Computer Physics Communications 31(1984)137—141 137 North-Holland, Amsterdam ON THE NUMERICAL APPROXIMATION OF A FREE BOUNDARY PROBLEM RELATED TO MHD EQUILIBRIA G. CALOZ and J. RAPPAZ Department of Mathematics, EPFL, 1015 Lausanne, Switzerland We are interested in the nonlinear problem — 4u = A{u~ + s(u~ )2) in SI, u — don OS?, — J~(au/an)ds = I where SI is a convex polygonal domain in R 2 with boundary 1)52; here I is a given positive number, the function u and the positive constant A ase the unknowns of the problem, the real numbers d and js are parameters. Using a variant of the implicit function theorem, we can prove the existence of a solution branch of this problem, depending on d and ~s;we investigate its approximation by a finite element method; the error analysis of the discrete problem is given. 1. Introduction ~Q. We impose the boundary conditions u=O onaQ~, (1.4) We consider the model of the ideal MHD equi- librium of a plasma confined in a toroidal cavity, u = —d on ~Q, (1.5) see for example the appendix in ref. [131. In a plane Orz, the meridian section of the tokamak is where d is an a priori unknown positive constant. a domain Q with boundary 3S~,and Oz is the axis This choice is justified if the equilibrium is ensured of symmetry. We suppose ~S?fl Oz = 4). The by currents in a conducting casing around the plasma fills an unknown subdomain f2,~, of Q with plasma [9]. The function u must satisfy the pres- boundary af2~. We assume that o~, = £2 — is sure balance relation across the plasma—vacuum vacuous. The space is referred to the cylindrical interface, i.e. coordinate system (r, 9, z). From the MHD equa- tions, we derive in terms of the meridian magnetic — is continuous on aQ~, (1.6) stream function u, the relations where a/an is the outward normal derivative on = g(r, u) in (1.1) as2~. Moreover, if u is a physical solution .2’u=O inQ~, (1.2) u>O inQ~. (1.7) where One current profile which occurs in ref. [6] is a /1 a \ i a2 + u \Umax )2}; and g(r, u) is the toroidal current density whose here u expression is in principle not known. We also max — max(rz)~ u(r, z), a is a given posi- impose the total toroidal current i + i.e. tive constant, 8> 0 and i~> —1 are real numbers. In fact, the physical problem consists in finding d, 1 au ~, 8, u, Q~, satisfying (1.1)—(1.7) and such that (1.3) passes through two fixed points A and B in £2. By using the above definition of g(r, u) and the where a/an is the outward normal derivative on maximum principle, the eqs. (1.1),(1.2), (1.4) and O010-4655/84/$03.0O © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)