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.

中文翻译：

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从多边形子域扩展的 Grad-Shafranov 方程的自适应混合不连续 Galerkin 离散化

我们为模拟轴对称约束装置中的磁等离子体平衡的半线性椭圆边界值问题提出了一种高阶自适应数值求解器。在固定边界的情况下，方程是在具有分段平滑曲线边界的曲线域上提出的，这些曲线边界可能存在拐角。我们提出的求解方法基于可混合的不连续伽辽金方法，并且由于允许仅使用多边形子集作为计算域来近似求解的传输技术，因此不需要符合几何形状的三角剖分。此外，求解器具有由基于残差的后验误差估计器驱动的自动网格细化功能。由于网格被局部细化，计算域会自动更新，以始终保持实际边界与局部网格直径数量级的计算边界之间的距离。数值证据表明，估计器适用于与压力基座、内部传输障碍和现实几何结构上的电流孔的物理相关平衡的近似误差度量。