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Author: Tonatiuh Sanchez-Vizuet
Requested Type: Poster
Submitted: 2018-02-27 22:20:22

Co-authors: M. E. Solano, A. J. Cerfon

Contact Info:
Courant Institute of Mathematical Sciences, NYU.
251 Mercer Street.
New York, NY   10012

Abstract Text:
In axisymmetric fusion reactors, the equilibrium magnetic configuration can be expressed in terms of the solution to a semi-linear elliptic equation known as the Grad-Shafranov equation. When the geometry of the confinement region is known, the problem becomes an interior boundary value problem posed in domains with piecewise-smooth boundaries which can present x-points.

We propose a high order solver based on the Hybridizable Discontinuous Galerkin method [1]. The resulting solver provides high order of convergence for the flux function and its partial derivatives, can handle piecewise smooth curved geometries by extension from polygonal meshes using a high-order transfer technique [2], and deals with the semi-linearity through an accelerated two-grid fixed-point/Newton iteration [3,4].

The underlying Discontinuous Galerkin solver makes the method robust with respect to the geometry of the plasma region and ideally suited for parallel implementations, resulting in matrices of the same size and sparsity as those of hybridized or mixed Finite Element discretizations [5].

[1] B. Cockburn. Proc. International Congress of Mathematicians Vol. IV pp. 2749--2775, 2010.
[2] B. Cockburn, M. E. Solano. SIAM J. Sci. Comput. Vol. 34 pp. A497--A519, 2012.
[3] A. Toth, C.T. Kelley. SIAM J. Numer. Anal. Vol. 53 pp.805--819, 2015.
[4] J. Xu. SIAM J. Sci. Comput. Vol.15. pp. 231--237, 1994.
[5] R. M. Kirby, S. J. Sherwin, B. Cockburn. J. Sci. Comp. Vol. 51. pp. 183--212, 2012.

* A. J. Cerfon and T. Sanchez-Vizuet have been partially funded by the US Department of Energy. Grant No. DE-FG02-86ER53233.
** M. E. Solano has been partially funded by by CONICYT--Chile through FONDECYT project No. 1160320, by BASAL project CMM, Universidad de Chile and by Centro de Investigacion en Ingenieria Matematica, Universidad de Concepcion.