Posteriori Error Indicator Research Articles

Boundary Integral Equations and A Posteriori Error Estimates

Adaptive methods have been rapidly developed and applied in many fields of scientific and engineering computing. Reliable and efficient a posteriori error estimates play key roles for both adaptive finite element and boundary element methods. The aim of this paper is to develop a posteriori error estimates for boundary element methods. The standard a posteriori error estimates for boundary element methods are obtained from the classical boundary integral equations. This paper presents hyper-singular a posteriori error estimates based on the hyper-singular integral equations. Three kinds of residuals are used as the estimates for boundary element errors. The theoretical analysis and numerical examples show that the hyper-singular residuals are good a posteriori error indicators in many adaptive boundary element computations.

Computational Error Approximation and H-Adaptive Algorithm for the 3-D Generalized Finite Difference Method

This paper shows the explicit formulae for the generalized finite difference method in the three-dimensional case. The method solves different types of partial differential equations for irregular clouds of nodes. The explicit difference formulae obtained make it possible to propose a posteriori error indicator (taking more terms of the Taylor's series expansion). This serves as a starting point for an h-adaptive method to improve the solution by selectively adding nodes to the domain. The h-adaptive method consists in marking the nodes with an indicated error higher than a provided limit. Then, several nodes can be added to each star with a marked central node and placed centrally between this and the star nodes with greater indicator errors. This paper analyses the results obtained using the h-adaptive method in several cases and shows how the solution error decreases. It also analyses the influence of key parameters involved, and the quality of the error estimate.

ADER schemes on adaptive triangular meshes for scalar conservation laws

ADER schemes are recent finite volume methods for hyperbolic conservation laws, which can be viewed as generalizations of the classical first order Godunov method to arbitrary high orders. In the ADER approach, high order polynomial reconstruction from cell averages is combined with high order flux evaluation, where the latter is done by solving generalized Riemann problems across cell interfaces. Currently available nonlinear ADER schemes are restricted to Cartesian meshes. This paper proposes an adaptive nonlinear finite volume ADER method on unstructured triangular meshes for scalar conservation laws, which works with WENO reconstruction. To this end, a customized stencil selection scheme is developed, and the flux evaluation of previous ADER schemes is extended to triangular meshes. Moreover, an a posteriori error indicator is used to design the required adaption rules for the dynamic modification of the triangular mesh during the simulation. The expected convergence orders of the proposed ADER method are confirmed by numerical experiments for linear and nonlinear scalar conservation laws. Finally, the good performance of the adaptive ADER method, in particular its robustness and its enhanced flexibility, is further supported by numerical results concerning Burgers equation.

A recipe to detect the error in discretization schemes

AbstractThe necessity for a reliable measure of the discretization error arises in adaptive mesh refinement and in moving mesh adaptation. The present work discusses a detector of the discretization error based on the interpolation reconstruction of the operators. The technique presented here is named operator recovery error source detector (ORESD). Its main features are: First, the technique is based on the operators being discretized and does not require any user intervention or any a priori knowledge of the solution or its properties. Second, the ORESD is an a posteriori error indicator, but it is shown to be consistent with the a priori error provided by the modified equation approach. Third, the technique is based on the operators being solved and is tailored to the specific problem at hand. Four, the technique is simple and is based on a small stencil, resulting in a very inexpensive error detection.In the present work, the ORESD is derived and applied to two tutorial examples: divergence and gradient. With the aid of the two examples and using the general derivation, the ORESD is then applied to the gas dynamics equations. Two benchmarks are used to test the performance. First, a shock tube problem is solved (Sod's benchmark) in a Lagrangian and in a Eulerian frame. Second, the Colella's wedge problem is solved using CLAWPACK. Finally, the ORESD is applied to the 2D Poisson equation on a uniform and on a non‐uniform grid to test the application to elliptic problems. In all examples the operator recovery error source detector succeeds in detecting the real sources of error. Copyright © 2004 John Wiley & Sons, Ltd.

Numerical Entropy Production for Central Schemes

In this work a detailed study of the numerical entropy production for both low- and high-order central schemes is carried out. Our data show that entropy production can be used to signal the presence of shocks. Moreover, once shocked cells have been identified, the spurious entropy production occurring in the remaining cells mimics the behavior of the local error, and therefore it can be used as an a posteriori error indicator. Applications to shock tracking and scheme adaptivity are given. We also investigate numerically the entropic properties of third and fourth order Central WENO schemes. To our knowledge, this is the first study of the entropic behavior of such high-order schemes.

A posteriori error estimates for finite element discretizations of the heat equation

We consider discretizations of the heat equation by A-stable θ-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.

Adaptive F.E. method for the shakedown and limit analysis of pressure vessels

Upper bound estimates of limit and shakedown loads for pressure vessels are calculated by using the technique described in this paper. These have been achieved by applying Koiter's theorem and by discretizing the shell into finite elements. The flow law associated with an hexagonal prism yield surface, relates the plastic strain increments and curvatures to plastic multipliers. A suitable matrix also relates such a plastic strain field to a displacement field through a classical relation. A novel method enforces a consistent relationship between nodal displacements and nodal plastic multipliers by minimizing the residual between the two independent descriptions of the plastic increments, measured with respect to the energy norm. The discretized problem is then reduced to a minimization problem and solved by linear programming. An a posteriori error indicator in the energy norm is derived with and adaptive mesh refinement scheme.

COMPUTING HIGHLY ACCURATE SOLUTIONS OF A TUMOUR ANGIOGENESIS MODEL

In this paper we describe and implement a numerical method which provides highly accurate solutions of a generic two-dimensional model for the formation of capillary networks as a partial process in tumour angiogenesis. The model includes effects due to diffusion, chemotaxis, haptotaxis and cell proliferation. The governing partial differential equation is a diffusion-advection-reaction equation of parabolic type. In order to achieve high accuracy in space, we use a semi-discretisation based on the spectral element method. The resulting system of stiff ordinary differential equations is advanced forward in time using one-step explicit higher order time integrators based on Taylor series expansions. The high accuracy in space is monitored by a residual based a posteriori error indicator while the high accuracy in time is guaranteed by the local and global truncation errors of the higher order Taylor series method.

Adaptive finite element procedures for elastoplastic problems at finite strains

AbstractA major difficulty in the context of adaptive analysis of geometrically nonlinear problems is to provide a robust remeshing procedure that accounts both for the error caused by the spatial discretization and for the error due to the time discretization. For stability problems, such as strain localization and necking, it is essential to provide a step–size control in order to get a robust algorithm for the solution of the boundary value problem. For this purpose we developed an easy to implement step–size control algorithm. In addition we will consider possible a posteriori error indicators for the spatial error distribution of elastoplastic problems at finite strains. This indicator is adopted for a density–function–based adaptive remeshing procedure. Both error indicators are combined for the adaptive analysis in time and space. The performance of the proposed method is documented by means of representative numerical examples.

An h-adaptive method in the generalized finite differences

This paper describes an h-adaptive method in generalized finite difference (GFD) to solve second-order partial differential equations. These equations representing the behaviour of many physical processes. The explicit difference formulae obtained make it possible to propose an a posteriori error indicator which serves as starting point for an h-adaptive method to improve the solution by selectively adding nodes to the domain. This paper also analyses the influence of key parameters, as the number of nodes to add in each step or the minimum distance between nodes, through the analysis of the obtained solutions for different types of differential equations.

A first error indicator for micro–macro simulations of viscoelastic fluids having closed-form constitutive equations

An a posteriori error indicator for viscoelastic flow calculations is proposed by the author for a recently introduced method. This new method is derived from the Brownian configuration field approach and can be applied to models for which closed-form expressions already exist. Numerical results are presented in the context of the spectral element method to show the reliability of the error indicator. The benchmark problem of the flow of an Oldroyd-B fluid past a cylinder in a channel is used and we show that this new method is always more accurate than using a constitutive equation.

On the Efficiency of Adaptive Finite Element Methods for Elliptic Problems with Discontinuous Coefficients

The successful implementation of adaptive finite element methods based on a posteriori error estimates depends on several ingredients: an a posteriori error indicator, a refinement/coarsening strategy, and the choice of various parameters. The objective of the paper is to examine the influence of these factors on the performance of adaptive finite element methods for a model problem: the linear elliptic equation with strongly discontinuous coefficients. We derive a new a posteriori error estimator which depends locally on the oscillations of the coefficients around singular points. Extensive numerical experiments are reported to support our theoretical results and to show the competitive behaviors of the proposed adaptive algorithm.

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 Numerical Treatment of Elliptic Systems on Manifolds

Adaptive multilevel finite element methods are developed and analyzed for certain elliptic systems arising in geometric analysis and general relativity. This class of nonlinear elliptic systems of tensor equations on manifolds is first reviewed, and then adaptive multilevel finite element methods for approximating solutions to this class of problems are considered in some detail. Two a posteriori error indicators are derived, based on local residuals and on global linearized adjoint or dual problems. The design of Manifold Code (MC) is then discussed; MC is an adaptive multilevel finite element software package for 2- and 3-manifolds developed over several years at Caltech and UC San Diego. It employs a posteriori error estimation, adaptive simplex subdivision, unstructured algebraic multilevel methods, global inexact Newton methods, and numerical continuation methods for the numerical solution of nonlinear covariant elliptic systems on 2- and 3-manifolds. Some of the more interesting features of MC are described in detail, including some new ideas for topology and geometry representation in simplex meshes, and an unusual partition of unity-based method for exploiting parallel computers. A short example is then given which involves the Hamiltonian and momentum constraints in the Einstein equations, a representative nonlinear 4-component covariant elliptic system on a Riemannian 3-manifold which arises in general relativity. A number of operator properties and solvability results recently established in [55] are first summarized, making possible two quasi-optimal a priori error estimates for Galerkin approximations which are then derived. These two results complete the theoretical framework for effective use of adaptive multilevel finite element methods. A sample calculation using the MC software is then presented; more detailed examples using MC for this application may be found in [26].

On Mixed Error Estimates for Elliptic Obstacle Problems

We establish in this paper sharp error estimates of residual type for finite element approximation to elliptic obstacle problems. The estimates are of mixed nature, which are neither of a pure a priori form nor of a pure a posteriori form but instead they are combined by an a priori part and an a posteriori part. The key ingredient in our derivation for the mixed error estimates is the use of a new interpolator which enables us to eliminate inactive data from the error estimators. One application of our mixed error estimates is to construct a posteriori error indicators reliable and efficient up to higher order terms, and these indicators are useful in mesh-refinements and adaptive grid generations. In particular, by approximating the a priori part with some a posteriori quantities we can successfully track the free boundary for elliptic obstacle problems.

Comparison of different error estimators for contact problems

Several a posteriori error indicators and error estimators for frictionless contact problems are compared. In detail, residual based error estimators, error indicators relying on superconvergence properties and error estimators based on duality principles are investigated. Applications are to 2D solids under the hypothesis of nonlinear 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.

An h–p adaptive spectral element method for Stokes flow

An h–p adaptive spectral element method for three-dimensional axisymmetric domains featuring an element residual a posteriori error indicator, equidistribution of the error amongst the elements in the initial mesh and simultaneous h and p refinements is presented. The mortar element method is used to efficiently handle the nonconforming decompositions that arise in the adaptive process. Results obtained by applying the adaptive strategy to Stokes flow for different orders of regularity of the righthand side function are compared with the results obtained using a uniform p refinement. The results obtained show the advantages of using the proposed strategy.

A posteriori error estimates and grid adjustment for a nonlinear parabolic equation

The paper is concerned with a posteriori error estimates needed for the adaptive construction of a space grid in solving an initial-boundary value one-dimensional problem for a nonlinear parabolic partial differential equation by the method of lines. A posteriori error indicators are introduced and studied for the semidiscrete case, and the statement on their convergence rate is presented. Under certain conditions, it adds some more results to those of Moore [SIAM J. Numer. Anal. 31 (1994) 149–164] who treats only a linear equation in the semidiscrete case. Complete statements including proofs will appear as a paper [K. Segeth, A posteriori error estimation with the finite element method of lines for a nonlinear parabolic equation in one space dimension, to appear]. A simple numerical example illustrates the capabilities of a posteriori error estimates when used for grid adjustments in the finite element method of lines. The example fully confirms the theoretical statement.

Multilevel solution of the time-harmonic Maxwell's equations based on edge elements

A widely used approach for the computation of time-harmonic electromagnetic fields is based on the well-known double-curl equation for either E or H, where edge elements are an appealing choice for finite element discretizations. Yet, the nullspace of the curl-operator comprises a considerable part of all spectral modes on the finite element grid. Thus standard multilevel solvers are rendered inefficient, as they essentially hinge on smoothing procedures like Gauss–Seidel relaxation, which cannot provide a satisfactory error reduction for modes with small or even negative eigenvalues. We propose to remedy this situation by an extended multilevel algorithm which relies on corrections in the space of discrete scalar potentials. After every standard V-cycle with respect to the canonical basis of edge elements, error components in the nullspace are removed by an additional projection step. Furthermore, a simple criterion for the coarsest mesh is derived to guarantee both stability and efficiency of the iterative multilevel solver. For the whole scheme we observe convergence rates independent of the refinement level of the mesh. The sequence of nested meshes required for our multilevel techniques is constructed by adaptive refinement. To this end we have devised an a posteriori error indicator based on stress recovery. Copyright © 1999 John Wiley & Sons, Ltd.

Error estimation and adaptive meshing in strongly nonlinear dynamic problems

We present work aimed at developing a general framework for mesh adaption in strongly nonlinear, possibly dynamic, problems. We begin by showing that the solutions of the incremental boundary value problem for a wide class of materials, including nonlinear elastic materials, compressible Newtonian fluids and viscoplastic solids, obey a minimum principle, provided that the constitutive updates are formulated appropriately. This minimum principle can be taken as a basis for asymptotic error estimation. In particular, we chose to monitor the error of a lower-order projection of the finite element solution. The optimal mesh size distribution then follows from a posteriori error indicators which are purely local, i.e. can be computed element-by-element. We demonstrate the robustness and versatility of the computational framework with the aid of convergence studies and selected examples of application.