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A posteriori error estimation for extended finite elements by an extended global recovery. (English) Zbl 1195.74171

Summary: This contribution presents an extended global derivative recovery for enriched finite element methods (FEMs), such as the extended FEM along with an associated error indicator. Owing to its simplicity, the proposed scheme is ideally suited to industrial applications. The procedure is based on global minimization of the ${L}_{2}$ norm of the difference between the raw strain field $\left({C}_{-1}\right)$ and the recovered $\left({C}_{0}\right)$ strain field. The methodology engineered in this paper extends the ideas of J. T. Oden and H. J. Brauchli [Internat. J. Numer. Methods Engin. 3, 317–325 (1971; Zbl 0251.73056)] E. Hinton and J.S. Campbell [Int. J. Numer. Methods Eng. 8, 461-480 (1974; Zbl 0286.73066)] by enriching the approximation used for the construction of the recovered derivatives (strains) with the gradients of the functions employed to enrich the approximation employed for the primal unknown (displacements). We show linear elastic fracture mechanics examples, both in simple two-dimensional settings, and for a three-dimensional structure. Numerically, we show that the effectivity index of the proposed indicator converges to unity upon mesh refinement. Consequently, the approximate error converges to the exact error, indicating that the error indicator is valid.

Additionally, the numerical examples suggest a novel adaptive strategy for enriched approximations in which the dimensions of the enrichment zone are first increased, before standard $h$- and $p$-adaptivities are applied; we suggest to coin this methodology $e-adaptivity$.

##### MSC:
 74S05 Finite element methods in solid mechanics 74R10 Brittle fracture