A posteriori finite element error estimators for indefinite elliptic boundary value problems ∗

A general construction technique is presented for a posteriori error estimators of finite element solutions of elliptic boundary value problems that satisfy a Gång inequality. The estimators are obtained by an element–by–element solution of ‘weak residual’ with or without considering element boundary residuals. There is no order restriction on the finite element spaces used for the approximate solution or the error estimation; that is, the design of the estimators is applicable in connection with either one of the h–p–, or hp– formulations of the finite element method. Under suitable assumptions it is shown that the estimators are bounded by constant multiples of the true error in a suitable norm. Some numerical results are given to demonstrate the effectiveness and efficiency of the approach.     

A posteriori error estimates for hp finite element solutions of convex optimal control problems

In this paper, we present a posteriori error analysis for hp finite element approximation of convex optimal control problems. We derive a new quasi-interpolation operator of Clément type and a new quasi-interpolation operator of Scott-Zhang type that preserves homogeneous boundary condition. The Scott-Zhang type quasi-interpolation is suitable for an application in bounding the errors in L²-norm. Then hp a posteriori error estimators are obtained for the coupled state and control approximations. Such estimators can be used to construct reliable adaptive finite elements for the control problems.     

A posteriori error estimators in the finite element method

Summary In the paper we develop a structured approach to thea posteriori estimation of the error in the approximation obtained via the finite element method. This aids the classification of existing estimators as well as allowing new estimators to be proposed for new situations. A class of abstract estimators for finite elements of orderp>1 in ,n=2, 3 based on exploiting the superconvergence phenomenon are analyzed.     

Interpolation error estimates in W 1,p for degenerate Q 1 isoparametric elements

Optimal order error estimates in H ¹, for the Q ₁ isoparametric interpolation were obtained in Acosta and Durán (SIAM J Numer Anal37, 18–36, 1999) for a very general class of degenerate convex quadrilateral elements. In this work we show that the same conlusions are valid in W ^1,p for 1≤ p < 3 and we give a counterexample for the case p ≥ 3, showing that the result cannot be generalized for more regular functions. Despite this fact, we show that optimal order error estimates are valid for any p ≥ 1, keeping the interior angles of the element bounded away from 0 and π, independently of the aspect ratio. We also show that the restriction on the maximum angle is sharp for p ≥ 3.     

<Emphasis Type="Italic">hp</Emphasis>-Discontinuous Galerkin methods for strongly nonlinear elliptic boundary value problems

In this paper, we have analyzed a one parameter family of hp-discontinuous Galerkin methods for strongly nonlinear elliptic boundary value problems with Dirichlet boundary conditions. These methods depend on the values of the parameter , where θ = + 1 corresponds to the nonsymmetric and θ = −1 corresponds to the symmetric interior penalty methods when and f(u,∇u) = −f, that is, for the Poisson problem. The error estimate in the broken H ¹ norm, which is optimal in h (mesh size) and suboptimal in p (degree of approximation) is derived using piecewise polynomials of degree p ≥ 2, when the solution . In the case of linear elliptic problems also, this estimate is optimal in h and suboptimal in p. Further, optimal error estimate in the L ² norm when θ = −1 is derived. Numerical experiments are presented to illustrate the theoretical results. Supported by DST-DAAD (PPP-05) project.     

Besov regularity and new error estimates for finite volume approximations of the p-laplacian

Summary. In [1], we have constructed a family of finite volume schemes on rectangular meshes for the p-laplacian and we proved error estimates in case the exact solution lies in W^2,p. Actually, W^2,p is not a natural space for solutions of the p-laplacian in the case p>2. Indeed, for general L^p data it can be shown that the solution only belongs to the Besov space In this paper, we prove Besov kind a priori estimates on the approximate solution for any data in L^p. We then obtain new error estimates for such solutions in the case of uniform meshes     

A p-version finite element method for nonlinear elliptic variational inequalities in 2D

This article introduces and analyzes a p-version FEM for variational inequalities resulting from obstacle problems for some quasi-linear elliptic partial differential operators. We approximate the solution by controlling the obstacle condition in images of the Gauss–Lobatto points. We show existence and uniqueness for the discrete solution u _p from the p-version for the obstacle problem. We prove the convergence of u _p towards the solution with respect to the energy norm, and assuming some additional regularity for the solution we derive an a priori error estimate. In numerical experiments the p-version turns out to be superior to the h-version concerning the convergence rate and the number of unknowns needed to achieve a certain exactness of the approximation.     

ELLIPTIC H^2-VOLTERRA PROJECTION AND THE H^1-GALERKIN METHODS FOR THE INTEGRO-DIFFERENTIAL EQUATIONS OF EVOLUTION

This paper studies H^1-Galerkin methods for the integro-differential equations of evolution. The elliptic H^2-Volterra projection is induced and then used in the derivations of error estimates for semi-discrete and full-discrete H^1-Galerkin methods.The optimal L^2, H^1 and H^2 norm error estimates are obtained.     

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.     

Quadratic spline collocation methods for elliptic partial differential equations

We consider Quadratic Spline Collocation (QSC) methods for linear second order elliptic Partial Differential Equations (PDEs). The standard formulation of these methods leads to non-optimal approximations. In order to derive optimal QSC approximations, high order perturbations of the PDE problem are generated. These perturbations can be applied either to the PDE problem operators or to the right sides, thus leading to two different formulations of optimal QSC methods. The convergence properties of the QSC methods are studied. OptimalO(h ^3–j ) global error estimates for thejth partial derivative are obtained for a certain class of problems. Moreover,O(h ^4–j ) error bounds for thejth partial derivative are obtained at certain sets of points. Results from numerical experiments verify the theoretical behaviour of the QSC methods. Performance results also show that the QSC methods are very effective from the computational point of view. They have been implemented efficiently on parallel machines.This research was supported in part by David Ross Foundation (U.S.A) and NSERC (Natural Sciences and Engineering Research Council of Canada).     

A posteriori error estimators for the first-order least-squares finite element method

In this paper, we propose a posteriori error estimators for certain quantities of interest for a first-order least-squares finite element method. In particular, we propose an a posteriori error estimator for when one is interested in where . Our a posteriori error estimators are obtained by assigning proper weight (in terms of local mesh size h_T) to the terms of the least-squares functional. An a posteriori error analysis yields reliable and efficient estimates based on residuals. Numerical examples are presented to show the effectivity of our error estimators.     

L p estimates of boundary integral equations for some nonlinear boundary value problems

Summary Recently, Galerkin and collocation methods have been analyzed for boundary integral equation formulations of some potential problems in the plane with nonlinear boundary conditions, and stability results and error estimates in theH ^1/2-norm have been proved (Ruotsalainen and Wendland, and Ruotsalainen and Saranen). We show that these results extend toL ^p setting without any extra conditions. These extensions are proved by studying the uniform boundedness of the inverses of the linearized integral operators, and then considering the nonlinear equations. The fact that inH ^1/2 setting the nonlinear operator is a homeomorphism with Lipschitz continuous inverse plays a crucial role. Optimal error estimates for the Galerkin and collocation method inL ^p space then follow.This research was performed while the second author was visiting professor at the University of Delaware, spring 1989     

Error estimators for advection–reaction–diffusion equations based on the solution of local problems

This paper deals with a posteriori error estimates for advection–reaction–diffusion equations. In particular, error estimators based on the solution of local problems are derived for a stabilized finite element method. These estimators are proved to be equivalent to the error, with equivalence constants eventually depending on the physical parameters. Numerical experiments illustrating the performance of this approach are reported.     

Error estimates for a mixed finite volume method for the p-Laplacian problem

In this work we propose and analyze a mixed finite volume method for the p-Laplacian problem which is based on the lowest order Raviart–Thomas element for the vector variable and the P1 nonconforming element for the scalar variable. It is shown that this method can be reduced to a P1 nonconforming finite element method for the scalar variable only. One can then recover the vector approximation from the computed scalar approximation in a virtually cost-free manner. Optimal a priori error estimates are proved for both approximations by the quasi-norm techniques. We also derive an implicit error estimator of Bank–Weiser type which is based on the local Neumann problems.This work was supported by the Post-doctoral Fellowship Program of Korea Science & Engineering Foundation (KOSEF).     

Operator-splitting galerkin methods for one kind of oin reaction model for the pollution in groundwater

Abstract. The operator-splitting methods for the mathematic model of one kind of oin reactionsfor the problem of groundwater are considered. Optimal error estimates in Lz and Hl norm areobtained for the approximation solution.     

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.     

On computing the pressure by thep version of the finite element method for Stokes problem

Summary This paper introduces and analyzes two ways of extracting the hydrostatic pressure when solving Stokes problem using thep version of the finite element method. When one uses a localH ¹ projection, we show that optimal rates of convergence for the pressure approximation is achieved. When the pressure is not inH ¹. or the value of the pressure is only needed at a few points, one may extract the pressure pointwise using e.g. a single layer potential recovery. Negative, zero, and higher norm estimates for the Stokes velocity are derived within the framework of thep version of the F.E.M.Partially supported by ONR grants N00014-87-K-0427 and N00014-90-J-1238     

A posteriori error estimates of constrained optimal control problem governed by convection diffusion equations

In this paper, we study a posteriori error estimates of the edge stabilization Galerkin method for the constrained optimal control problem governed by convection-dominated diffusion equations. The residual-type a posteriori error estimators yield both upper and lower bounds for control u measured in L ²-norm and for state y and costate p measured in energy norm. Two numerical examples are presented to illustrate the effectiveness of the error estimators provided in this paper.      

A posteriori error estimates for mixed finite element approximations of elliptic problems

We derive residual based a posteriori error estimates of the flux in L ²-norm for a general class of mixed methods for elliptic problems. The estimate is applicable to standard mixed methods such as the Raviart–Thomas–Nedelec and Brezzi–Douglas–Marini elements, as well as stabilized methods such as the Galerkin-Least squares method. The element residual in the estimate employs an elementwise computable postprocessed approximation of the displacement which gives optimal order.     

The influence and selection of subspaces for a posteriori error estimators

Summary. The element residual method for a posteriori error estimation is analyzed for degree finite element approximation on quadrilateral elements. The influence of the choice of subspace used to solve the element residual problem is studied. It is shown that the resulting estimators will be consistent (or asymptotically exact) for all if and only if the mesh is parallel. Moreover, even if the mesh consists of rectangles, then the estimators can be inconsistent when . The results provide concrete guidelines for the selection of a posteriori error estimators and establish the limits of their performance. In particular, the use of the element residual method for high orders of approximation (such as those arising in the - version finite element method) is vindicated. The mechanism behind the rather poor performance of the estimators is traced back to the basic formulation of the residual problem. The investigations reveal a deficiency in the formulation, leading, as it does, to spurious modes in the true solution of the residual problem. The recommended choice of subspaces may be viewed as being sufficient to guarantee that the spurious modes are filtered out from the approximate solution while at the same time retaining a sufficient degree of approximation to represent the true modes. Received February 27, 1995 / Revised version received June 7, 1995