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**On the construction of blending elements for local partition of unity enriched finite elements.**
*(English)*
Zbl 1035.65122

Summary: For computational efficiency, partition of unity enrichments are preferably localized to the sub-domains where they are needed. It is shown that an appropriate construction of the elements in the blending area, the region where the enriched elements blend to unenriched elements, is often crucial for good performance of local partition of unity enrichments.

An enhanced strain formulation is developed which leads to good performance; the optimal rate of convergence is achieved. For polynomial enrichments, it is shown that a proper choice of the finite element shape functions and partition of unity shape functions also improves the accuracy and convergence.

The methods are illustrated by several examples. The examples deal primarily with the signed distance function enrichment for treating discontinuous derivatives inside an element, but other enrichments are also considered. Results show that both methods provide optimal rates of convergence.

An enhanced strain formulation is developed which leads to good performance; the optimal rate of convergence is achieved. For polynomial enrichments, it is shown that a proper choice of the finite element shape functions and partition of unity shape functions also improves the accuracy and convergence.

The methods are illustrated by several examples. The examples deal primarily with the signed distance function enrichment for treating discontinuous derivatives inside an element, but other enrichments are also considered. Results show that both methods provide optimal rates of convergence.

### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

35R05 | PDEs with low regular coefficients and/or low regular data |

### Keywords:

enriched finite element; blending elements; reproducing conditions; Laplace equation; discontinuous coefficients; numerical examples; partition of unity; convergence### Software:

XFEM
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\textit{J. Chessa} et al., Int. J. Numer. Methods Eng. 57, No. 7, 1015--1038 (2003; Zbl 1035.65122)

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