Abstract

Abstract

This work presents a ZZ-BD a posteriori error estimator tailored for 3-D linear elastic fracture mechanics problems that are approximated by second-order pFEM-GFEM formulations. The proposed error estimator is shown to estimate well discretization errors in the energy norm, with the estimated discretization error converging at the same rate as the exact discretization error. Also, the computed effectivity indexes are close to the optimal value of 1 for a LEFM problem that exhibits 3-D effects.

Abstract

In a posteriori error analysis, the relationship between error and estimator is usually spoiled by so-called oscillation terms, which cannot be bounded by the error. In order to remedy, we devise a new approach where the oscillation has the following two properties. First, it is dominated by the error, irrespective of mesh fineness and the regularity of data and the exact solution. Second, it captures in terms of data the part of the residual that, in general, cannot be quantified with finite information. The new twist in our approach is a locally stable projection onto discretized residuals. In the case of the Poisson problem, the approach leads to an improved standard residual estimator and allows tackling problems with sources in H^-1.

Abstract

 This work presents a ZZ-BD a posteriori error estimator tailored for 3-D linear elastic fracture mechanics problems that are approximated by second-order pFEM-GFEM formulations. The proposed error estimator is shown to estimate well discretization errors in the energy norm, with the estimated discretization error converging at the same rate as the exact discretization error. Also, the computed effectivity indexes are close to the optimal value of 1 for a LEFM problem that exhibits 3-D effects.