Local Error Estimates Research Articles

Doubly Adaptive Quadrature Routines Based on Newton–Cotes Rules

In this paper we test two recently published Matlab codes, adaptsim and adaptlob, using both a Lyness–Kaganove test and a battery type of test. Furthermore we modify these two codes using sequences of null rules in the error estimator with the intention to increase the reliability for both codes. In addition two new Matlab codes applying a locally and a globally adaptive strategy respectively are developed. These two new codes turn out to have very good properties both with respect to reliability and efficiency. Both algorithms are using sequences of null rules in their local error estimators. These error estimators allow us both to test if we are in the region of asymptotic behavior and thus increase reliability and to take advantage of the degree of precision of the basic quadrature rule. The new codes compare favorably to the two recently published adaptive codes both when we use a Lyness–Kaganove testing technique and by using a battery test.

Local error estimation for dimensionally reduced models of elliptic boundary value problems

In this paper, we present an approach to the local estimation of errors in dimensionally reduced models of elliptic boundary value problems posed on thin domains. In this approach, errors are measured in quantities of interest whose computation is typically the goal of the analysis. Explicit bounds and estimates are derived for errors in quantities characterized by continuous linear functionals. A representative numerical example showing the accuracy of the bounds and estimates is also presented.

Local residual error estimator and adaptive finite element analysis of Poisson’s problems

A local residual error estimator based on the least-square residuals of the balance equation, constitutive relation and irrotationality condition is presented to enable adaptive finite element analyses of Poisson’s problems. The efficiency of the residual error indicator and the quality of the L 2 error indicator are investigated in problems with sharp and relatively sharp concentration of gradients. Both error estimators are highly dependent on the accuracy of the recovered or post-processed derivatives. Two types of post-processing are adopted: the L 2 projection and a local macroelement technique. In the proposed macroelement post-processing technique, the derivatives are recovered by solving local variational problems involving the residuals of the balance equation irrotationality condition and constitutive relation at special superconvergence points.

Adaptive Grid Refinement in Numerical Models for Water Flow and Chemical Transport in Soil: A Review

Insufficient spatial or temporal resolution is a common source of errors in numerical solutions for both water flow and solute transport in the variably unsaturated vadose zone. Evaporation near the surface, as well as infiltration into initially dry soil profiles, typically create mobile local regions with large gradients of the pressure head. Convection-dominant transport of solutes during water flow in soil also tends to create steep moving fronts of concentration with large localized concentration gradients. Groundwater flow and solute transport in highly heterogeneous aquifers similarly tend to be preferentially channeled through regions of high flow rates. Without due consideration of special resolution requirements for such critical cases of flow and transport, simulations using traditional finite difference (FDM) and finite element (FEM) numerical methods typically provide inaccurate solutions characterized by undesirable features such as oscillation and numerical dispersion. Incorporation of local adaptive grid refinement (LAGR) algorithms in numerical models for solving such cases is an effective approach that has been used to provide accurate numerical approximations by automated adjustment of local spatial resolution. Local error estimates are typically utilized to optimize spatial resolution. Definite advantages, as well as some limitations, exist for using LAGR algorithms in FDM and FEM numerical models for flow and transport in soils.

Adaptive Grid Refinement in Numerical Models for Water Flow and Chemical Transport in Soil: A Review

Insufficient spatial or temporal resolution is a common source of errors in numerical solutions for both water flow and solute transport in the variably unsaturated vadose zone. Evaporation near the surface, as well as infiltration into initially dry soil profiles, typically create mobile local regions with large gradients of the pressure head. Convection‐dominant transport of solutes during water flow in soil also tends to create steep moving fronts of concentration with large localized concentration gradients. Groundwater flow and solute transport in highly heterogeneous aquifers similarly tend to be preferentially channeled through regions of high flow rates. Without due consideration of special resolution requirements for such critical cases of flow and transport, simulations using traditional finite difference (FDM) and finite element (FEM) numerical methods typically provide inaccurate solutions characterized by undesirable features such as oscillation and numerical dispersion. Incorporation of local adaptive grid refinement (LAGR) algorithms in numerical models for solving such cases is an effective approach that has been used to provide accurate numerical approximations by automated adjustment of local spatial resolution. Local error estimates are typically utilized to optimize spatial resolution. Definite advantages, as well as some limitations, exist for using LAGR algorithms in FDM and FEM numerical models for flow and transport in soils.

New approaches for error estimation and adaptivity for 2D potential boundary element methods

AbstractThis work presents two new error estimation approaches for the BEM applied to 2D potential problems. The first approach involves a local error estimator based on a gradient recovery procedure in which the error function is generated from differences between smoothed and non‐smoothed rates of change of boundary variables in the local tangential direction. The second approach involves the external problem formulation and gives both local and global measures of error, depending on a choice of the external evaluation point. These approaches are post‐processing procedures. Both estimators show consistency with mesh refinement and give similar qualitative results. The error estimator using the gradient recovery approach is more general, as this formulation does not rely on an ‘optimal’ choice of an external parameter. This work presents also the use of a local error estimator in an adaptive mesh refinement procedure. This r‐refinement approach is based on the minimization of the standard deviation of the local error estimate. A non‐linear programming procedure using a feasible‐point method is employed using Lagrange multipliers and a set of active constraints. The optimization procedure produces finer meshes close to a singularity and results that are consistent with the problem physics. Copyright © 2002 John Wiley & Sons, Ltd.

Local and Parallel Finite Element Algorithms for Eigenvalue Problems

Some new local and parallel finite element algorithms are proposed and analyzed in this paper for eigenvalue problems. With these algorithms, the solution of an eigenvalue problem on a fine grid is reduced to the solution of an eigenvalue problem on a relatively coarse grid together with solutions of some linear algebraic systems on fine grid by using some local and parallel procedure. A theoretical tool for analyzing these algorithms is some local error estimate that is also obtained in this paper for finite element approximations of eigenvectors on general shape-regular grids.

Derivation and implementation of Two-Step Runge-Kutta pairs

Explicit Runge-Kutta pairs are known to provide efficient solutions to initial value differential equations with inexpensive derivative evaluations. Two-step Runge-Kutta methods strive to improve the efficiency by utilizing approximations to the solution and its derivatives from the previous step. This article suggests a strategy for computing embeddedpairs of such two-step methods using a smaller number of function evaluations than that required for traditional Runge-Kutta methods of the same order. This leads to the efficient and reliable estimation of local discretization error and a robust step control strategy. The change of stepsize is achieved by a suitable interpolation of stage values from the previous step and does not require any additional function evaluations. Two examples illustrate the features of these pairs.

Adaptive boundary element method of time-harmonic exterior acoustics in two dimensions

In this paper we carry out boundary element computations of the Helmholtz equation in two dimensions, in the context of time-harmonic exterior acoustics. The purpose is to demonstrate cost savings engendered through adaptivity for propagating solutions at moderate wave numbers. The computation are performed on meshes of constant boundary elements, and are adapted to the solution by locally changing element sizes (h-version). Burton and Miller approach is employed to solve the exterior problems for all wave numbers. Two error indicators obtained from the dual integral equations in conjunction with the exact error indicator are used for local error estimation, which are essential ingredients for all adaptive mesh schemes in BEM. Computational experiments are performed for the two-dimensional exterior acoustics. The three error tracking curves are in good agreement with their shapes. Three examples show that the adaptive mesh based on the error indicators converge to the exact solution more efficiently using the same number of elements than does uniform mesh discretization.

Extension Theorems for Spaces Arising from Approximation by Translates of a Basic Function

We establish extension theorems for functions in spaces which arise naturally in studying interpolation by radial basic functions. These spaces are akin in some way to the non-integer-valued Sobolev spaces, although they are considerably more general. Such extensions allow us to establish local error estimates in a way which we make precise in the introductory section of our paper. There are many other applications of these fundamental results, including improved Lp error estimates for interpolation by shifts of a single basic function, but these applications have been left to a later paper.

Grid-free adaptive semi-Lagrangian advection using radial basis functions

This paper proposes a new grid-free adaptive advection scheme. The resulting algorithm is a combination of the semi-Lagrangian method (SLM) and the grid-free radial basis function interpolation (RBF). The set of scattered interpolation nodes is subject to dynamic changes at run time. Based on a posteriori local error estimates, a self-adaptive local refinement and coarsening of the nodes serves to obtain enhanced accuracy at reasonable computational costs. Due to well-known features of SLM and RBF, the method is guaranteed to be stable, it has good approximation behaviour, and it works for arbitrary space dimension. Numerical examples in two dimensions illustrate the performance of the method in comparison with existing grid-based advection schemes.

A Wigner-Measure Analysis of the Dufort--Frankel Scheme for the Schrödinger Equation

We apply Wigner transform techniques to the analysis of the Dufort--Frankel difference scheme for the Schrodinger equation and to the continuous analogue of the scheme in the case of a small (scaled) Planck constant (semiclassical regime). In this way we are able to obtain sharp conditions on the spatial-temporal grid which guarantee convergence for average values of observables as the Planck constant tends to zero. The theory developed in this paper is not based on local and global error estimates and does not depend on whether or not caustics develop. Numerical test examples are presented to help interpret the theory and to compare the Dufort--Frankel scheme to other difference schemes for the Schr{odinger equation.

Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials: Part II: a computational environment for adaptive modeling of heterogeneous elastic solids

This paper addresses a classical and largely unsolved problem: given a structural component constructed of a heterogeneous elastic material that is in equilibrium under the action of applied loads, determine local micromechanical features of its response (e.g., local stresses and displacements in or around phase boundaries or in inclusions) to an arbitrary preset level of accuracy, it being understood that the microstructure is a priori unknown, may be randomly distributed, may exist at multiple spatial scales, and may contain millions, even billions, of microscale components. The approach described in this work begins with a mathematical abstraction of this problem in which the material body is modeled as an elastic solid with highly variable, possibly randomly distributed, elastic properties. Information on the actual character of the microstructure of given material bodies is determined by computerized tomography (CT) imaging. A procedure is given for determining the effective material properties from imaging data, using either deterministic or stochastic methods. An algorithm is then described for determining local quantities of interest, such as average stresses on inclusion boundaries, to arbitrary accuracy relative to the fine-scale model. A new computational environment for implementing such analyses is presented which employs parallel, adaptive, hp finite element methods, CT interfaces, automatic meshing procedures, and, effectively, adaptive modeling schemes. Within the basic premises on which the approach is based, results of any specified accuracy can be obtained, independently of the number of microscale components and constituents. The results of several numerical experiments are presented.

Some mixed finite element methods on anisotropic meshes

The paper deals with some mixed finite element methods on a class of anisotropic meshes based on tetrahedra and prismatic (pentahedral) elements. Anisotropic local interpolation error estimates are derived in some anisotropic weighted Sobolev spaces. As particular applications, the numerical approximation by mixed methods of the Laplace equation in domains with edges is investigated where anisotropic finite element meshes are appropriate. Optimal error estimates are obtained using some anisotropic regularity results of the solutions.

Local error estimates of FEM for displacements and stresses in linear elasticity by solving local Neumann problems

AbstractIn this paper we present two types of local error estimators for the primal finite‐element‐method (FEM) by duality arguments. They are first derived from the (explicit) residual error estimation method (REM) and then—as a new contribution—from the (implicit) posterior equilibrium method (PEM) using improved boundary tractions, gained by local post‐processing with local Neumann problems, with applications in elastic problems. For the displacements a local error estimator with an upper bound is derived and also a local estimator for stresses. Furthermore—for better numerical efficiency—the residua are projected energy‐invariant onto reference elements, where the local Neumann problems have to be solved. Comparative examples between REM‐ and PEM‐type local estimators show superior effectivity indices for the latter one. Copyright © 2001 John Wiley & Sons, Ltd.

Restricted Padé scheme in computational structural dynamics

Restricted Padé schemes R mp where the denominator is a complete power p of a linear function have been examined analytically by Nörsett. Here this method is evaluated with paying special attention to numerical aspects when solving typical finite-element systems in structural dynamics with n degrees of freedom and with essential properties like sparsety and symmetry. This paper presents an efficient implicit restricted time-stepping scheme R 22 with a final coefficient matrix of order n and a leading error term proportional to h 4. This is a significant improvement compared with the most popular Padé- P 11 -Newmark scheme with matrix order n, too, but error term h 3 on the one hand and the well established Padé- P 12 -discontinuous Hamiltonian approach with error term h 4 but matrix order 2× n on the other hand. The classical P 11 -scheme and the new restricted R 22 -scheme are compared in a modal manner and by assessing the correlation between local error estimator and global error.

Crouzeix-Raviart type finite elements on anisotropic meshes

The paper deals with a non-conforming finite element method on a class of anisotropic meshes. The Crouzeix-Raviart element is used on triangles and tetrahedra. For rectangles and prismatic (pentahedral) elements a novel set of trial functions is proposed. Anisotropic local interpolation error estimates are derived for all these types of element and for functions from classical and weighted Sobolev spaces. The consistency error is estimated for a general differential equation under weak regularity assumptions. As a particular application, an example is investigated where anisotropic finite element meshes are appropriate, namely the Poisson problem in domains with edges. A numerical test is described.

Enhancing structural topology optimization

The goal of structural optimization is to find the best possible configuration that minimizes the objective function and satisfies a set of constraints. Here we present a method based on the evolutionary structural optimization method, where the quality of the solution is improved by avoiding the chain‐like sets of elements which are sources of potential kinematic instabilities, and by including local error estimators. Both of these enhancements are employed to activate refining the mesh so as to obtain accurate and stable solutions as the volume removal proceeds. Several related contributions of Professor E. Hinton are cited.

Increasing reliability of Gauss–Kronrod quadrature by Eratosthenes' sieve method

The reliability of the local error estimates returned by the Gauss–Kronrod quadrature rules can be raised up to the theoretical 100% rate of success, under error estimate sharpening, provided a number of natural validating conditions are required. The self-validating scheme of the local error estimates, which is easy to implement and adds little supplementary computing effort, strengthens considerably the correctness of the decisions within the automatic adaptive quadrature.

A Local Problem Error Estimator for Anisotropic Tetrahedral Finite Element Meshes

The Poisson problem is solved by the finite element method on anisotropic tetrahedral or triangular meshes. The focus is on adaptive algorithms and, in particular, on a posteriori error estimators based on the solution of a local problem. On anisotropic meshes, such estimators cannot be analyzed in the common way known from isotropic meshes. The first estimator proposed here is a slight modification of a familiar isotropic counterpart. By a rigorous analysis it is proven that this estimator is equivalent to a known anisotropic residual error estimator. Hence the local problem error estimator also yields reliable upper and lower error bounds on anisotropic meshes. The local problems are shown to be well-conditioned. Two further local problem error estimators originally defined for isotropic meshes are investigated in the anisotropic context here. The results reveal the significance of the proper choice of local problem. Furthermore the analysis covers Dirichlet, Neumann, and Robin boundary conditions. Particular attention is paid to the Robin boundary conditions since they require a different treatment than the Neumann boundary conditions. A numerical example supports the error analysis.