A Hybridizable Discontinuous Galerkin solver for the Grad–Shafranov equation

Sánchez-Vizuet, Tonatiuh; Solano, Manuel E.

doi:10.1016/j.cpc.2018.09.013

Title: A Hybridizable Discontinuous Galerkin solver for the Grad–Shafranov equation

Abstract

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, the solution of which determines the poloidal component of the magnetic field. When the geometry of the confinement region is known, the problem becomes an interior Dirichlet boundary value problem. We propose a high order solver based on the Hybridizable Discontinuous Galerkin method. The resulting algorithm (1) provides high order of convergence for the flux function and its gradient, (2) incorporates a novel method for handling piecewise smooth geometries by extension from polygonal meshes, (3) can handle geometries with non-smooth boundaries and x-points, and (4) deals with the semi-linearity through an accelerated two-grid fixed-point iteration. The effectiveness of the algorithm is verified with computations for cases where analytic solutions are known on configurations similar to those of actual devices (ITER with single null and double null divertor, NSTX, ASDEX upgrade, and Field Reversed Configurations).

Authors:

^[1]; Solano, Manuel E. ^[2]

New York Univ. (NYU), NY (United States)
Univ. de Concepción (Chile)

Publication Date:: Fri Sep 28 00:00:00 EDT 2018

Research Org.:: New York Univ. (NYU), NY (United States)

Sponsoring Org.:: USDOE

OSTI Identifier:: 1610077

Alternate Identifier(s):: OSTI ID: 1635825

Grant/Contract Number:: FG02-86ER53233

Resource Type:: Accepted Manuscript

Journal Name:: Computer Physics Communications

Additional Journal Information:: Journal Volume: 235; Journal ID: ISSN 0010-4655

Publisher:: Elsevier

Country of Publication:: United States

Language:: English

Subject:: 70 PLASMA PHYSICS AND FUSION TECHNOLOGY; Hybridizable Discontinuous Galerkin (HDG); curved boundary; Grad–Shafranov; Anderson acceleration; plasma equilibrium; magnetohydrodynamics (MHD)

Citation Formats


                    Sánchez-Vizuet, Tonatiuh, and Solano, Manuel E. A Hybridizable Discontinuous Galerkin solver for the Grad–Shafranov equation.  United States: N. p., 2018. 
Web.  doi:10.1016/j.cpc.2018.09.013.

Copy to clipboard


                    Sánchez-Vizuet, Tonatiuh, & Solano, Manuel E. A Hybridizable Discontinuous Galerkin solver for the Grad–Shafranov equation.  United States.  https://doi.org/10.1016/j.cpc.2018.09.013

Copy to clipboard


                    Sánchez-Vizuet, Tonatiuh, and Solano, Manuel E. Fri .  
"A Hybridizable Discontinuous Galerkin solver for the Grad–Shafranov equation".  United States.  https://doi.org/10.1016/j.cpc.2018.09.013.  https://www.osti.gov/servlets/purl/1610077.

Copy to clipboard


                    
@article{osti_1610077,

  title        = {A Hybridizable Discontinuous Galerkin solver for the Grad–Shafranov equation},

  author       = {Sánchez-Vizuet, Tonatiuh and Solano, Manuel E.},

  abstractNote = {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, the solution of which determines the poloidal component of the magnetic field. When the geometry of the confinement region is known, the problem becomes an interior Dirichlet boundary value problem. We propose a high order solver based on the Hybridizable Discontinuous Galerkin method. The resulting algorithm (1) provides high order of convergence for the flux function and its gradient, (2) incorporates a novel method for handling piecewise smooth geometries by extension from polygonal meshes, (3) can handle geometries with non-smooth boundaries and x-points, and (4) deals with the semi-linearity through an accelerated two-grid fixed-point iteration. The effectiveness of the algorithm is verified with computations for cases where analytic solutions are known on configurations similar to those of actual devices (ITER with single null and double null divertor, NSTX, ASDEX upgrade, and Field Reversed Configurations).},

  doi          = {10.1016/j.cpc.2018.09.013},

  journal      = {Computer Physics Communications},

  number       = ,

  volume       = 235,

  place        = {United States},

  year         = {Fri Sep 28 00:00:00 EDT 2018},

  month        = {Fri Sep 28 00:00:00 EDT 2018}

}

Copy to clipboard

Journal Article:

Free Publicly Available Full Text

Accepted Manuscript (Publisher)

Accepted Manuscript (DOE)

Publisher's Version of Record

https://doi.org/10.1016/j.cpc.2018.09.013

Other availability

Search WorldCat to find libraries that may hold this journal

Citation Metrics:

Cited by: 10 works

Citation information provided by
Web of Science

Save / Share:

Export Metadata

Save to My Library

Works referenced in this record:

Computation of MHD equilibrium of tokamak plasma
journal, March 1991

Takeda, Tatsuoki; Tokuda, Shinji
Journal of Computational Physics, Vol. 93, Issue 1
DOI: 10.1016/0021-9991(91)90074-U

Axialsymmetrische magnetohydrodynamische Gleichgewichtskonfigurationen
journal, October 1957

Lüst, R.; Schlüter, A.
Zeitschrift für Naturforschung A, Vol. 12, Issue 10
DOI: 10.1515/zna-1957-1014

The ITER divertor concept
journal, April 1995

Janeschitz, G.; Borrass, K.; Federici, G.
Journal of Nuclear Materials, Vol. 220-222
DOI: 10.1016/0022-3115(94)00447-1

Engineering and design aspects related to the development of the ITER divertor
journal, March 1995

Dietz, K.
Fusion Engineering and Design, Vol. 27, Issue 1-2
DOI: 10.1016/0920-3796(94)00242-Y

Isoparametric Bicubic Hermite Elements for Solution of the Grad-Shafranov Equation
journal, March 1991

Huysmans, G. T. A.; Goedbloed, J. P.; Kerner, W.
International Journal of Modern Physics C, Vol. 02, Issue 01
DOI: 10.1142/S0129183191000512

Solving the Grad–Shafranov equation with spectral elements
journal, May 2014

Howell, E. C.; Sovinec, C. R.
Computer Physics Communications, Vol. 185, Issue 5
DOI: 10.1016/j.cpc.2014.02.008

A mimetic spectral element solver for the Grad–Shafranov equation
journal, July 2016

Palha, A.; Koren, B.; Felici, F.
Journal of Computational Physics, Vol. 316
DOI: 10.1016/j.jcp.2016.04.002

Quasi-static free-boundary equilibrium of toroidal plasma with CEDRES++: Computational methods and applications
journal, January 2015

Heumann, H.; Blum, J.; Boulbe, C.
Journal of Plasma Physics, Vol. 81, Issue 3
DOI: 10.1017/S0022377814001251

A finite element method with overlapping meshes for free-boundary axisymmetric plasma equilibria in realistic geometries
journal, April 2017

Heumann, Holger; Rapetti, Francesca
Journal of Computational Physics, Vol. 334
DOI: 10.1016/j.jcp.2017.01.006

Application of Parallelized Multigrid Method to Solution of MHD Equilibrium With MPI
journal, March 2004

Hanawa, T.; Ikuno, S.; Kamitani, A.
IEEE Transactions on Magnetics, Vol. 40, Issue 2
DOI: 10.1109/TMAG.2004.825440

High-resolution magnetohydrodynamic equilibrium code for unity beta plasmas
journal, July 2006

Gourdain, P. -A.; Leboeuf, J. -N.; Neches, R. Y.
Journal of Computational Physics, Vol. 216, Issue 1
DOI: 10.1016/j.jcp.2005.12.005

A fast, high-order solver for the Grad–Shafranov equation
journal, June 2013

Pataki, Andras; Cerfon, Antoine J.; Freidberg, Jeffrey P.
Journal of Computational Physics, Vol. 243
DOI: 10.1016/j.jcp.2013.02.045

ECOM: A fast and accurate solver for toroidal axisymmetric MHD equilibria
journal, May 2015

Lee, Jungpyo; Cerfon, Antoine
Computer Physics Communications, Vol. 190
DOI: 10.1016/j.cpc.2015.01.015

Meshless method for solving fixed boundary problem of plasma equilibrium
journal, July 2015

Imazawa, Ryota; Kawano, Yasunori; Itami, Kiyoshi
Journal of Computational Physics, Vol. 292
DOI: 10.1016/j.jcp.2015.03.035

Numerical solution of ideal MHD equilibrium via radial basis functions collocation and moving least squares approximation methods
journal, June 2016

Ghasemi, Maryam; Amrollahi, Reza
Engineering Analysis with Boundary Elements, Vol. 67
DOI: 10.1016/j.enganabound.2016.02.008

Computation of fixed boundary tokamak equilibria using a method based on approximate particular solutions
journal, September 2015

Nath, D.; Kalra, M. S.; Munshi, P.
Computers & Mathematics with Applications, Vol. 70, Issue 5
DOI: 10.1016/j.camwa.2015.07.013

Solution of Grad–Shafranov equation by the method of fundamental solutions
journal, February 2014

Nath, D.; Kalra, M. S.
Journal of Plasma Physics, Vol. 80, Issue 3
DOI: 10.1017/S0022377814000026

Accurate derivative evaluation for any Grad–Shafranov solver
journal, January 2016

Ricketson, L. F.; Cerfon, A. J.; Rachh, M.
Journal of Computational Physics, Vol. 305
DOI: 10.1016/j.jcp.2015.11.015

To CG or to HDG: A Comparative Study
journal, July 2011

Kirby, Robert M.; Sherwin, Spencer J.; Cockburn, Bernardo
Journal of Scientific Computing, Vol. 51, Issue 1
DOI: 10.1007/s10915-011-9501-7

Solving Dirichlet Boundary-value Problems on Curved Domains by Extensions from Subdomains
journal, January 2012

Cockburn, Bernardo; Solano, Manuel
SIAM Journal on Scientific Computing, Vol. 34, Issue 1
DOI: 10.1137/100805200

Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems
journal, January 2009

Cockburn, Bernardo; Gopalakrishnan, Jayadeep; Lazarov, Raytcho
SIAM Journal on Numerical Analysis, Vol. 47, Issue 2
DOI: 10.1137/070706616

Optimal Isoparametric Finite Elements and Error Estimates for Domains Involving Curved Boundaries
journal, June 1986

Lenoir, M.
SIAM Journal on Numerical Analysis, Vol. 23, Issue 3
DOI: 10.1137/0723036

A finite element method for interface problems in domains with smooth boundaries and interfaces
journal, December 1996

Bramble, James H.; King, J. Thomas
Advances in Computational Mathematics, Vol. 6, Issue 1
DOI: 10.1007/BF02127700

Flow patterns around heart valves: A numerical method
journal, October 1972

Peskin, Charles S.
Journal of Computational Physics, Vol. 10, Issue 2
DOI: 10.1016/0021-9991(72)90065-4

A priori error analysis for HDG methods using extensions from subdomains to achieve boundary conformity
journal, July 2013

Cockburn, Bernardo; Qiu, Weifeng; Solano, Manuel
Mathematics of Computation, Vol. 83, Issue 286
DOI: 10.1090/S0025-5718-2013-02747-0

A projection-based error analysis of HDG methods
journal, March 2010

Cockburn, Bernardo; Gopalakrishnan, Jayadeep; Sayas, Francisco-Javier
Mathematics of Computation, Vol. 79, Issue 271
DOI: 10.1090/S0025-5718-10-02334-3

A Characterization of Hybridized Mixed Methods for Second Order Elliptic Problems
journal, January 2004

Cockburn, Bernardo; Gopalakrishnan, Jayadeep
SIAM Journal on Numerical Analysis, Vol. 42, Issue 1
DOI: 10.1137/S0036142902417893

Algorithm 949: MATLAB Tools for HDG in Three Dimensions
journal, June 2015

Fu, Zhixing; Gatica, Luis F.; Sayas, Francisco-javier
ACM Transactions on Mathematical Software, Vol. 41, Issue 3
DOI: 10.1145/2658992

Reduction of stiffness and mass matrices
journal, February 1965

Guyan, Robert J.
AIAA Journal, Vol. 3, Issue 2
DOI: 10.2514/3.2874

To CG or to HDG: A Comparative Study in 3D
journal, July 2015

Yakovlev, Sergey; Moxey, David; Kirby, Robert M.
Journal of Scientific Computing, Vol. 67, Issue 1
DOI: 10.1007/s10915-015-0076-6

Conditions for superconvergence of HDG methods for second-order elliptic problems
journal, September 2012

Cockburn, Bernardo; Qiu, Weifeng; Shi, Ke
Mathematics of Computation, Vol. 81, Issue 279
DOI: 10.1090/S0025-5718-2011-02550-0

A Novel Two-Grid Method for Semilinear Elliptic Equations
journal, January 1994

Xu, Jinchao
SIAM Journal on Scientific Computing, Vol. 15, Issue 1
DOI: 10.1137/0915016

Two-Grid Discretization Techniques for Linear and Nonlinear PDE s
journal, October 1996

Xu, Jinchao
SIAM Journal on Numerical Analysis, Vol. 33, Issue 5
DOI: 10.1137/S0036142992232949

Iterative Procedures for Nonlinear Integral Equations
journal, October 1965

Anderson, Donald G.
Journal of the ACM, Vol. 12, Issue 4
DOI: 10.1145/321296.321305

Convergence Analysis for Anderson Acceleration
journal, January 2015

Toth, Alex; Kelley, C. T.
SIAM Journal on Numerical Analysis, Vol. 53, Issue 2
DOI: 10.1137/130919398

Anderson Acceleration for Fixed-Point Iterations
journal, January 2011

Walker, Homer F.; Ni, Peng
SIAM Journal on Numerical Analysis, Vol. 49, Issue 4
DOI: 10.1137/10078356X

“One size fits all” analytic solutions to the Grad–Shafranov equation
journal, March 2010

Cerfon, Antoine J.; Freidberg, Jeffrey P.
Physics of Plasmas, Vol. 17, Issue 3
DOI: 10.1063/1.3328818

Analytical solutions to the Grad–Shafranov equation for tokamak equilibrium with dissimilar source functions
journal, September 1999

Mc Carthy, P. J.
Physics of Plasmas, Vol. 6, Issue 9
DOI: 10.1063/1.873630

A family of analytic equilibrium solutions for the Grad–Shafranov equation
journal, November 2007

Guazzotto, L.; Freidberg, J. P.
Physics of Plasmas, Vol. 14, Issue 11
DOI: 10.1063/1.2803759

Noncircular, finite aspect ratio, local equilibrium model
journal, April 1998

Miller, R. L.; Chu, M. S.; Greene, J. M.
Physics of Plasmas, Vol. 5, Issue 4
DOI: 10.1063/1.872666

Axisymmetric MHD equilibrium solver with bicubic Hermite elements
journal, March 1992

Lütjens, H.; Bondeson, A.; Roy, A.
Computer Physics Communications, Vol. 69, Issue 2-3
DOI: 10.1016/0010-4655(92)90167-w

Hydromagnetic equilibria and force-free fields
journal, September 1958

Grad, Harold; Rubin, Hanan
Journal of Nuclear Energy (1954), Vol. 7, Issue 3-4
DOI: 10.1016/0891-3919(58)90139-6

ECOM: A fast and accurate solver for toroidal axisymmetric MHD equilibria
journal, May 2015

Lee, Jungpyo; Cerfon, Antoine
Computer Physics Communications, Vol. 190
DOI: 10.1016/j.cpc.2015.01.015

A mimetic spectral element solver for the Grad–Shafranov equation
journal, July 2016

Palha, A.; Koren, B.; Felici, F.
Journal of Computational Physics, Vol. 316
DOI: 10.1016/j.jcp.2016.04.002

Works referencing / citing this record:

Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity
text, January 2019

Giacomini, Matteo; Sevilla, Ruben
arXiv
DOI: 10.48550/arxiv.1908.06604

A priori and a posteriori error analysis of an unfitted HDG method for semi-linear elliptic problems
preprint, January 2019

Sánchez, Nestor; Sánchez-Vizuet, Tonatiuh; Solano, Manuel E.
arXiv
DOI: 10.48550/arxiv.1911.12298

A parallel cut-cell algorithm for the free-boundary Grad-Shafranov problem
preprint, January 2020

Liu, Shuang; Tang, Qi; Tang, Xian-Zhu
arXiv
DOI: 10.48550/arxiv.2012.06015

Similar Records in DOE PAGES and OSTI.GOV collections:

Adaptive Hybridizable Discontinuous Galerkin discretization of the Grad–Shafranov equation by extension from polygonal subdomains

Journal Article Sánchez-Vizuet, Tonatiuh ; Solano, Manuel E. ; Cerfon, Antoine J. - Computer Physics Communications

In this work, we propose a high-order adaptive numerical solver for the semilinear elliptic boundary value problem modeling magnetic plasma equilibrium in axisymmetric confinement devices. In the fixed boundary case, the equation is posed on curved domains with piecewise smooth curved boundaries that may present corners. The solution method we present is based on the hybridizable discontinuous Galerkin method and sidesteps the need for geometry-conforming triangulations thanks to a transfer technique that allows to approximate the solution using only a polygonal subset as computational domain. Moreover, the solver features automatic mesh refinement driven by a residual-based a posteriori error estimator.more »« less
Cited by 7
https://doi.org/10.1016/j.cpc.2020.107239

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

Journal Article Peng, Zhichao ; Tang, Qi ; Tang, Xian-Zhu - SIAM Journal on Scientific Computing

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 systemmore »« less
https://doi.org/10.1137/19M1309894

Full Text Available
A Parallel Cut-Cell Algorithm for the Free-Boundary Grad--Shafranov Problem

Journal Article Liu, Shuang ; Tang, Qi ; Tang, Xian-zhu - SIAM Journal on Scientific Computing

A parallel cut-cell algorithm is described to solve the free-boundary problem of the Grad--Shafranov equation. The algorithm reformulates the free-boundary problem in an irregular bounded domain and its important aspects include a searching algorithm for the magnetic axis and separatrix, a surface integral along the irregular boundary to determine the boundary values, an approach to optimize the coil current based on a targeting plasma shape, Picard iterations with Aitken's acceleration for the resulting nonlinear problem, and a Cartesian grid embedded boundary method to handle the complex geometry. Here the algorithm is implemented in parallel using a standard domain-decomposition approach andmore »« less
https://doi.org/10.1137/20m1385470

Full Text Available
Accurate derivative evaluation for any Grad–Shafranov solver

Journal Article Ricketson, L. F. ; Cerfon, A.J., E-mail: cerfon@cims.nyu.edu ; Rachh, M. ; ... - Journal of Computational Physics

We present a numerical scheme that can be combined with any fixed boundary finite element based Poisson or Grad–Shafranov solver to compute the first and second partial derivatives of the solution to these equations with the same order of convergence as the solution itself. At the heart of our scheme is an efficient and accurate computation of the Dirichlet to Neumann map through the evaluation of a singular volume integral and the solution to a Fredholm integral equation of the second kind. Our numerical method is particularly useful for magnetic confinement fusion simulations, since it allows the evaluation of quantitiesmore »« less
https://doi.org/10.1016/J.JCP.2015.11.015
A semi-analytical solver for the Grad-Shafranov equation

Journal Article Ciro, D., E-mail: dciro@if.usp.br ; Caldas, I. L., E-mail: ibere@if.usp.br - Physics of Plasmas

In toroidally confined plasmas, the Grad-Shafranov equation, in general, a non-linear partial differential equation, describes the hydromagnetic equilibrium of the system. This equation becomes linear when the kinetic pressure is proportional to the poloidal magnetic flux and the squared poloidal current is a quadratic function of it. In this work, the eigenvalue of the associated homogeneous equation is related with the safety factor on the magnetic axis, the plasma beta, and the Shafranov shift; then, the adjustable parameters of the particular solution are bounded through physical constrains. The poloidal magnetic flux becomes a linear superposition of independent solutions and itsmore »« less
https://doi.org/10.1063/1.4901036

Similar Records

Title: A Hybridizable Discontinuous Galerkin solver for the Grad–Shafranov equation

Abstract

Citation Formats

Computation of MHD equilibrium of tokamak plasma journal, March 1991

Axialsymmetrische magnetohydrodynamische Gleichgewichtskonfigurationen journal, October 1957

The ITER divertor concept journal, April 1995

Engineering and design aspects related to the development of the ITER divertor journal, March 1995

Isoparametric Bicubic Hermite Elements for Solution of the Grad-Shafranov Equation journal, March 1991

Solving the Grad–Shafranov equation with spectral elements journal, May 2014

A mimetic spectral element solver for the Grad–Shafranov equation journal, July 2016

Quasi-static free-boundary equilibrium of toroidal plasma with CEDRES++: Computational methods and applications journal, January 2015

A finite element method with overlapping meshes for free-boundary axisymmetric plasma equilibria in realistic geometries journal, April 2017

Application of Parallelized Multigrid Method to Solution of MHD Equilibrium With MPI journal, March 2004

High-resolution magnetohydrodynamic equilibrium code for unity beta plasmas journal, July 2006

A fast, high-order solver for the Grad–Shafranov equation journal, June 2013

ECOM: A fast and accurate solver for toroidal axisymmetric MHD equilibria journal, May 2015

Meshless method for solving fixed boundary problem of plasma equilibrium journal, July 2015

Numerical solution of ideal MHD equilibrium via radial basis functions collocation and moving least squares approximation methods journal, June 2016

Computation of fixed boundary tokamak equilibria using a method based on approximate particular solutions journal, September 2015

Solution of Grad–Shafranov equation by the method of fundamental solutions journal, February 2014

Accurate derivative evaluation for any Grad–Shafranov solver journal, January 2016

To CG or to HDG: A Comparative Study journal, July 2011

Solving Dirichlet Boundary-value Problems on Curved Domains by Extensions from Subdomains journal, January 2012

Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems journal, January 2009

Optimal Isoparametric Finite Elements and Error Estimates for Domains Involving Curved Boundaries journal, June 1986

A finite element method for interface problems in domains with smooth boundaries and interfaces journal, December 1996

Flow patterns around heart valves: A numerical method journal, October 1972

A priori error analysis for HDG methods using extensions from subdomains to achieve boundary conformity journal, July 2013

A projection-based error analysis of HDG methods journal, March 2010

A Characterization of Hybridized Mixed Methods for Second Order Elliptic Problems journal, January 2004

Algorithm 949: MATLAB Tools for HDG in Three Dimensions journal, June 2015

Reduction of stiffness and mass matrices journal, February 1965

To CG or to HDG: A Comparative Study in 3D journal, July 2015

Conditions for superconvergence of HDG methods for second-order elliptic problems journal, September 2012

A Novel Two-Grid Method for Semilinear Elliptic Equations journal, January 1994

Two-Grid Discretization Techniques for Linear and Nonlinear PDE s journal, October 1996

Iterative Procedures for Nonlinear Integral Equations journal, October 1965

Convergence Analysis for Anderson Acceleration journal, January 2015

Anderson Acceleration for Fixed-Point Iterations journal, January 2011

“One size fits all” analytic solutions to the Grad–Shafranov equation journal, March 2010

Analytical solutions to the Grad–Shafranov equation for tokamak equilibrium with dissimilar source functions journal, September 1999

A family of analytic equilibrium solutions for the Grad–Shafranov equation journal, November 2007

Noncircular, finite aspect ratio, local equilibrium model journal, April 1998

Axisymmetric MHD equilibrium solver with bicubic Hermite elements journal, March 1992

Hydromagnetic equilibria and force-free fields journal, September 1958

ECOM: A fast and accurate solver for toroidal axisymmetric MHD equilibria journal, May 2015

A mimetic spectral element solver for the Grad–Shafranov equation journal, July 2016

Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity text, January 2019

A priori and a posteriori error analysis of an unfitted HDG method for semi-linear elliptic problems preprint, January 2019

A parallel cut-cell algorithm for the free-boundary Grad-Shafranov problem preprint, January 2020

Computation of MHD equilibrium of tokamak plasma
journal, March 1991

Axialsymmetrische magnetohydrodynamische Gleichgewichtskonfigurationen
journal, October 1957

The ITER divertor concept
journal, April 1995

Engineering and design aspects related to the development of the ITER divertor
journal, March 1995

Isoparametric Bicubic Hermite Elements for Solution of the Grad-Shafranov Equation
journal, March 1991

Solving the Grad–Shafranov equation with spectral elements
journal, May 2014

A mimetic spectral element solver for the Grad–Shafranov equation
journal, July 2016

Quasi-static free-boundary equilibrium of toroidal plasma with CEDRES++: Computational methods and applications
journal, January 2015

A finite element method with overlapping meshes for free-boundary axisymmetric plasma equilibria in realistic geometries
journal, April 2017

Application of Parallelized Multigrid Method to Solution of MHD Equilibrium With MPI
journal, March 2004

High-resolution magnetohydrodynamic equilibrium code for unity beta plasmas
journal, July 2006

A fast, high-order solver for the Grad–Shafranov equation
journal, June 2013

ECOM: A fast and accurate solver for toroidal axisymmetric MHD equilibria
journal, May 2015

Meshless method for solving fixed boundary problem of plasma equilibrium
journal, July 2015

Numerical solution of ideal MHD equilibrium via radial basis functions collocation and moving least squares approximation methods
journal, June 2016

Computation of fixed boundary tokamak equilibria using a method based on approximate particular solutions
journal, September 2015

Solution of Grad–Shafranov equation by the method of fundamental solutions
journal, February 2014

Accurate derivative evaluation for any Grad–Shafranov solver
journal, January 2016

To CG or to HDG: A Comparative Study
journal, July 2011

Solving Dirichlet Boundary-value Problems on Curved Domains by Extensions from Subdomains
journal, January 2012

Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems
journal, January 2009

Optimal Isoparametric Finite Elements and Error Estimates for Domains Involving Curved Boundaries
journal, June 1986

A finite element method for interface problems in domains with smooth boundaries and interfaces
journal, December 1996

Flow patterns around heart valves: A numerical method
journal, October 1972

A priori error analysis for HDG methods using extensions from subdomains to achieve boundary conformity
journal, July 2013

A projection-based error analysis of HDG methods
journal, March 2010

A Characterization of Hybridized Mixed Methods for Second Order Elliptic Problems
journal, January 2004

Algorithm 949: MATLAB Tools for HDG in Three Dimensions
journal, June 2015

Reduction of stiffness and mass matrices
journal, February 1965

To CG or to HDG: A Comparative Study in 3D
journal, July 2015

Conditions for superconvergence of HDG methods for second-order elliptic problems
journal, September 2012

A Novel Two-Grid Method for Semilinear Elliptic Equations
journal, January 1994

Two-Grid Discretization Techniques for Linear and Nonlinear PDE s
journal, October 1996

Iterative Procedures for Nonlinear Integral Equations
journal, October 1965

Convergence Analysis for Anderson Acceleration
journal, January 2015

Anderson Acceleration for Fixed-Point Iterations
journal, January 2011

“One size fits all” analytic solutions to the Grad–Shafranov equation
journal, March 2010

Analytical solutions to the Grad–Shafranov equation for tokamak equilibrium with dissimilar source functions
journal, September 1999

A family of analytic equilibrium solutions for the Grad–Shafranov equation
journal, November 2007

Noncircular, finite aspect ratio, local equilibrium model
journal, April 1998

Axisymmetric MHD equilibrium solver with bicubic Hermite elements
journal, March 1992

Hydromagnetic equilibria and force-free fields
journal, September 1958

ECOM: A fast and accurate solver for toroidal axisymmetric MHD equilibria
journal, May 2015

A mimetic spectral element solver for the Grad–Shafranov equation
journal, July 2016

Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity
text, January 2019

A priori and a posteriori error analysis of an unfitted HDG method for semi-linear elliptic problems
preprint, January 2019

A parallel cut-cell algorithm for the free-boundary Grad-Shafranov problem
preprint, January 2020