A posteriori error analysis for a boundary element method with nonconforming domain decomposition

We present and analyze an a posteriori error estimator based on mesh refinement for the solution of the hypersingular boundary integral equation governing the Laplacian in three dimensions. The discretization under consideration is a nonconforming domain decomposition method based on the Nitsche technique. Assuming a saturation property, we establish quasireliability and efficiency of the error estimator in comparison with the error in a natural (nonconforming) norm. Numerical experiments with uniform and adaptively refined meshes confirm our theoretical results. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 947–963, 2014     

A posteriori error estimation for elliptic eigenproblems

An a posteriori error estimator is presented for a subspace implementation of preconditioned inverse iteration, which derives from the well‐known inverse iteration in such a way that the associated system of linear equations is solved approximately by using a preconditioner. The error estimator is integrated in an adaptive multigrid algorithm to compute approximations of a modest number of the smallest eigenvalues together with the eigenfunctions of an elliptic differential operator. Error estimation is applied both within the actual finite element space (in order to estimate the iteration error) as well as in its hierarchical refinement of higher‐order elements (to estimate the discretization error) which gives rise to a balanced reduction of the iteration error and of the discretization error in the adaptive multigrid algorithm. Copyright © 2002 John Wiley & Sons, Ltd.     

On a hierarchical error estimator combined with a stabilized method for the Navier–Stokes equations

This work combines two complementary strategies for solving the steady incompressible Navier–Stokes model with a zeroth‐order term, namely, a stabilized finite element method and a mesh–refinement approach based on an error estimator. First, equal order interpolation spaces are adopted to approximate both the velocity and the pressure while stability is recovered within the stabilization approach. Also designed to handle advection dominated flows under zeroth‐order term influence, the stabilized method incorporates a new parameter with a threefold asymptotic behavior. Mesh adaptivity driven by a new hierarchical error estimator and built on the stabilized method is the second ingredient. The estimator construction process circumvents the saturation assumption by using an enhancing space strategy which is shown to be equivalent to the error. Several numerical tests validate the methodology. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Adaptive numerical solution of thick plates using first‐order shear deformation theory. Part I: Error estimates

A posteriori error estimation employing both a residual based estimator and a recovery based estimator is discussed. Interest is focused upon the application to Reissner‐Mindlin type thick plates modeled using first‐order shear deformation theory, and our investigation is limited to uniform meshes of bilinear quadrilateral elements. Numerical results for selected test problems are presented for the resulting error estimators and discussed. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 44–66, 2003     

An adaptive edge element method and its convergence for a Saddle‐Point problem from magnetostatics

Reliable and efficient a posteriori error estimates are derived for the edge element discretization of a saddle‐point Maxwell's system. By means of the error estimates, an adaptive edge element method is proposed and its convergence is rigorously demonstrated. The algorithm uses a marking strategy based only on the error indicators, without the commonly used information on local oscillations and the refinement to meet the standard interior node property. Some new ingredients in the analysis include a novel quasi‐orthogonality and a new inf‐sup inequality associated with an appropriately chosen norm. It is shown that the algorithm is a contraction for the sum of the energy error plus the error indicators after each refinement step. Numerical experiments are presented to show the robustness and effectiveness of the proposed adaptive algorithm. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012     

Convergence of adaptive 3D BEM for weakly singular integral equations based on isotropic mesh‐refinement

We consider the adaptive lowest‐order boundary element method based on isotropic mesh refinement for the weakly‐singular integral equation for the three‐dimensional Laplacian. The proposed scheme resolves both, possible singularities of the solution as well as of the given data. The implementation thus only deals with discrete integral operators, that is, matrices. We prove that the usual adaptive mesh‐refining algorithm drives the corresponding error estimator to zero. Under an appropriate saturation assumption which is observed empirically, the sequence of discrete solutions thus tends to the exact solution within the energy norm. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A posteriori error estimates for mixed FEM in elasticity

A residue based reliable and efficient error estimator is established for finite element solutions of mixed boundary value problems in linear, planar elasticity. The proof of the reliability of the estimator is based on Helmholtz type decompositions of the error in the stress variable and a duality argument for the error in the displacements. The efficiency follows from inverse estimates. The constants in both estimates are independent of the Lamé constant , and so locking phenomena for are properly indicated. The analysis justifies a new adaptive algorithm for automatic mesh–refinement. Received July 17, 1997     

On the subdomain‐Galerkin/least squares method for 2‐ and 3‐D mixed elliptic problems with reaction terms

In this article we apply the subdomain‐Galerkin/least squares method, which is first proposed by Chang and Gunzburger for first‐order elliptic systems without reaction terms in the plane, to solve second‐order non‐selfadjoint elliptic problems in two‐ and three‐dimensional bounded domains with triangular or tetrahedral regular triangulations. This method can be viewed as a combination of a direct cell vertex finite volume discretization step and an algebraic least‐squares minimization step in which the pressure is approximated by piecewise linear elements and the flux by the lowest order Raviart‐Thomas space. This combined approach has the advantages of both finite volume and least‐squares methods. Among other things, the combined method is not subject to the Ladyzhenskaya‐Babus?ka‐Brezzi condition, and the resulting linear system is symmetric and positive definite. An optimal error estimate in the H¹(Ω) × H(div; Ω) norm is derived. An equivalent residual‐type a posteriori error estimator is also given. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 738–751, 2002; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/num.10030.     

A posteriori error estimator for expanded mixed hybrid methods

In this article, we construct an a posteriori error estimator for expanded mixed hybrid finite‐element methods for second‐order elliptic problems. An a posteriori error analysis yields reliable and efficient estimate based on residuals. Several numerical examples are presented to show the effectivity of our error indicators. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 330–349, 2007     

Robust hierarchical a posteriori error estimators for stabilized convection–diffusion problems

We construct a hierarchical a posteriori error estimator for a stabilized finite element discretization of convection‐diffusion equations with height Péclet number. The error estimator is derived without the saturation assumption and without any comparison with the classical residual estimator. Besides, it is robust, such that the equivalence between the norm of the exact error and the error estimator is independent of the meshsize or the diffusivity parameter. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012     

Even‐hole‐free graphs part I: Decomposition theorem

We prove a decomposition theorem for even‐hole‐free graphs. The decompositions used are 2‐joins and star, double‐star and triple‐star cutsets. This theorem is used in the second part of this paper to obtain a polytime recognition algorithm for even‐hole‐free graphs. © 2002 John Wiley & Sons, Inc. J Graph Theory 39: 6–49, 2002     

Least‐squares mixed finite element methods for the incompressible Navier‐Stokes equations

Least‐squares mixed finite element schemes are formulated to solve the evolutionary Navier‐Stokes equations and the convergence is analyzed. We recast the Navier‐Stokes equations as a first‐order system by introducing a vorticity flux variable, and show that a least‐squares principle based on L² norms applied to this system yields optimal discretization error estimates. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 441–453, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10015     

Decay rates of adaptive finite elements with Dörfler marking

We investigate the decay rate for an adaptive finite element discretization of a second order linear, symmetric, elliptic PDE. We allow for any kind of estimator that is locally equivalent to the standard residual estimator. This includes in particular hierarchical estimators, estimators based on the solution of local problems, estimators based on local averaging, equilibrated residual estimators, the ZZ-estimator, etc. The adaptive method selects elements for refinement with Dörfler marking and performs a minimal refinement in that no interior node property is needed. Based on the local equivalence to the residual estimator we prove an error reduction property. In combination with minimal Dörfler marking this yields an optimal decay rate in terms of degrees of freedom.     

Toward anisotropic mesh construction and error estimation in the finite element method

Directional, anisotropic features like layers in the solution of partial differential equations can be resolved favorably by using anisotropic finite element meshes. An adaptive algorithm for such meshes includes the ingredients Error estimation and Information extraction/Mesh refinement. Related articles on a posteriori error estimation on anisotropic meshes revealed that reliable error estimation requires an anisotropic mesh that is aligned with the anisotropic solution. To obtain anisotropic meshes the so‐called Hessian strategy is used, which provides information such as the stretching direction and stretching ratio of the anisotropic elements. This article combines the analysis of anisotropic information extraction/mesh refinement and error estimation (for several estimators). It shows that the Hessian strategy leads to well‐aligned anisotropic meshes and, consequently, reliable error estimation. The underlying heuristic assumptions are given in a stringent yet general form. Numerical examples strengthen the exposition. Hence the analysis provides further insight into a particular aspect of anisotropic error estimation. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 625–648, 2002; DOI 10.1002/num.10023     

A recovery‐based error estimator for finite volume methods of interface problems: Conforming linear elements

In this article, we present a recovery‐based a posteriori error estimator for finite volume methods which use the conforming linear trial functions to approximate elliptic interface problems. The reliability and efficiency bounds for the error estimator are established by recovering the flux from a weighted L² projection to H(div) conforming finite element spaces. Numerical experiments are given to support the conclusions. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Optimal volume-corrected laplace-metropolis method

The article provides a refinement for the volume-corrected Laplace-Metropolis estimator of the marginal likelihood of DiCiccioet al. The correction volume of probability α in DiCiccioet al. is fixed and suggested to take the value α=0.05. In this article α is selected based on an asymptotic analysis to minimize the mean square relative error (MSRE). This optimal choice of α is shown to be invariant under linear transformations. The invariance property leads to easy implementation for multivariate problems. An implementation procedure is provided for practical use. A simulation study and a real data example are presented.     

An adaptive least squares mixed finite element method for the stress‐displacement formulation of linear elasticity

A least‐squares mixed finite element method for linear elasticity, based on a stress‐displacement formulation, is investigated in terms of computational efficiency. For the stress approximation quadratic Raviart‐Thomas elements are used and these are coupled with the quadratic nonconforming finite element spaces of Fortin and Soulie for approximating the displacement. The local evaluation of the least‐squares functional serves as an a posteriori error estimator to be used in an adaptive refinement algorithm. We present computational results for a benchmark test problem of planar elasticity including nearly incompressible material parameters in order to verify the effectiveness of our adaptive strategy. For comparison, conforming quadratic finite elements are also used for the displacement approximation showing convergence orders similar to the nonconforming case, which are, however, not independent of the Lamé parameters. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Adaptive meshless Galerkin boundary node methods for hypersingular integral equations

Adaptive refinement techniques are developed in this paper for the meshless Galerkin boundary node method for hypersingular boundary integral equations. Two types of error estimators are derived. One is a perturbation error estimator that is formulated based on the difference between numerical solutions obtained using two consecutive nodal arrangements. The other is a projection error estimator that is formulated based on the difference between the numerical solution itself and its projection. These error estimators are proven to have an upper and a lower bound by the constant multiples of the exact error in the energy norm. A localization scheme is presented to accomodate the non-local property of hypersingular integral operators for the needed computable local error indicators. The convergence of the adaptive meshless techniques is verified theoretically. To confirm the theoretical results and to show the efficiency of the adaptive techniques, numerical examples in 2D and 3D with high singularities are provided.     

A Posteriori Error Estimates and a Local Refinement Strategy for a Finite Element Method to Solve Structural-Acoustic Vibration Problems

This paper deals with an adaptive technique to compute structural-acoustic vibration modes. It is based on an a posteriori error estimator for a finite element method free of spurious or circulation nonzero-frequency modes. The estimator is shown to be equivalent, up to higher order terms, to the approximate eigenfunction error, measured in a useful norm; moreover, the equivalence constants are independent of the corresponding eigenvalue, the physical parameters, and the mesh size. This a posteriori error estimator yields global upper and local lower bounds for the error and, thus, it may be used to design adaptive algorithms. We propose a local refinement strategy based on this estimator and present a numerical test to assess the efficiency of this technique.     

Projection stabilized nonconforming finite element methods for the stokes problem

In this article we study a projection‐stabilized nonconforming finite element discretization of the Stokes problem. We present a priori error analysis and give a recovery‐based a posteriori error estimator for the considered problem. Numerical results illustrate the theoretical performance of the error estimator. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 218–240, 2017