Posteriori Error Indicator Research Articles

A posteriori analysis for dynamic model adaptation in convection-dominated problems

In this work, we present an a posteriori error indicator for approximation schemes of Runge–Kutta–discontinuous–Galerkin type arising in applications of compressible fluid flows. The purpose of this indicator is not only for mesh adaptivity, we also make use of this to drive model adaptivity. This is perhaps where a costly complex model and a cheaper simple model are solved over different parts of the domain. The a posteriori bound we derive indicates the regions where the complex model can be relatively well approximated with the cheaper one. One such example which we choose to highlight is that of the Navier–Stokes–Fourier equations approximated by Euler’s equations.

Aspects of an adaptive finite element method for the fractional Laplacian: A priori and a posteriori error estimates, efficient implementation and multigrid solver

We develop all of the components needed to construct an adaptive finite element code that can be used to approximate fractional partial differential equations, on non-trivial domains in d≥1 dimensions. Our main approach consists of taking tools that have been shown to be effective for adaptive boundary element methods and, where necessary, modifying them so that they can be applied to the fractional PDE case. Improved a priori error estimates are derived for the case of quasi-uniform meshes which are seen to deliver sub-optimal rates of convergence owing to the presence of singularities. Attention is then turned to the development of an a posteriori error estimate and error indicators which are suitable for driving an adaptive refinement procedure. We assume that the resulting refined meshes are locally quasi-uniform and develop efficient methods for the assembly of the resulting linear algebraic systems and their solution using iterative methods, including the multigrid method. The storage of the dense matrices along with efficient techniques for computing the dense matrix–vector products needed for the iterative solution is also considered. The performance and efficiency of the resulting algorithm is illustrated for a variety of examples.

C 0IPG adaptive algorithms for the biharmonic eigenvalue problem

This paper focuses on C 0IPG adaptive algorithms for the biharmonic eigenvalue problem with the clamped boundary condition. We prove the reliability and efficiency of the a posteriori error indicator of the approximating eigenfunctions and analyze the reliability of the a posteriori error indicator of the approximating eigenvalues. We present two adaptive algorithms, and numerical experiments indicate that both algorithms are efficient.

An ultraweak DPG method for viscoelastic fluids

We explore a vexing benchmark problem for viscoelastic fluid flows with the discontinuous Petrov-Galerkin (DPG) finite element method of Demkowicz and Gopalakrishnan [1,2]. In our analysis, we develop an intrinsic a posteriori error indicator which we use for adaptive mesh generation. The DPG method is useful for the problem we consider because the method is inherently stable—requiring no stabilization of the linearized discretization in order to handle the advective terms in the model. Because stabilization is a pressing issue in these models, this happens to become a very useful property of the method which simplifies our analysis. This built-in stability at all length scales and the a posteriori error indicator additionally allows for the generation of parameter-specific meshes starting from a common coarse initial mesh. A DPG discretization always produces a symmetric positive definite stiffness matrix. This feature allows us to use the most efficient direct solvers for all of our computations. We use the Camellia finite element software package [3,4] for all of our analysis.

Simulation of two‐ and three‐dimensional viscoplastic flows using adaptive mesh refinement

SummaryThis paper presents a finite element solver for the simulation of steady non‐Newtonian flow problems, using a regularized Bingham model, with adaptive mesh refinement capabilities.The solver is based on a stabilized formulation derived from the variational multiscale framework. This choice allows the introduction of an a posteriori error indicator based on the small scale part of the solution, which is used to drive a mesh refinement procedure based on element subdivision.This approach applied to the solution of a series of benchmark examples, which allow us to validate the formulation and assess its capabilities to model 2D and 3D non‐Newtonian flows.

A new accurate residual-based a posteriori error indicator for the BEM in 2D-acoustics

In this work we construct a new reliable, efficient and local a posteriori error estimate for the single layer and hyper-singular boundary integral equations associated to the Helmholtz equation in two dimensions. It uses a localization technique based on a generic operator Λ which is used to transport the residual into L2. Under appropriate conditions on the construction of Λ, we show that it is asymptotically exact with respect to the energy norm of the error. The single layer equation and the hyper-singular equation are treated separately. While the current analysis requires the boundary to be smooth, numerical experiments show that the new error estimators also perform well for non-smooth boundaries.

An a posteriori error analysis for an optimal control problem involving the fractional Laplacian

In a previous work, we introduced a discretization scheme for a constrained optimal control problem involving the fractional Laplacian. For such a control problem, we derived optimal a priori error estimates that demand the convexity of the domain and some compatibility conditions on the data. To relax such restrictions, in this paper, we introduce and analyze an efficient and, under certain assumptions, reliable a posteriori error estimator. We realize the fractional Laplacian as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi--infinite cylinder in one more spatial dimension. This extra dimension further motivates the design of an posteriori error indicator. The latter is defined as the sum of three contributions, which come from the discretization of the state and adjoint equations and the control variable. The indicator for the state and adjoint equations relies on an anisotropic error estimator in Muckenhoupt weighted Sobolev spaces. The analysis is valid in any dimension. On the basis of the devised a posteriori error estimator, we design a simple adaptive strategy that exhibits optimal experimental rates of convergence.

Reduced basis approximation anda posteriorierror estimates for parametrized elliptic eigenvalue problems

We develop a new reduced basis (RB) method for the rapid and reliable approximation of parametrized elliptic eigenvalue problems. The method hinges upon dual weighted residual type a posteriori error indicators which estimate, for any value of the parameters, the error between the high-fidelity finite element approximation of the first eigenpair and the corresponding reduced basis approximation. The proposed error estimators are exploited not only to certify the RB approximation with respect to the high-fidelity one, but also to set up a greedy algorithm for the offline construction of a reduced basis space. Several numerical experiments show the overall validity of the proposed RB approach.

3-D local mesh refinement XFEM with variable-node hexahedron elements for extraction of stress intensity factors of straight and curved planar cracks

A novel local mesh refinement approach for fracture analysis of three-dimensional (3-D) linear elastic solids is developed, considering both 3-D straight and curved planar cracks. The present local mesh refinement formulation is a combination of the extended finite element method (XFEM), variable-node hexahedron elements, and a posteriori error indicator. Our 3-D formulation using hexahedron elements rigorously embraces a posteriori error estimation scheme, a structural coupling scale-meshes strategy and an enrichment technique. Local mesh refinement is only performed where it is needed, e.g., a vicinity of crack, through an error estimator based on the recovery stress procedure. To treat the mismatching problem induced by different scale-meshes in the domain, a structural coupling scheme employing variable-node transition hexahedron elements based on the generic point interpolation with an arbitrary number of nodes on each of their faces is presented. The 3-D finite element approximations of field variables are enhanced by enrichments so that the mesh is fully independent of the crack geometry. The displacement extrapolation method is taken for the evaluation of linear elastic fracture parameters (e.g., stress intensity factors—SIFs). To show the accuracy and performance of our proposed 3-D formulation, six numerical examples of planar 3-D straight and curved shaped cracks with single and mixed-mode fractures and different configurations are considered and analyzed. The SIFs computed by the developed method are validated with respect to analytical solutions and the ones derived from the conventional XFEM. Associated with an adaptive process, the present 3-D formulation allows the analysts to gain a desirable accuracy with a few trials, which is suited for practices purpose.

Interpolation in Jacobi-weighted spaces and its application to a posteriori error estimations of the p-version of the finite element method

The goal of this work is to introduce a local and a global interpolator in Jacobi-weighted spaces, with optimal order of approximation in the context of the p-version of finite element methods. Then, an a posteriori error indicator of the residual type is proposed for a model problem in two dimensions and, in the mathematical framework of the Jacobi-weighted spaces, the equivalence between the estimator and the error is obtained on appropriate weighted norm.

An Adaptive Finite Element Method for the Transmission Eigenvalue Problem

The classical weak formulation of the Helmholtz transmission eigenvalue problem can be linearized into an equivalent nonsymmetric eigenvalue problem. Based on this nonsymmetric eigenvalue problem, we first discuss the a posteriori error estimates and adaptive algorithm of conforming finite elements for the Helmholtz transmission eigenvalue problem. We give the a posteriori error indicators for primal eigenfunction, dual eigenfunction and eigenvalue. Theoretical analysis shows that the indicators for both primal eigenfunction and dual eigenfunction are reliable and efficient and that the indicator for eigenvalue is reliable. Numerical experiments confirm our theoretical analysis.

The backward euler anisotropic a posteriori error analysis for parabolic integro-differential equations

We derive two optimal a posteriori error estimators for an implicit fully discrete approximation to the solutions of linear integro-differential equations of the parabolic type. A continuous, piecewise linear finite element space is used for the space discretization and the time discretization is based on an implicit backward Euler method. The a posteriori error indicator corresponding to space discretization is derived using the anisotropic interpolation estimates in conjunction with a Zienkiewicz-Zhu error estimator to approach the error gradient. The error due to time discretization is derived using continuous, piecewise linear polynomial in time. We use the linear approximation of the Volterra integral term to estimate the quadrature error in the second estimator. Numerical experiments are performed on the isotropic mesh to validate the derived results.© 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1309–1330, 2016

An adaptive coupling method for exterior anisotropic elliptic problems

In this paper, we propose an adaptive coupling method for solving anisotropic elliptic PDEs in unbounded domains. Firstly, the existence and the uniqueness of the solution for the coupling method are proven, and the a priori error estimates in H1-norm and L2-norm that depend on the size of the FEM mesh, the location of the elliptic artificial boundary and the truncation of the infinite series in the artificial boundary integral condition are derived. Secondly, the a posteriori error estimates and the a posteriori error indicator of the coupling method are obtained. Finally, the adaptive coupling method refines the mesh distribution by the arc-length equidistribution principle and the a posteriori error indicator successively. Numerical examples confirm the advantage in accuracy and efficiency for the proposed method.

An Adaptive Approach for the Segmentation and the TV-Filtering in the Optic Flow Estimation

We consider a variational model for the joint total variation filtering (TV) and the segmentation in the optic flow estimation. The model is based on a functional with a spatially varying regularization parameter to solve such an ill-posed problem. We present an adaptive approach based on a posteriori error indicators which allows us to select locally the optimal values of the diffusion coefficient in the functional. We show that the adaptive process applied to the linear part of the functional (with respect to the optic flow variable) fulfills the segmentation objective. Moreover, this adaptive approach provides an approximation of the Mumford---Shah functional in the sense of the $$\Gamma $$Γ-convergence of a family of discrete energies. The simultaneous filtering and segmentation are achieved within this approach with accuracy and a reduced number of degrees of freedom, which improves each task to obtain a reliable optic flow estimation. We present some numerical simulations to show the performances of the method for the segmentation and the simultaneous segmentation-filtering.

Error estimates of spectral Legendre–Galerkin methods for the fourth-order equation in one dimension

We employ spectral Legendre–Galerkin and mixed Legendre–Galerkin approximations to solve the first bi-harmonic equation in one dimension, respectively. By orthogonal properties of Legendre polynomials, we obtain an explicit a posteriori error indicator for spectral Legendre–Galerkin methods. Furthermore, in virtue of an auxiliary variable, we present spectral mixed Legendre–Galerkin methods and study the a priori estimate and a posteriori error indicator. Especially, these indicators only depend on the expansions of the right-hand item. Numerical examples are presented to verify our theoretical analysis.

The a Posteriori Error Estimates for Chebyshev-Galerkin Spectral Methods in One Dimension

AbstractIn this paper, the Chebyshev-Galerkin spectral approximations are employed to investigate Poisson equations and the fourth order equations in one dimension. Meanwhile, p-version finite element methods with Chebyshev polynomials are utilized to solve Poisson equations. The efficient and reliable a posteriori error estimators are given for different models. Furthermore, the a priori error estimators are derived independently. Some numerical experiments are performed to verify the theoretical analysis for the a posteriori error indicators and a priori error estimations.

A posteriori error analysis for nonconforming approximation of multiple eigenvalues

In this paper, we study an a posteriori error indicator introduced in E. Dari, R.G. Durán, and C. Padra, Appl. Numer. Math., 2012, for the approximation of the Laplace eigenvalue problem with Crouzeix–Raviart nonconforming finite elements. In particular, we show that the estimator is robust also in presence of eigenvalues of multiplicity greater than one. Some numerical examples confirm the theory and illustrate the convergence of an adaptive algorithm when dealing with multiple eigenvalues. Copyright © 2015 John Wiley & Sons, Ltd.

A posteriori analysis of iterative algorithms for a nonlinear problem

A posteriori error indicators have been studied in recent years owing to their remarkable capacity to enhance both speed and accuracy in computing. This work deals with a posteriori error estimation for the finite element discretization of a nonlinear problem. For a given nonlinear equation considering finite elements we solve the discrete problem using iterative methods involving some kind of linearization. For each of them, there are actually two sources of error, namely discretization and linearization. Balancing these two errors can be very important, since it avoids performing an excessive number of iterations. Our results lead to the construction of computable upper indicators for the full error. Several numerical tests are provided to evaluate the efficiency of our indicators.

Residual-Based Adaptivity and PWDG Methods for the Helmholtz Equation

We present a study of two residual a posteriori error indicators for the plane wave discontinuous Galerkin (PWDG) method for the Helmholtz equation. In particular, we study the $h$-version of PWDG in which the number of plane wave directions per element is kept fixed. First, we use a slight modification of the appropriate a priori analysis to determine a residual indicator. Numerical tests show that this is reliable but pessimistic in that the ratio between the true error and the indicator increases as the mesh is refined. We therefore introduce a new analysis based on the observation that sufficiently many plane waves can approximate piecewise linear functions as the mesh is refined. Numerical results demonstrate an improvement in the efficiency of the indicators.

A posteriori error estimators for linear reduced‐order models using Krylov‐based integrators

SummaryReduced‐order models for linear time‐invariant dynamical systems are considered, and the error between the full‐order model and the reduced‐order model solutions is characterized. Based on the analytical representation of the error, an a posteriori error indicator is proposed that combines a Krylov‐based exponential integrator and an a posteriori residual‐based estimate. Numerical experiments illustrate the quality of the error estimator. Copyright © 2014 John Wiley & Sons, Ltd.