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

TL;DR

This paper introduces an adaptive discontinuous Petrov-Galerkin method for solving the nonlinear Grad-Shafranov equation, featuring an ultraweak formulation, residual-based adaptive refinement, and efficient nonlinear solvers, demonstrating high accuracy and efficiency.

Contribution

It develops a novel high-order adaptive DPG scheme with residual-based refinement and parallel implementation for the Grad-Shafranov equation, enhancing accuracy and computational efficiency.

Findings

The method achieves higher accuracy than conventional finite element methods.

Adaptive mesh refinement improves solution quality dynamically.

Parallel implementation demonstrates scalability and efficiency.

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.