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

## TL;DR

This paper introduces an adaptive hybridizable discontinuous Galerkin method for solving the Grad-Shafranov equation on curved domains, using polygonal subdomains and automatic mesh refinement to accurately model plasma equilibria.

## Contribution

It presents a novel high-order solver that avoids complex geometry-conforming meshes by employing a transfer technique on polygonal subdomains with adaptive mesh refinement.

## Key findings

- Effective error estimator for plasma equilibria
- Automatic boundary updating maintains geometric accuracy
- Suitable for realistic plasma confinement geometries

## Abstract

We propose a high-order adaptive numerical solver for the semilinear elliptic boundary value problem modelling 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. As the mesh is locally refined, the computational domain is automatically updated in order to always maintain the distance between the actual boundary and the computational boundary of the order of the local mesh diameter. Numerical evidence is presented of the suitability of the estimator as an approximate error measure for physically relevant equilibria with pressure pedestals, internal transport barriers, and current holes on realistic geometries.