Goal-Oriented Error Estimates Based on Different FE-Spaces for the Primal and the Dual Problem with Applications to Fracture Mechanics

Abstract

The objective of this paper is to derive goal-oriented a posteriori error estimators for the error obtained while approximately evaluating the nonlinear J -integral as a fracture criterion in linear elastic fracture mechanics (LEFM) using the finite element method. Such error estimators are based on the well-established technique of solving an auxiliary dual problem. In a straightforward fashion, the solution to the discretized dual problem is sought in the same FEspace as the solution to the original (primal) problem, i.e. on the same mesh, although it merely acts as a weight of the discretization error only. In this paper, we follow the strategy recently proposed by Korotov et al. [J Numer Math 11:33-59, 2003; Comp Lett (in press)] and derive goal-oriented error estimators of the averaging type, where the discrete dual solution is computed on a different mesh than the primal solution. On doing so, the FE-solutions to the primal and the dual problems need to be transferred from one mesh to the other. The necessary algorithms are briefly explained and finally some illustrative numerical examples are presented.

Two-sided a posteriori error estimates for linear elliptic problems with mixed boundary conditions

“…Such a verification is the main purpose of a posteriori error estimation methods. Several approaches for deriving various a posteriori estimates for elliptic problems for errors measured in global (energy) norms ( [1], [2], [3], [10], [13], [19], [24], [31], [32], [33]), or in terms of various local quantities ( [4], [8], [11], [12], [16], [17], [23], [25], [29], [30]) have been suggested (see also references in the above mentioned works).…”

Sharp upper global a posteriori error estimates for nonlinear elliptic variational problems

“…For piecewise linear FEM, if (4.24) is solved numerically on the same mesh as used for u h , then its right-hand side −div z * = −div G h (∇u h ) is constant on each element, hence it requires minimal numerical integration and is therefore a cheap auxiliary problem. On the other hand, using a finer (or just different) mesh for (4.24) than the one used for u h may considerably increase the accuracy of the estimate, similarly as for adjoint problems for linear equations [19,25] (see also [45]), with low extra cost due to the linearity of (4.24).…”

“…(b) Interface problems. Let Γ int be a piecewise smooth surface lying in the interior of Ω, and let us consider the problem 45) provided that the exact solution satisfies u…”

“…Many physical phenomena can be described by means of mathematical models presenting linear and nonlinear boundary value problems of elliptic type [10,13,28,36,45]. Various numerical techniques (such as the finite difference method, the finite element method, the finite volume method etc) are well developed for finding approximate solutions for such problems, see, e.g., [10,28] and references therein.…”

“…Such a verification is the main purpose of a posteriori error estimation methods. Several approaches for deriving various a posteriori estimates for elliptic problems for errors measured in global (energy) norms ( [1], [4], [5], [17], [20], [39], [46], [47], [50]), or in terms of various local quantities ( [6], [12], [19], [25], [40], [45]) have been suggested (see also references in the above mentioned works). However, most of the estimates proposed there strongly use the fact that the computed solutions are true finite element (FE) approximations which, in fact, rarely happens in real computations, e.g., due to quadrature rules, forcibly stopped iterative processes, various round-off errors, or even bugs in computer codes.…”