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- Edge based Schwarz methods for the Crouzeix-Raviart finite volume element discretization of elliptic problems(arXiv)
Author : Atle Loneland, Leszek Marcinkowski, Talal Rahman
Abstract : In this paper, we present two variants of the Additive Schwarz Method for a Crouzeix-Raviart finite volume element (CRFVE) discretization of second order elliptic problems with discontinuous coefficients where the discontinuities are only across subdomain boundaries. One preconditioner is symmetric while the other is nonsymmetric. The proposed methods are almost optimal, in the sense that the residual error estimates for the GMRES iteration in the both cases depend only polylogarithmically on the mesh parameters.
2.A new quadratic and cubic polynomial enrichment of the Crouzeix-Raviart finite element (arXiv)
Author : Francesco Dell’Accio, Allal Guessab, Federico Nudo
Abstract : In this paper, we introduce quadratic and cubic polynomial enrichments of the classical Crouzeix — Raviart finite element, with the aim of constructing accurate approximations in such enriched elements. To achieve this goal, we respectively add three and seven weighted line integrals as enriched degrees of freedom. For each case, we present a necessary and sufficient condition under which these augmented elements are well-defined. For illustration purposes, we then use a general approach to define two-parameter families of admissible degrees of freedom. Additionally, we provide explicit expressions for the associated basis functions and subsequently introduce new quadratic and cubic approximation operators based on the proposed admissible elements. The efficiency of the enriched methods is compared to the triangular Crouzeix — Raviart element. As expected, the numerical results exhibit a significant improvement, confirming the effectiveness of the developed enrichment strategy