Flux-corrected transport algorithms for continuous Galerkin methods based on high order Bernstein finite elements
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Date
2016-12
Journal Title
Journal ISSN
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Publisher
Lehrstuhl für Angewandte Mathematik und Numerik
Abstract
This work extends the flux-corrected transport (FCT) methodology to arbitrary-order continuous finite element discretizations
of scalar conservation laws on simplex meshes. Using Bernstein polynomials as local basis functions, we
constrain the total variation of the numerical solution by imposing local discrete maximum principles on the Bézier
net. The design of accuracy-preserving FCT schemes for high order Bernstein-Bézier finite elements requires the development
of new algorithms and/or generalization of limiting techniques tailored for linear and multilinear Lagrange
elements. In this paper, we propose (i) a new discrete upwinding strategy leading to local extremum bounded low
order approximations with compact stencils, (ii) a high order stabilization operator based on gradient recovery, and
(iii) new localized limiting techniques for antidi usive element contributions. The optional use of a smoothness indicator,
based on a second derivative test, makes it possible to potentially avoid unnecessary limiting at smooth extrema
and achieve optimal convergence rates for problems with smooth solutions. The accuracy of the proposed schemes is
assessed in numerical studies for the linear transport equation in 1D and 2D.
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Keywords
Bernstein-Bézier Finite Elements, continuous Galerkin method, flux-corrected transport, artificial diffusion, local discrete maximum principles, total variation diminishing property