A mimetic spectral element solver for the Grad-Shafranov equation

Abstract

In this work we present a robust and accurate arbitrary order solver for the fixed-boundary plasma equilibria in toroidally axisymmetric geometries. To achieve this we apply the mimetic spectral element formulation presented in [56] to the solution of the Grad-Shafranov equation. This approach combines a finite volume discretization with the mixed finite element method. In this way the discrete differential operators ($\nabla$, $\nabla \times$, $\nabla \cdot$) can be represented exactly and metric and all approximation errors are present in the constitutive relations. The result of this formulation is an arbitrary order method even on highly curved meshes. Additionally, the integral of the \reviewerone{toroidal current $J_{\varphi }$} is exactly equal to the boundary integral of the poloidal field over the plasma boundary. This property can play an important role in the coupling between equilibrium and transport solvers. The proposed solver is tested on a varied set of plasma cross-sections (smooth and with an X-point) and also for a wide range of pressure and \reviewertwo{toroidal magnetic flux profiles}. Equilibria accurate up to machine precision are obtained. Optimal algebraic convergence rates of order $(p+1)$ and geometric convergence rates are shown for Soloviev solutions (including high Shafranov shifts), field-reversed configuration (FRC) solutions and spheromak analytical solutions. The robustness of the method is demonstrated for non-linear test cases, in particular on an equilibrium solution with a pressure pedestal.

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