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arXiv · 2001.04524

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

Zhichao Peng·Qi Tang·Xian-Zhu Tang
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Abstract

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.

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Zhichao Peng, Qi Tang, Xian-Zhu Tang. 2020-01-13. An adaptive discontinuous Petrov-Galerkin method for the Grad-Shafranov equation. https://arxiv.org/abs/2001.04524

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