Computation of MHD equilibrium of Tokamak Plasma

This paper is concerned with an optimal control problem governed by a non-smooth semilinear elliptic equation. We show that the control-to-state mapping is directionally differentiable and precisely characterize its Bouligand subdifferential. By means of a suitable regularization, first-order optimality conditions including an adjoint equation are derived and afterwards interpreted in light of the previously obtained characterization. In addition, the directional derivative of the control-to-state mapping is used to establish strong stationarity conditions. While the latter conditions are shown to be stronger, we demonstrate by numerical examples that the former conditions are amenable to numerical solution using a semi-smooth Newton method.

Optimal control of a non-smooth semilinear elliptic equation

We propose and analyze a primal-dual active set method for discretized versions of the local and nonlocal Allen–Cahn variational inequalities. An existence result for the nonlocal variational inequality is shown in a formulation involving Lagrange multipliers for local and nonlocal constraints. Local convergence of the discrete method is shown by interpreting the approach as a semismooth Newton method. Properties of the method are discussed and several numerical simulations demonstrate its efficiency. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

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Primal-dual active set methods for Allen–Cahn variational inequalities with nonlocal constraints

This work is concerned with the iterative regularization of a non-smooth nonlinear ill-posed problem where the forward mapping is merely directionally but not Gâteaux differentiable. Using a Bouligand subderivative of the forward mapping, a modified Landweber method can be applied; however, the standard analysis is not applicable since the Bouligand subderivative mapping is not continuous unless the forward mapping is Gâteaux differentiable. We therefore provide a novel convergence analysis of the modified Landweber method that is based on the concept of asymptotic stability and merely requires a generalized tangential cone condition. These conditions are verified for an inverse source problem for an elliptic PDE with a non-smooth Lipschitz continuous nonlinearity, showing that the corresponding Bouligand–Landweber iteration converges strongly for exact data as well as in the limit of vanishing data if the iteration is stopped according to the discrepancy principle. This is illustrated with a numerical example.

Bouligand–Landweber iteration for a non-smooth ill-posed problem

A mimetic spectral element solver for the Grad-Shafranov equation

This paper presents numerical analysis of finite element approximation to a nonlinear eigenvalue problem: −Δu=λu + in Ω,u=−1 on Γ=∂Ω, where Ω is a bounded domain inR n (n=1, 2, 3). This problem arises in MHD (magnetohydrodynamics) equilibria and some other important physical phenomena. We consider a simple finite element scheme, and perform its error analysis. We also discuss a lumped finite element scheme, which is introduced to simplify the computations of the original scheme. Some numerical results are illustrated to show the validity of the analysis.

Kenji Nakazato

Papers

Finite element approximation of a nonlinear eigenvalue problem related to MHD equilibria