A posteriori error estimates for elliptic problems with Dirac delta source terms

The aim of this paper is to introduce residual type a posteriori error estimators for a Poisson problem with a Dirac delta source term, in L ^p norm and W^1,p seminorm. The estimators are proved to yield global upper and local lower bounds for the corresponding norms of the error. They are used to guide adaptive procedures, which are experimentally shown to lead to optimal orders of convergence.     

A posteriori error analysis of the discontinuous finite element methods for first order hyperbolic problems

In this paper, we consider the a posteriori error analysis of discontinuous Galerkin finite element methods for the steady and nonsteady first order hyperbolic problems with inflow boundary conditions. We establish several residual-based a posteriori error estimators which provide global upper bounds and a local lower bound on the error. Further, for nonsteady problem, we construct a fully discrete discontinuous finite element scheme and derive the a posteriori error estimators which yield global upper bound on the error in time and space. Our a posteriori error analysis is based on the mesh-dependent a priori estimates for the first order hyperbolic problems. These a posteriori error analysis results can be applied to develop the adaptive discontinuous finite element methods.     

Robust a posteriori error estimators for a singularly perturbed reaction-diffusion equation

We derive robust a posteriori error estimators for a singularly perturbed reaction-diffusion equation. Here, robust means that the estimators yield global upper and local lower bounds on the error measured in the energy norm such that the ratio of the upper and lower bounds is bounded from below and from above by constants which do neither depend on any meshsize nor on the perturbation parameter. The estimators are based either on the evaluation of local residuals or on the solution of discrete local Dirichlet or Neumann problems. Received June 5, 1996     

A posteriori error estimates for H 1-Galerkin mixed finite-element method for parabolic problems

The purpose of this article is to derive a posteriori error estimates for the H ¹-Galerkin mixed finite element method for parabolic problems. We study both semidiscrete and fully discrete a posteriori error analyses using standard energy argument. A fully discrete a posteriori error analysis based on the backward Euler method is analysed and upper bounds for the errors are derived. The estimators yield upper bounds for the errors which are global in space and time. Our analysis is based on residual approach and the estimators are free from edge residuals.     

A posteriori error estimators for convection-diffusion equations

Summary. We derive a posteriori error estimators for convection-diffusion equations with dominant convection. The estimators yield global upper and local lower bounds on the error measured in the energy norm such that the ratio of the upper and lower bounds only depends on the local mesh-Peclet number. The estimators are either based on the evaluation of local residuals or on the solution of discrete local Dirichlet or Neumann problems. Received February 10, 1997 / Revised version received November 4, 1997     

A posteriori error estimation for fully discrete hierarchic models of elliptic boundary value problems on thin domains

Summary. A posteriori error estimators for fully discrete hierarchic modelling on thin domains are derived and are shown to provide computable upper bounds on the discretization error and on the total error. The estimators are shown to be robust and do not degenerate as the thickness of the domain tends to zero. If the discretization part of the error is negligible, the estimator for the modelling error reduces to the one recently obtained for semi-discrete hierarchical modelling by Babuska and Schwab. Received July 25, 1996 / Revised version received July 31, 1997     

A posteriori error estimation and adaptive mesh-refinement techniques

We analyse three different a posteriori error estimators for elliptic partial differential equations. They are based on the evaluation of local residuals with respect to the strong form of the differential equation, on the solution of local problems with Neumann boundary conditions, and on the solution of local problems with Dirichlet boundary conditions. We prove that all three are equivalent and yield global upper and local lower bounds for the true error. Thus adaptive mesh-refinement techniques based on these estimators are capable to detect local singularities of the solution and to appropriately refine the grid near these singularities. Some numerical examples prove the efficiency of the error estimators and the mesh-refinement techniques.     

Local a posteriori error estimates and adaptive control of pollution effects

Local a posteriori error estimators are derived for linear elliptic problems over general polygonal domains in 2d. The estimators lead to a sharp upper bound for the energy error in a local region of interest. This upper bound consists of H¹‐type local error indicators in a slightly larger subdomain, plus weighted L²‐type local error indicators outside this subdomain, which account for the pollution effects. This constitutes the basis of a local adaptive refinement procedure. Numerical experiments show a superior performance than the standard global procedure as well as the generation of locally quasi‐optimal meshes. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 421–442, 2003     

Guaranteed a posteriori error estimates for nonconforming finite element approximations to a singularly perturbed reaction–diffusion problem

We derive guaranteed a posteriori error estimates for nonconforming finite element approximations to a singularly perturbed reaction–diffusion problem. First, an abstract a posteriori error bound is derived under a special equilibration condition. Based on conservative flux reconstruction, two error estimators are proposed and provide actual upper error bounds in the usual energy norm without unknown constants, one of which can be directly constructed without solving local Neumann problems and provide practical computable error bounds. The error estimators also provide local lower bounds but with the multiplicative constants dependent on the diffusion coefficient and mesh size, where the constants can be bounded for enough small mesh size comparable with the square root of the diffusion coefficient. By adding edge jumps with weights to the energy norm, two modified error estimators with additional edge tangential jumps are shown to be robust with respect to the diffusion coefficient and provide guaranteed upper bounds on the error in the modified norm. Finally, the performance of the estimators are illustrated by the numerical results.     

A posteriori error estimators for nonconforming finite element methods of the linear elasticity problem

In this work we derive and analyze a posteriori error estimators for low-order nonconforming finite element methods of the linear elasticity problem on both triangular and quadrilateral meshes, with hanging nodes allowed for local mesh refinement. First, it is shown that equilibrated Neumann data on interelement boundaries are simply given by the local weak residuals of the numerical solution. The first error estimator is then obtained by applying the equilibrated residual method with this set of Neumann data. From this implicit estimator we also derive two explicit error estimators, one of which is similar to the one proposed by Dörfler and Ainsworth (2005) [24] for the Stokes problem. It is established that all these error estimators are reliable and efficient in a robust way with respect to the Lamé constants. The main advantage of our error estimators is that they yield guaranteed, i.e., constant-free upper bounds for the energy-like error (up to higher order terms due to data oscillation) when a good estimate for the inf-sup constant is available, which is confirmed by some numerical results.     

Discontinuous Galerkin finite element method for a nonlinear boundary value problem

In this paper, we investigate the a priori and a posteriori error estimates for the discontinuous Galerkin finite element approximation to a regularization version of the variational inequality of the second kind. We show the optimal error estimates in the DG-norm （stronger than the H1 norm） and the L2 norm, respectively. Furthermore, some residual-based a posteriori error estimators are established which provide global upper bounds and local lower bounds on the discretization error. These a posteriori analysis results can be applied to develop the adaptive DG methods.     

A type of new posteriori error estimators for stokes problems

1. IntroductionIn the numerical approximation of PDE, it is often very importals to detect regionswhere the accuracy of the numerical solution is degraded by local singularities of the solutionof the continuous problem such as the singularity near the re-entrant corller. An obviousremedy is to refine the discretization in the critical regions, i.e., to place more gridpointswhere the solution is less regular. The question is how to identify these regions automdticallyand how to determine a goo…     

Sharp a posteriori error estimate for elliptic equation with singular data

We introduce two residual type a posteriori error estimators for second-order elliptic partial differential equations with its right-hand side in L ^p (1 < p ⩽ 2) space. Both estimators are proved to yield global upper and local lower bounds for the W ^1,p seminorm of the error. We adopt the estimators as the indicators in h-mesh adaptive method to solve two typical model problems. It is verified by the numerical results that the estimators lead to optimal orders of convergence.     

Local a posteriori error estimators for variational inequalities

Local a posteriori error estimators for finite element approximation of variational inequalities are derived. These are shown to provide upper bounds on the discretization error. Numerical examples are given illustrating the theoretical results. © 1993 John Wiley & Sons, Inc.     

Global Error Estimators for Order 7, 8 Runge–Kutta Pairs

Dormand, Prince and their colleagues [3–5] showed in a sequence of papers that the approximation of an initial value differential system propagated by a Runge–Kutta pair, together with a continuous approximation obtained using additional derivative values could be utilized to obtain estimates of the global error. They illustrated the results using pairs of orders p–1 and p for several values of p. The current authors [13] have developed a more direct representation of the order conditions, characterized families of global error estimators for Runge–Kutta pairs of arbitrary values of p, and showed that efficient global error estimating Runge–Kutta methods are based on the nodes of a Lobatto quadrature formula. Here, formulas for a good 7, 8 pair, interpolants of each of orders 7 and 8, and global error estimators of orders 10 and 12 illustrate how to obtain global error estimates of orders 9, 10, or 11, for arbitrary initial value systems. One set of graphs indicates that the stated order of the global error estimators is achieved numerically, and a second set illustrates the relative efficiency for several global error estimators when the approximation is propagated with a variable stepsize.     

Adaptation under probabilistic error for estimating linear functionals

The problem of estimating linear functionals based on Gaussian observations is considered. Probabilistic error is used as a measure of accuracy and attention is focused on the construction of adaptive estimators which are simultaneously near optimal under probabilistic error over a collection of convex parameter spaces. In contrast to mean squared error it is shown that fully rate optimal adaptive estimators can be constructed for probabilistic error. A general construction of such estimators is provided and examples are given to illustrate the general theory.     

A posteriori FE error control for p-Laplacian by gradient recovery in quasi-norm

A posteriori error estimators based on quasi-norm gradient recovery are established for the finite element approximation of the p-Laplacian on unstructured meshes. The new a posteriori error estimators provide both upper and lower bounds in the quasi-norm for the discretization error. The main tools for the proofs of reliability are approximation error estimates for a local approximation operator in the quasi-norm.

     

Optimal Control of Population Dynamics

This paper studies the role of projection algorithms in conditional set membership estimation. These algorithms are known to be suboptimal in terms of the worst-case estimation error. A tight upper bound on the error of central projection estimators and interpolatory projection estimators is computed as a function of the conditional radius of information. Since the radius of information represents the minimum achievable error, the derived bound provides a measure of the reliability level of the suboptimal algorithms. The results are derived in a general deterministic setting, which allows the consideration of linearly parametrized approximations of a compact set of feasible problem elements.     

A posteriori error estimates of stabilized finite volume method for the Stokes equations

In this work, the residual‐type posteriori error estimates of stabilized finite volume method are studied for the steady Stokes problem based on two local Gauss integrations. By using the residuals between the source term and numerical solutions, the computable global upper and local lower bounds for the errors of velocity in H¹ norm and pressure in L² norm are derived. Furthermore, a global upper bound of u ? u_h in L²‐norm is also derived. Finally, some numerical experiments are provided to verify the performances of the established error estimators. Copyright © 2015 John Wiley & Sons, Ltd.     

AN ADAPTIVE VERSION OF GLIMM＇S SCHEME

This article describes a local error estimator for Glimm＇s scheme for hyperbolic systems of conservation laws and uses it to replace the usual random choice in Glimm＇s scheme by an optimal choice. As a by-product of the local error estimator, the procedure provides a global error estimator that is shown numerically to be a very accurate estimate of the error in L1 （R） for all times. Although there is partial mathematical evidence for the error estimator proposed, at this stage the error estimator must be considered ad- hoc. Nonetheless, the error estimator is simple to compute, relatively inexpensive, without adjustable parameters and at least as accurate as other existing error estimators. Numerical experiments in 1-D for Burgers＇ equation and for Euler＇s system are performed to measure the asymptotic accuracy of the resulting scheme and of the error estimator.