Finite Element Methods in Linear Ideal Magnetohydrodynamics

Inhaltsverzeichnis

• 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-Lüst 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. Secrétan).
• 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-Schlüter-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 Alfvén 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.