Solution of Grad-Shafranov equation with linear triangular finite elements providing magnetic field continuity with a second order accuracy

Abstract

The formulation presented here is applied to the Grad-Shafranov equation, which describes the equilibrium of two-dimensional axisymmetric fusion plasmas. The equilibrium configurations, obtained via standard finite element formulations in terms of poloidal flux using linear triangles, provide the magnetic flux with an accuracy O(h2). However, the poloidal magnetic field is obtained with a lower accuracy O(h) and is discontinuous across adjacent elements, with consequent problems in the derivation of linearized models for Magneto-Hydro-Dynamic stability analysis. This requires the calculation of terms related to magnetic field derivatives. The method presented here, based on Helmholtz's theorem, achieves continuity and convergence rate of order O(h2) for the magnetic field by using three-node linear triangular finite elements. It can be implemented as a postprocessor of the magnetic flux solution obtained with linear triangles, in such a way to provide the magnetic field with an accuracy O(h2). The continuity properties of the magnetic field provide a high level of accuracy and the possibility of deriving reliable linearized models with a limited computational effort. The accuracy is highly reliable for some applications requiring a high precision in the vicinity of magnetic poloidal field null points, like for instance breakdown analysis, or control of the X-point of diverted configurations in a tokamak. The effectiveness of the method is shown in linear and nonlinear cases for which analytical solutions are available.