An adaptive discontinuous Petrov-Galerkin method for the Grad-Shafranov equation

Peng, Zhichao; Tang, Qi; Tang, Xian-Zhu

doi:10.1137/19M1309894

An adaptive discontinuous Petrov-Galerkin method for the Grad-Shafranov equation

Journal Article · Thu Oct 08 00:00:00 EDT 2020 · SIAM Journal on Scientific Computing

DOI:https://doi.org/10.1137/19M1309894· OSTI ID:1726182

Peng, Zhichao ^[1]; Tang, Qi ^[2]; Tang, Xian-Zhu ^[2]

Rensselaer Polytechnic Inst., Troy, NY (United States)
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

In this work, we propose and develop an arbitrary-order adaptive discontinuous Petrov--Galerkin (DPG) method for the nonlinear Grad--Shafranov equation. An ultraweak formulation of the DPG scheme for the equation is given based on a minimal residual method. The DPG scheme has the advantage of providing more accurate gradients compared to conventional finite element methods, which is desired for numerical solutions to the Grad--Shafranov equation. The numerical scheme is augmented with an adaptive mesh refinement approach, and a criterion based on the residual norm in the minimal residual method is developed to achieve dynamic refinement. Nonlinear solvers for the resulting system are explored and a Picard iteration with Anderson acceleration is found to be efficient to solve the system. Finally, the proposed algorithm is implemented in parallel on MFEM using a domain-decomposition approach, and our implementation is general, supporting arbitrary order of accuracy and general meshes. Furthermore, numerical results are presented to demonstrate the efficiency and accuracy of the proposed algorithm.

View Accepted Manuscript (DOE)

Research Organization:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Organization:: USDOE Office of Science (SC). Advanced Scientific Computing Research (ASCR)

Grant/Contract Number:: 89233218CNA000001

OSTI ID:: 1726182

Report Number(s):: LA-UR--19-31260

Journal Information:: SIAM Journal on Scientific Computing, Journal Name: SIAM Journal on Scientific Computing Journal Issue: 5 Vol. 42; ISSN 1064-8275

Publisher:: SIAMCopyright Statement

Country of Publication:: United States

Language:: English

References (38)

A hybridized discontinuous Petrov-Galerkin scheme for scalar conservation laws: HDPG SCHEME FOR SCALAR CONSERVATION LAWS Moro, D.; Nguyen, N. C.; Peraire, J. International Journal for Numerical Methods in Engineering, Vol. 91, Issue 9 https://doi.org/10.1002/nme.4300	journal	June 2012
A class of discontinuous Petrov-Galerkin methods. II. Optimal test functions Demkowicz, L.; Gopalakrishnan, J. Numerical Methods for Partial Differential Equations, Vol. 27, Issue 1 https://doi.org/10.1002/num.20640	journal	October 2010
Mixed finite elements in ?3 Nedelec, J. C. Numerische Mathematik, Vol. 35, Issue 3 https://doi.org/10.1007/BF01396415	journal	September 1980
Nonlinear discontinuous Petrov–Galerkin methods Carstensen, C.; Bringmann, P.; Hellwig, F. Numerische Mathematik, Vol. 139, Issue 3 https://doi.org/10.1007/s00211-018-0947-5	journal	March 2018
The CHEASE code for toroidal MHD equilibria Lütjens, H.; Bondeson, A.; Sauter, O. Computer Physics Communications, Vol. 97, Issue 3 https://doi.org/10.1016/0010-4655(96)00046-X	journal	September 1996
A class of discontinuous Petrov–Galerkin methods. Part III: Adaptivity Demkowicz, Leszek; Gopalakrishnan, Jay; Niemi, Antti H. Applied Numerical Mathematics, Vol. 62, Issue 4 https://doi.org/10.1016/j.apnum.2011.09.002	journal	April 2012
The DPG method for the Stokes problem Roberts, Nathan V.; Bui-Thanh, Tan; Demkowicz, Leszek Computers & Mathematics with Applications, Vol. 67, Issue 4 https://doi.org/10.1016/j.camwa.2013.12.015	journal	March 2014
Construction of DPG Fortin operators for second order problems Nagaraj, Sriram; Petrides, Socratis; Demkowicz, Leszek F. Computers & Mathematics with Applications, Vol. 74, Issue 8 https://doi.org/10.1016/j.camwa.2017.05.030	journal	October 2017
An adaptive DPG method for high frequency time-harmonic wave propagation problems Petrides, Socratis; Demkowicz, Leszek F. Computers & Mathematics with Applications, Vol. 74, Issue 8 https://doi.org/10.1016/j.camwa.2017.06.044	journal	October 2017
A class of discontinuous Petrov–Galerkin methods. Part I: The transport equation Demkowicz, L.; Gopalakrishnan, J. Computer Methods in Applied Mechanics and Engineering, Vol. 199, Issue 23-24 https://doi.org/10.1016/j.cma.2010.01.003	journal	April 2010
A PDE-constrained optimization approach to the discontinuous Petrov–Galerkin method with a trust region inexact Newton-CG solver Bui-Thanh, Tan; Ghattas, Omar Computer Methods in Applied Mechanics and Engineering, Vol. 278 https://doi.org/10.1016/j.cma.2014.04.018	journal	August 2014
Discrete least-squares finite element methods Keith, Brendan; Petrides, Socratis; Fuentes, Federico Computer Methods in Applied Mechanics and Engineering, Vol. 327 https://doi.org/10.1016/j.cma.2017.08.043	journal	December 2017
A DPG method for steady viscous compressible flow Chan, Jesse; Demkowicz, Leszek; Moser, Robert Computers & Fluids, Vol. 98 https://doi.org/10.1016/j.compfluid.2014.02.024	journal	July 2014
Solving the Grad–Shafranov equation with spectral elements Howell, E. C.; Sovinec, C. R. Computer Physics Communications, Vol. 185, Issue 5 https://doi.org/10.1016/j.cpc.2014.02.008	journal	May 2014
ECOM: A fast and accurate solver for toroidal axisymmetric MHD equilibria Lee, Jungpyo; Cerfon, Antoine Computer Physics Communications, Vol. 190 https://doi.org/10.1016/j.cpc.2015.01.015	journal	May 2015
A Hybridizable Discontinuous Galerkin solver for the Grad–Shafranov equation Sánchez-Vizuet, Tonatiuh; Solano, Manuel E. Computer Physics Communications, Vol. 235 https://doi.org/10.1016/j.cpc.2018.09.013	journal	February 2019
Adaptive Hybridizable Discontinuous Galerkin discretization of the Grad–Shafranov equation by extension from polygonal subdomains Sánchez-Vizuet, Tonatiuh; Solano, Manuel E.; Cerfon, Antoine J. Computer Physics Communications, Vol. 255 https://doi.org/10.1016/j.cpc.2020.107239	journal	October 2020
A class of discontinuous Petrov–Galerkin methods. Part IV: The optimal test norm and time-harmonic wave propagation in 1D Zitelli, J.; Muga, I.; Demkowicz, L. Journal of Computational Physics, Vol. 230, Issue 7 https://doi.org/10.1016/j.jcp.2010.12.001	journal	April 2011
A fast, high-order solver for the Grad–Shafranov equation Pataki, Andras; Cerfon, Antoine J.; Freidberg, Jeffrey P. Journal of Computational Physics, Vol. 243 https://doi.org/10.1016/j.jcp.2013.02.045	journal	June 2013
A discontinuous Petrov–Galerkin methodology for adaptive solutions to the incompressible Navier–Stokes equations Roberts, Nathan V.; Demkowicz, Leszek; Moser, Robert Journal of Computational Physics, Vol. 301 https://doi.org/10.1016/j.jcp.2015.07.014	journal	November 2015
A mimetic spectral element solver for the Grad–Shafranov equation Palha, A.; Koren, B.; Felici, F. Journal of Computational Physics, Vol. 316 https://doi.org/10.1016/j.jcp.2016.04.002	journal	July 2016
A finite element method with overlapping meshes for free-boundary axisymmetric plasma equilibria in realistic geometries Heumann, Holger; Rapetti, Francesca Journal of Computational Physics, Vol. 334 https://doi.org/10.1016/j.jcp.2017.01.006	journal	April 2017
“One size fits all” analytic solutions to the Grad–Shafranov equation Cerfon, Antoine J.; Freidberg, Jeffrey P. Physics of Plasmas, Vol. 17, Issue 3 https://doi.org/10.1063/1.3328818	journal	March 2010
Models of primary runaway electron distribution in the runaway vortex regime Guo, Zehua; Tang, Xian-Zhu; McDevitt, Christopher J. Physics of Plasmas, Vol. 24, Issue 11 https://doi.org/10.1063/1.5006917	journal	November 2017
Equilibrium properties of spherical torus plasmas in NSTX Sabbagh, S. A.; Kaye, S. M.; Menard, J. Nuclear Fusion, Vol. 41, Issue 11 https://doi.org/10.1088/0029-5515/41/11/309	journal	November 2001
MHD stability in X-point geometry: simulation of ELMs Huysmans, G. T. A.; Czarny, O. Nuclear Fusion, Vol. 47, Issue 7 https://doi.org/10.1088/0029-5515/47/7/016	journal	June 2007
The ITER design Aymar, R.; Barabaschi, P.; Shimomura, Y. Plasma Physics and Controlled Fusion, Vol. 44, Issue 5 https://doi.org/10.1088/0741-3335/44/5/304	journal	April 2002
An analysis of the practical DPG method Gopalakrishnan, J.; Qiu, W. Mathematics of Computation, Vol. 83, Issue 286 https://doi.org/10.1090/S0025-5718-2013-02721-4	journal	May 2013
Nodal Auxiliary Space Preconditioning in H(curl) and H(div) Spaces Hiptmair, Ralf; Xu, Jinchao SIAM Journal on Numerical Analysis, Vol. 45, Issue 6 https://doi.org/10.1137/060660588	journal	January 2007
A Flexible Inner-Outer Preconditioned GMRES Algorithm Saad, Youcef SIAM Journal on Scientific Computing, Vol. 14, Issue 2 https://doi.org/10.1137/0914028	journal	March 1993
Anderson Acceleration for Fixed-Point Iterations Walker, Homer F.; Ni, Peng SIAM Journal on Numerical Analysis, Vol. 49, Issue 4 https://doi.org/10.1137/10078356X	journal	January 2011
Analysis of the DPG Method for the Poisson Equation Demkowicz, L.; Gopalakrishnan, J. SIAM Journal on Numerical Analysis, Vol. 49, Issue 5 https://doi.org/10.1137/100809799	journal	January 2011
A Posteriori Error Control for DPG Methods Carstensen, Carsten; Demkowicz, Leszek; Gopalakrishnan, Jay SIAM Journal on Numerical Analysis, Vol. 52, Issue 3 https://doi.org/10.1137/130924913	journal	January 2014
Composing Scalable Nonlinear Algebraic Solvers Brune, Peter R.; Knepley, Matthew G.; Smith, Barry F. SIAM Review, Vol. 57, Issue 4 https://doi.org/10.1137/130936725	journal	January 2015
Nonconforming Mesh Refinement for High-Order Finite Elements Červený, Jakub; Dobrev, Veselin; Kolev, Tzanio SIAM Journal on Scientific Computing, Vol. 41, Issue 4 https://doi.org/10.1137/18M1193992	journal	January 2019
Isoparametric Bicubic Hermite Elements for Solution of the Grad-Shafranov Equation Huysmans, G. T. A.; Goedbloed, J. P.; Kerner, W. International Journal of Modern Physics C, Vol. 02, Issue 01 https://doi.org/10.1142/S0129183191000512	journal	March 1991
Iterative Procedures for Nonlinear Integral Equations Anderson, Donald G. Journal of the ACM, Vol. 12, Issue 4 https://doi.org/10.1145/321296.321305	journal	October 1965
Parallel Auxiliary Space AMG for H(Curl) Problems Vassilevski, Tzanio V. Kolev Panayot S. Journal of Computational Mathematics, Vol. 27, Issue 5 https://doi.org/10.4208/jcm.2009.27.5.013	journal	June 2009

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