Finite Element Methods in Linear Ideal Magnetohydrodynamics

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-Lust 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.- 3.6.3 Lagrange Cubic Elements.- 3.6.4 Hermite Cubic Elements with Collocation.- 3.7 Applications and Comparison Studies (with M.-A. Secretan).- 3.7.1 Application to Test Case A.- 3.7.2 Application to Test Case B.- 3.7.3 Application to Test Case C.- 3.7.4 Application to Test Case F.- 3.8 Discussion and Conclusion.- 4. Two-Dimensional Finite Elements Applied to Cylindrical Geometry.- 4.1 Conforming Finite Elements.- 4.1.1 Conforming Triangular Finite Elements.- 4.1.2 Conforming Lowest-Order Quadrangular Finite Elements.- 4.2 Non-Conforming, Finite Hybrid Elements.- 4.2.1 Finite Hybrid Elements Formulation.- 4.2.2 Lowest-Order Finite Hybrid Elements.- 4.2.3 Application to the Test Cases.- 4.2.4 Explanation of the Spectral Shift.- 4.2.5 Convergence Properties.- 4.3 Discussion.- 5. ERATO: Application to Toroidal Geometry.- 5.1 Static Equilibrium.- 5.1.1 Grad-Schluter-Shafranov Equation.- 5.1.2 Weak Formulation.- 5.2 Mapping of (?, ?) into (?, ?) Coordinates in ?p.- 5.3 Variational Formulation of the Potential and Kinetic Energies..- 5.4 Variational Formulation of the Vacuum Energy.- 5.5 Finite Hybrid Elements.- 5.6 Extraction of the Rapid Angular Variation.- 5.7 Calculation of ?-Limits (with F. Troyon).- 6. HERA: Application to Helical Geometry (Peter Merkel, IPP Garching).- 6.1 Equilibrium.- 6.2 Variational Formulation of the Stability Problem.- 6.3 Applications.- 6.3.1 Straight Heliac.- 6.3.2 Straight Heliotron Equilibria.- 6.3.3 Large-k Ballooning Modes.- 6.3.4 Conclusion.- 7. Similar Problems.- 7.1 Similar Problems in Plasma Physics.- 7.1.1 Resistive Spectrum in a Cylinder.- 7.1.2 Non-Linear Plasma Wave Equation (with M. C. Festeau-Barrioz).- 7.1.3 Alfven and ICRF Heating in a Tokamak (with K. Appert, T. Hellsten, J. Vaclavik, and L. Villard).- 7.2 Similar Problems in Other Domains.- 7.2.1 Stability of a Compressible Gas in a Rotating Cylinder..- 7.2.2 Normal Modes in the Oceans.- Appendices.- A: Variational Formulation of the Ballooning Mode Criterion.- B.1 The Problem.- B.2 Two Numberings of the Components.- B.3 Resolution for Numbering (D1).- B.4 Resolution for Numbering (D2).- B.5 Higher Order Finite Elements.- C: Organization of ERATO.- D: Listing of ERATO 3 (with R. Iacono).- References.