A posteriori error estimation for finite element solutions of Helmholtz' equation—Part II: estimation of the pollution error

In part I of this investigation, we proved that the standard a posteriori estimates, based only on local computations, may severely underestimate the exact error for the classes of wave-numbers and the types of meshes employed in engineering analyses. We showed that this is due to the fact that the local estimators do not measure the pollution effect inherent to the FE-solutions of Helmholtz' equation with large wavenumber. Here, we construct a posteriori estimates of the pollution error. We demonstrate that these estimates are reliable and can be used to correct the standard a posteriori error estimates in any patch of elements of interest. © 1997 John Wiley & Sons, Ltd.     

A posteriori error estimates and adaptivity for finite element solutions in finite elasticity

Methods for a posteriori error estimation for finite element solutions are well established and widely used in engineering practice for linear boundary value problems. In contrast here we are concerned with finite elasticity and error estimation and adaptivity in this context. In the paper a brief outline of continuum theory of finite elasticity is first given. Using the residuals in the equilibrium conditions the discretization error of the finite element solution is estimated both locally and globally. The proposed error estimator is physically interpreted in the energy sense. We then present and discuss the convergence behaviour of the discretization error in uniformly and adaptively refined finite element sequences.     

Statically admissible finite element solution of the thin plate bending problem with a posteriori error estimation

Two triangular elements of class C⁰ developed on the basis of the principle of complementary work are applied in the static analysis of a thin plate. Some techniques to widen the versatility of the equilibrium approach for the finite element method are presented. Plates of various shapes subjected to diverse types of loading are considered. The results are compared with outcomes obtained by use of the displacement-based finite element method. By use of these two dual types of solutions, the error of the approximate solution is calculated. The lower and upper bounds for the strain energy are found.     

Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation

When applying numerical methods for the computation of stationary waves from the Helmholtz equation, one obtains ‘numerical waves’ that are dispersive also in non-dispersive media. The numerical wave displays a phase velocity that depends on the parameter k of the Helmholtz equation. In dispersion analysis, the phase difference between the exact and the numerical solutions is investigated. In this paper, the authors' recent result on the phase difference for one-dimensional problems is numerically evaluated and discussed in the context of other work directed to this topic. It is then shown that previous error estimates in H¹-norm are of nondispersive character but hold for medium or high wavenumber on extremely refined mesh only. On the other hand, recently proven error estimates for constant resolution contain a pollution term. With certain assumptions on the exact solution, this term is of the order of the phase difference. Thus a link is established between the results of dispersion analysis and the results of numerical analysis. Throughout the paper, the presentation and discussion of theoretical results is accompanied by numerical evaluation of several model problems. Special attention is given to the performance of the Galerkin method with a higher order of polynomial approximation p(h-p-version).     

A posteriori estimation and adaptive control of the pollution error in the h-version of the finite element method

In References 1 and 2 we showed that the error in the finite-element solution has two parts, the local error and the pollution error, and we studied the effect of the pollution error on the quality of the local error-indicators and the quality of the derivatives recovered by local post-processing. Here we show that it is possible to construct a posteriori estimates of the pollution error in any patch of elements by employing the local error-indicators over the mesh outside the patch. We also give an algorithm for the adaptive control of the pollution error in any patch of elements of interest.     

Influence of the pollution on the admissible field error estimation for FE solutions of the Helmholtz equation

The a posteriori error estimation in constitutive law has already been extensively developed and applied to finite element solutions of structural analysis problems. The paper presents an extension of this estimator to problems governed by the Helmholtz equation (e.g. acoustic problems) that we have already partially reported, this paper containing informations about the construction of the admissible fields for acoustics. Moreover, it has been proven that the upper bound property of this estimator applied to elasticity problems (the error in constitutive law bounds from above the exact error in energy norm) does not generally apply to acoustic formulations due to the presence of the specific pollution error. The numerical investigations of the present paper confirm that the upper bound property of this type of estimator is verified only in the case of low (non‐dimensional) wave numbers while it is violated for high wave numbers due to the pollution effect. Copyright © 1999 John Wiley & Sons, Ltd.     

A posteriori error estimation based on the superconvergent Recovery by Compatibility in Patches

The present work deals with an a posteriori error estimator for linear finite element analysis, based on a stress recovery procedure called Recovery by Compatibility in Patches. The key idea of this procedure is to recover improved stresses by minimizing the complementary energy over patches of elements. Displacements computed by the finite element analysis are prescribed on the boundary of the patch. Here, a new form of this recovery procedure is presented. Adopting a different patch configuration, centred upon an element instead of a node, allows to drastically simplify the recovery process, thus improving efficiency and making the implementation in finite element codes much easier. The robustness tests demonstrate that the error estimator associated to the new form of the recovery procedure retains the very good properties of the original one, such as superconvergence. The numerical results on two common benchmark problems confirm the effectiveness of the proposed error estimator, which appears to be competitive with those currently available. Copyright © 2006 John Wiley & Sons, Ltd.     

Estimation of the dispersion error in the numerical wave number of standard and stabilized finite element approximations of the Helmholtz equation

An estimator for the error in the wave number is presented in the context of finite element approximations of the Helmholtz equation. The proposed estimate is an extension of the ideas introduced in Steffens and D'?ez (Comput. Methods Appl. Mech. Engng 2009; 198 :1389–1400). In the previous work, the error assessment technique was developed for standard Galerkin approximations. Here, the methodology is extended to deal also with stabilized approximations of the Helmholtz equation. Thus, the accuracy of the stabilized solutions is analyzed, including also their sensitivity to the stabilization parameters depending on the mesh topology. The procedure builds up an inexpensive approximation of the exact solution, using post‐processing techniques standard in error estimation analysis, from which the estimate of the error in the wave number is computed using a simple closed expression. The recovery technique used in Steffens and Díez (Comput. Methods Appl. Mech. Engng 2009; 198 :1389–1400) is based in a polynomial least‐squares fitting. Here a new recovery strategy is introduced, using exponential (in a complex setup, trigonometric) local approximations respecting the nature of the solution of the wave problem. Copyright © 2011 John Wiley & Sons, Ltd.     

Adaptive selection of polynomial degrees on a finite element mesh

The problem of finding a nearly optimal distribution of polynomial degrees on a fixed finite element mesh is discussed. An a posteriori error estimator based on the minimum complementary energy principle is proposed which utilizes the displacement vector field computed from the finite element solution. This estimator, designed for p- and hp-extensions, is conceptually different from estimators based on residuals or patch recovery which are designed for h-extension procedures. The quality of the error estimator is demonstrated by examples. The results show that the effectivity index is reasonably close to unity and the sequences of p-distributions obtained with the error indicators closely follow the optimal trajectory. © 1998 John Wiley & Sons, Ltd.     

An a posteriori error estimate for the generalized finite element method for transient heat diffusion problems

We propose the study of a posteriori error estimates for time‐dependent generalized finite element simulations of heat transfer problems. A residual estimate is shown to provide reliable and practically useful upper bounds for the numerical errors, independent of the heuristically chosen enrichment functions. Two sets of numerical experiments are presented. First, the error estimate is shown to capture the decrease in the error as the number of enrichment functions is increased or the time discretization refined. Second, the estimate is used to predict the behaviour of the error where no exact solution is available. It also reflects the errors incurred in the poorly conditioned systems typically encountered in generalized finite element methods. Finally, we study local error indicators in individual time steps and elements of the mesh. This creates a basis towards the adaptive selection and refinement of the enrichment functions. Copyright © 2016 John Wiley & Sons, Ltd.     

Guaranteed a posteriori error estimation for fully discrete solutions of parabolic problems

This paper addresses the construction of guaranteed computable estimates for fully discrete solutions of parabolic problems. Copyright © 2003 John Wiley & Sons, Ltd.     

Partition of unity for localization in implicit a posteriori finite element error control for linear elasticity

The partition of unity for localization in adaptive finite element method (FEM) for elliptic partial differential equations has been proposed in Carstensen and Funken (SIAM J. Sci. Comput. 2000; 21 : 1465–1484) and is applied therein to the Laplace problem. A direct adaptation to linear elasticity in this paper yields a first estimator η_L based on patch‐oriented local‐weighted interface problems. The global Korn inequality with a constant C_Korn yields reliability for any finite element approximation u_h to the exact displacement u. In order to localize this inequality further and so to involve the global constant C_Korn directly in the local computations, we deduce a new error estimator µ_L. The latter estimator is based on local‐weighted interface problems with rigid body motions (RBM) as a kernel and so leads to effective estimates only if RBM are included in the local FE test functions. Therefore, the excluded first‐order FEM has to be enlarged by RBM, which leads to a partition of unit method (PUM) with RBM, called P₁+RBM or to second‐order FEMs, called P₂ FEM. For P₁+RBM and P₂ FEM (or even higher‐order schemes) one obtains the sharper reliability estimate . Efficiency holds in the strict sense of . The local‐weighted interface problems behind the implicit error estimators η_L and µ_L are usually not exactly solvable and are rather approximated by some FEM on a refined mesh and/or with a higher‐order FEM. The computable approximations are shown to be reliable in the sense of . The oscillations are known functions of the given data and higher‐order terms if the data are smooth for first‐order FEM. The mathematical proofs are based on weighted Korn inequalities and inverse estimates combined with standard arguments. The numerical experiments for uniform and adapted FEM on benchmarks such as an L‐shape problem, Cook's membrane, or a slit problem validate the theoretical estimates and also concern numerical bounds for C_Korn and the locking phenomena. Copyright © 2007 John Wiley & Sons, Ltd.     

A posteriori error estimation for extended finite elements by an extended global recovery

This contribution presents an extended global derivative recovery for enriched finite element methods (FEMs), such as the extended FEM along with an associated error indicator. Owing to its simplicity, the proposed scheme is ideally suited to industrial applications. The procedure is based on global minimization of the L₂ norm of the difference between the raw strain field (C_?1) and the recovered (C₀) strain field. The methodology engineered in this paper extends the ideas of Oden and Brauchli (Int. J. Numer. Meth. Engng 1971; 3 ) and Hinton and Campbell (Int. J. Numer. Meth. Engng 1974; 8 ) by enriching the approximation used for the construction of the recovered derivatives (strains) with the gradients of the functions employed to enrich the approximation employed for the primal unknown (displacements). We show linear elastic fracture mechanics examples, both in simple two‐dimensional settings, and for a three‐dimensional structure. Numerically, we show that the effectivity index of the proposed indicator converges to unity upon mesh refinement. Consequently, the approximate error converges to the exact error, indicating that the error indicator is valid. Additionally, the numerical examples suggest a novel adaptive strategy for enriched approximations in which the dimensions of the enrichment zone are first increased, before standard h‐ and p‐adaptivities are applied; we suggest to coin this methodology e‐adaptivity. Copyright © 2008 John Wiley & Sons, Ltd.     

Guaranteed computable error bounds for conforming and nonconforming finite element analyses in planar elasticity

We obtain fully computable a posteriori error estimators for the energy norm of the error in second‐order conforming and nonconforming finite element approximations in planar elasticity. These estimators are completely free of unknown constants and give a guaranteed numerical upper bound on the norm of the error. The estimators are shown to also provide local lower bounds, up to a constant and higher‐order data oscillation terms. Numerical examples are presented illustrating the theory and confirming the effectiveness of the estimator. Copyright © 2009 John Wiley & Sons, Ltd.     

A two-level finite element method and its application to the Helmholtz equation

A two-level finite element method is introduced and its application to the Helmholtz equation is considered. The method retains the desirable features of the Galerkin method enriched with residual-free bubbles, while it is not limited to discretizations using elements with simple geometry. The method can be applied to other equations and to irregular-shaped domains. © 1998 John Wiley & Sons, Ltd.     

A posteriori error estimates of mixed methods for miscible displacement problems

The miscible displacement of one incompressible fluid by another in a porous medium is governed by a system of two equations. One is an elliptic equation of the pressure and the other is a parabolic equation of the concentration of one of the fluids. Since the pressure appears in the concentration only through its velocity field, we choose a mixed finite element method to approximate the pressure equation and for the concentration we use the standard Galerkin method. We shall obtain an explicit a posteriori error estimator in L²(L²) for the semi‐discrete scheme applied to the non‐linear coupled system. Copyright © 2007 John Wiley & Sons, Ltd.     

A boundary‐perturbation finite element approach for shape optimization

A numerical method is proposed for the efficient solution of shape optimization problems, which combines the boundary perturbation technique and finite element analysis. The method is computationally efficient in that it requires a number of finite element analyses with a fixed geometry, as opposed to standard shape optimization which requires re‐analysis with varying geometry. The application of the method to general shape optimization is considered. In addition, a special optimization scheme is devised for a class of problems governed by linear partial differential equations. The performance of the method is illustrated via an example which involves acoustic wave scattering from an obstacle. Copyright © 2000 John Wiley & Sons, Ltd.     

A residual‐based finite element method for the Helmholtz equation

A new residual‐based finite element method for the scalar Helmholtz equation is developed. This method is obtained from the Galerkin approximation by appending terms that are proportional to residuals on element interiors and inter‐element boundaries. The inclusion of residuals on inter‐element boundaries distinguishes this method from the well‐known Galerkin least‐squares method and is crucial to the resulting accuracy of this method. In two dimensions and for regular bilinear quadrilateral finite elements, it is shown via a dispersion analysis that this method has minimal phase error. Numerical experiments are conducted to verify this claim as well as test and compare the performance of this method on unstructured meshes with other methods. It is found that even for unstructured meshes this method retains a high level of accuracy. Copyright © 2000 John Wiley & Sons, Ltd.     

An automatic adaptive refinement procedure for the reproducing kernel particle method. Part I: Stress recovery and a posteriori error estimation

In this study, an adaptive refinement procedure using the reproducing kernel particle method (RKPM) for the solution of 2D elastostatic problems is suggested. This adaptive refinement procedure is based on the Zienkiewicz and Zhu (ZZ) error estimator for the a posteriori error estimation and an adaptive finite point mesh generator for new point mesh generation. The presentation of the work is divided into two parts. In Part I, concentration will be paid on the stress recovery and the a posteriori error estimation processes for the RKPM. The proposed error estimator is different from most recovery type error estimators suggested previously in such a way that, rather than using the least-squares fitting approach, the recovery stress field is constructed by an extraction function approach. Numerical studies using 2D benchmark boundary value problems indicated that the recovered stress field obtained is more accurate and converges at a higher rate than the RKPM stress field. In Part II of the study, concentration will be shifted to the development of an adaptive refinement algorithm for the RKPM.     

On error estimator and p‐adaptivity in the generalized finite element method

This paper addresses the issue of a p‐adaptive version of the generalized finite element method (GFEM). The technique adopted here is the equilibrated element residual method, but presented under the GFEM approach, i.e., by taking into account the typical nodal enrichment scheme of the method. Such scheme consists of multiplying the partition of unity functions by a set of enrichment functions. These functions, in the case of the element residual method are monomials, and can be used to build the polynomial space, one degree higher than the one of the solution, in which the error functions is approximated. Global and local measures are defined and used as error estimator and indicators, respectively. The error indicators, calculated on the element patches that surrounds each node, are used to control a refinement procedure. Numerical examples in plane elasticity are presented, outlining in particular the effectivity index of the error estimator proposed. Finally, the ‐adaptive procedure is described and its good performance is illustrated by the last numerical example. Copyright © 2004 John Wiley & Sons, Ltd.