Finite Element Methods in Linear Ideal Magnetohydrodynamics [Paperback]

1. Finite Element Methods for the Discretization of Differential Eigenvalue Problems.- 1.1 A Classical Model Problem.- 1.1.1 Exact Problem.- 1.1.2 Approximate Problem.- 1.1.3 Questions on Numerical Stability.- 1.2 A Non-Standard Model Problem.- 1.2.1 Exact Problem.- 1.2.2 Conforming Polluting Approximations.- 1.2.3 Non-Polluting Conforming Approximation.- 1.2.4 Non-Conforming Approximation.- 1.3 Spectral Stability.- 1.3.1 General Considerations.- 1.3.2 Stability Conditions.- 1.3.3 Order of Convergence.- 1.4 Finite Elements of Order p.- 1.4.1 Discontinuous Finite Elements S0p.- 1.4.2 Continuous Finite Elements S1p (Lagrange Elements).- 1.4.3 C1-Finite Elements S2p (Hermite Elements).- 1.4.4 Application to the Model Problems.- 1.4.5 Non-Conformmg Lagrange Elements.- 1.4.6 Non-Conforming Hermite Elements with Collocation.- 1.5 Some Comments.- 2. The Ideal MHD Model.- 2.1 Basic Equations.- 2.2 Static Equilibrium.- 2.3 Linearized MHD Equations.- 2.4 Variational Formulation.- 2.5 Stability Considerations.- 2.6 Mechanical Analogon.- 3. Cylindrical Geometry.- 3.1 MHD Equations in Cylindrical Geometry.- 3.1.1 The AGV and Hain-L?st Equations.- 3.1.2 Continuous Spectrum.- 3.1.3 An Analytic Solution.- 3.2 Six Test Cases.- 3.2.1 Test Case A: Homogeneous Currentless Plasma Cylinder..- 3.2.2 Test Case B: Continuous Spectrum.- 3.2.3 Test Case C: Particular Free Boundary Mode.- 3.2.4 Test Case D: Unstable Region for k = -0.2, m = 1.- 3.2.5 Test Case E: Unstable Region for k = -0.2, m = 2.- 3.2.6 Test Case F: Internal Kink Mode.- 3.3 Approximations.- 3.3.1 Conforming Finite Elements.- 3.3.2 Non-Conforming Finite Elements.- 3.4 Polluting Finite Elements.- 3.4.1 Hat Function Elements.- 3.4.2 Application to Test Case A.- 3.5 Conforming Non-Polluting Finite Elements.- 3.5.1 Linear Elements.- 3.5.2 Quadratic Elements.- 3.5.3 Third-Order Lagrange Elements.- 3.5.4 Cubic Hermite Elements.- 3.6 Non-Conforming Non-Polluting Elements.- 3.6.1 Linear Elements.- 3.6.2 Quadratic Elements.- 3.6.lÓ4