on some local fine grids, which can be implemented in parallel. In addition, we also propose a reliable local a posteriori error estimator and construct an adaptive algorithm based on the corresponding a posterior error estimate. Finally, some numerical examples are presented to verify the algorithm.     

Adaptive Finite Element Approximation for a Constrained Optimal Control Problem via Multi-meshes

In this paper, we study adaptive finite element approximation schemes for a constrained optimal control problem. We derive the equivalent a posteriori error estimators for both the state and the control approximation, which particularly suit an adaptive multi-mesh finite element scheme. The error estimators are then implemented and tested with promising numerical results.     

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。     

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.     

On the adaptive finite element method of steady-state rolling contact for hyperelasticity in finite deformations

In this paper, we present an adaptive finite element method for steady-state rolling contact in finite deformations along with a residual based a posteriori error estimator for rolling contact problem with Coulomb friction. A general formulation of rolling contact geometry is derived from the point of view of differential geometry. Solvability conditions for the rolling contact problems are discussed. We use Newton's method to solve variational equations derived from a penalty regularization of the finite element approximation of the rolling contact problem. We provide a numerical example to illustrate the method.     

Adaptive modeling of turbulent flow with residual based turbulent kinetic energy dissipation

In this paper we first review our recent work on a new framework for adaptive turbulence simulation: we model turbulence by weak solutions to the Navier–Stokes equations that are wellposed with respect to mean value output in the form of functionals, and we use an adaptive finite element method to compute approximations with a posteriori error control based on the error in the functional output. We then derive a local energy estimate for a particular finite element method, which we connect to related work on dissipative weak Euler solutions with kinetic energy dissipation due to lack of local smoothness of the weak solutions. The ideas are illustrated by numerical results, where we observe a law of finite dissipation with respect to a decreasing mesh size.