Finite Element Methods in Linear Ideal Magnetohydrodynamics

For more than ten years we have been working with the ideal linear MHD equations used to study the stability of thermonuc1ear plasmas. Even though the equations are simple and the problem is mathematically well formulated, the numerical problems were much harder to solve than anticipated. Already in the one-dimensional cylindrical case, what we called "spectral pollution" appeared. We were able to eliminate it by our "ecological solution". This solution was applied to the two-dimensional axisymmetric toroidal geometry. Even though the spectrum was unpolluted the precision was not good enough. Too many mesh points were necessary to obtain the demanded precision. Our solution was what we called the "finite hybrid elements". These elements are efficient and cheap. They have also proved their power when applied to calculating equilibrium solutions and will certainly penetrate into other domains in physics and engineering. During all these years, many colleagues have contributed to the construc­ tion, testing and using of our stability code ERATO. We would like to thank them here. Some ofthem gave partial contributions to the book. Among them we mention Dr. Kurt Appert, Marie-Christine Festeau-Barrioz, Roberto Iacono, Marie-Alix Secretan, Sandro Semenzato, Dr. Jan Vac1avik, Laurent Villard and Peter Merkel who kindly agreed to write Chap. 6. Special thanks go to Hans Saurenmann who drew most of the figures, to Dr.

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.