Eliminating Gibbs Phenomena for Best-Approximations in Finite Elements and Isogeometric Analysis

  • ten Eikelder, Marco (Technical University of Darmstadt)
  • Stoter, Stein (Eindhoven University of Technology)
  • Bazilevs, Yuri (Brown University)
  • Schillinger, Dominik (Technical University of Darmstadt)

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One of the biggest challenges in the design of numerical methods is the occurrence of non-physical Gibbs oscillations near sharp layers and discontinuities. Preventing Gibbs oscillations is particularly important for the simulation of multiphase flows. Namely, multiphase flow models typically contain an order parameter, e.g. a volume fraction or a concentration, that describes the topological changes of the interface. Gibbs oscillations may cause the numerical approximation of the order parameter to exceed its physical bounds, which could result in a break down of the simulation. This motivates the construction of a numerical method that precludes Gibbs oscillations [1, 2]. The identification of a proper set of constraints that eliminate Gibbs phenomena in numerical methods is still an open research area. In this talk we identify these constraints in the approximation of sharp layers and discontinuities in finite element spaces of arbitrary degree and continuity in the isogeometric analysis framework [3]. We show that by enforcing these constraints Gibbs phenomena are entirely removed in one dimension and severely mitigated in higher dimensions. These constraints can be applied in the design of numerical methods for multiphase flows to remove non-physical oscillations. REFERENCES [1] M.F.P. ten Eikelder, and I. Akkerman, Variation entropy: a continuous local generalization of the TVD property using entropy principles. Comput. Methods Appl. Mech. Eng., Vol. 355, pp. 261–283, 2005. [2] M.F.P. ten Eikelder, Y. Bazilevs, and I. Akkerman, A theoretical framework for discontinuity capturing: Joining variational multiscale analysis and variation entropy theory. Comput. Methods Appl. Mech. Eng., Vol. 359, 112664, 2020. [3] T.J.R. Hughes and J.A. Cottrell and Y. Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement. Comput. Methods Appl. Mech. Eng., Vol. 195, pp. 4135-4195, 2005.