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Center for Nonlinear Studies |
In this talk, we present and compare some multidimensional limiting techniques for enforcing geometric and algebraic maximum principles in finite element methods for convection-dominated transport equations. In the context of discontinuous Galerkin (DG) methods, undershoots and overshoots are eliminated by limiting the derivatives of the Taylor polynomial representing a finite element shape function. The antidiffusive part of a continuous (linear or bilinear) Galerkin approximation is constrained using flux-corrected transport (FCT) algorithms formulated in terms of edge or element contributions. We show that the element-based approach offers more flexibility in the choice of algorithmic ingredients (low-order scheme, high-order scheme, time stepping, limiting strategy) than its edge-based counterpart. In particular, we introduce a localized FCT limiter which has the same structure as the Barth-Jespersen limiter for DG and finite volume methods. An anisotropic version of this limiter may be used to constrain directional derivatives in situations when the use of a common correction factor for all components gives rise to strong smearing or nonphysical ripples.
Host: Misha Shashkov