Anisotropic Diffusion in Toroidal geometries

Abstract

Abstract. In this work, we present a new finite element framework for toroidal geometries based on a tensor product description of the 3D basis functions. In the poloidal plan, different discretizations, including B-splines and cubic Hermite-Bézier surfaces are defined, while for the toroidal direction both Fourier discretization and cubic Hermite-Bézier elements can be used. In this work, we study the MHD equilibrium by solving the Grad-Shafranov equation, which is the basis and the starting point of any MHD simulation. Then we study the anisotropic diffusion problem in both steady and unsteady states.

A Two Field Iterated Asymptotic-Preserving Method for Highly Anisotropic Elliptic Equations

“…Let us now prove that the iterative scheme (16)(17) converges and that the limit solution solves the original singular perturbation problem.…”

“…In this method, the original strongly anisotropic elliptic problem (10) is replaced by a set of two only mildly anisotropic equations parameterized by ε 0 ε. Moreover, the matrix to be inverted in the first step (16) of the iterative method is the same as in the final step (17), the only difference is in the right hand side of the equation. That is to say, the matrix has to be factorized only once, the rest of the iterative scheme is a fast triangular system solve.…”

“…This method does note require any discretization of the space G (functions constant in the direction of the anisotropy), which can be complicated for generic field b. To be complete, the variational formulation of the iterative scheme (16)(17) is stated:…”

“…Theorem 2. For any (q 0 , u 0 ) ∈ V × V, the sequence (q n , u n ) n>0 defined by the iterative method (16)(17) converges to a solution (q,ū). The componentū of the stationary point solves uniquely the initial singular perturbation problem (10) for ε > 0 and the limit problem (12) when ε = 0.…”

“…Lemma 5 (Orthogonality of q n+1 − q n with respect to w ∈ G). For any q 0 ∈ V all functions in the sequence (q n ) n≥0 issued from the iterative method (16)(17) differ from each other only by a function orthogonal to G, the space of functions constant in the direction of anisotropy with respect to the H 1 seminorm. That is to say, for any i, j ≥ 0 the difference q j − q i is orthogonal to G with respect to the H 1 seminorm.…”