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. Numerical results are presented to demonstrate the efficiency and accuracy of the proposed algorithm.
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 geom…
A fast, high-order solver for the Grad-Shafranov equation
Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）；Plasma Science and Fusion Center, Massachusetts Institute of Technology（麻省理工学院等离子体科学与聚变中心）
Accurate Derivative Evaluation for any Grad-Shafranov Solver
Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）；Plasma Science and Fusion Center, Massachusetts Institute of Technology（麻省理工学院等离子体科学与聚变中心）
Adaptive Hybridizable Discontinuous Galerkin discretization of the Grad-Shafranov equation by extension from polygonal subdomains
New York University, Courant Institute of Mathematical Sciences（纽约大学 库朗数学科学研究所）；Universidad de Concepción, Department of Mathematical Engineering（康塞普西翁大学 数学工程系）；Universidad de Concepción, Center for Research in Mathematical Engineering（康塞普西翁大学 数学工程研究中心）
In magnetic confinement fusion devices, the equilibrium configuration of a plasma is determined by the balance between the hydrostatic pressure in the fluid and the magnetic forces generated by an array of external coils and the plasma itself. The location of the plasma is not known a priori and must be obtained as the solution to a free boundary problem. The partial differential equation that determines the behavior of the combined magnetic field depends on a set of physical parameters (location of the coils, intensity of the electric currents going through them, magnetic permeability, etc.) …
University Paris-Sud（巴黎第十一大学）；Conservatoire National des Arts et Métiers（法国国立工艺院）；Laboratoire de Mathématiques et de leurs Applications, CNRS, Université de Pau（波尔多大学数学及其应用实验室，法国国家科学研究中心）；Université de Pau（波尔多大学）
Raviart-Thomas finite elements of Petrov-Galerkin type
Laboratoire de Mathématiques, Université Paris-Sud（巴黎南大学数学实验室）；Conservatoire National des Arts et Métiers（国立工艺学院）；Laboratoire de Mathématiques et de leurs Applications, UMR CNRS 5142 Université de Pau et des Pays de l’Adour（波城及阿杜尔地区大学数学及其应用实验室）
In this paper, we investigate a spectral Petrov-Galerkin method for an optimal control problem governed by a two-sided space-fractional diffusion-advection-reaction equation. Taking into account the effect of singularities near the boundary generated by the weak singular kernel of the fractional operator, we establish the regularity of the problem in weighted Sobolev space. Error estimates are provided for the presented spectral Petrov-Galerkin method and the convergence orders of the state and control variables are determined. Furthermore, a fast projected gradient algorithm with a quasi-line…
Upwinded mass fluxes are described and analysed for advection operators discretised using mixed mimetic spectral elements. This involves a Petrov-Galerkin formulation by which the mass flux test functions are evaluated at downstream locations along velocity characteristics. As for the original mixed mimetic spectral element advection operator, the upwinded mass flux advection operator is conservative, however unlike the original advection operator, which is purely hyperbolic, the upwinded advection operator adds dissipation which is biased towards high wave numbers. The upwinded advection oper…
We present a Petrov-Gelerkin (PG) method for a class of nonlocal convection-dominated diffusion problems. There are two main ingredients in our approach. First, we define the norm on the test space as induced by the trial space norm, i.e., the optimal test norm, so that the inf-sup condition can be satisfied uniformly independent of the problem. We show the well-posedness of a class of nonlocal convection-dominated diffusion problems under the optimal test norm with general assumptions on the nonlocal diffusion and convection kernels. Second, following the framework of Cohen et al.~(2012), we …
Two numerical methods with graded temporal grids are analyzed for fractional evolution equations. One is a low-order discontinuous Galerkin (DG) discretization in the case of fractional order $0<α<1$, and the other one is a low-order Petrov Galerkin (PG) discretization in the case of fractional order $1<α<2$. By a new duality technique, pointwise-in-time error estimates of first-order and $ (3-α) $-order temporal accuracies are respectively derived for DG and PG, under reasonable regularity assumptions on the initial value. Numerical experiments are performed to verify the theoretical results.
In this work, we propose a new quasi-optimal test norm for a discontinuous Petrov-Galerkin (DPG) discretization of the ultra-weak formulation of the convection-diffusion equation. We prove theoretically that the proposed test norm leads to bounds between the target norm and the energy norm induced by the test norm which are robust with respect to the diffusion parameter in the solution and gradient components and have favorable scalings in the trace components. We conclude with numerical experiments to confirm our theoretical results.
