Synopses & Reviews
This is the first book devoted to the least-squares finite element method (LSFEM), which is a simple, efficient and robust technique for the numerical solution of partial differential equations. The book demonstrates that the LSFEM can solve a broad range of problems in fluid dynamics and electromagnetics with only one mathematical/computational formulation. The book shows that commonly adopted special treatments in computational fluid dynamics and computational electromagnetics, such as upwinding, numerical dissipation, staggered grid, non-equal-order elements, operator splitting and preconditioning, edge elements, vector potential, and so on, are unnecessary. This book introduces the basic theory of the least-squares method for first-order PDE systems, particularly the div-curl system and the div-curl-grad system. It is applied to the study of permissible boundary conditions for the incompressible Navier--Stokes equations, to show that the divergence equations in the Maxwell equations are not redundant, and to derive equivalent second-order versions of the Navier--Stokes equations and the Maxwell equations. This book covers diverse applications such as incompressible viscous flows, rotational inviscid flows, low- or high-Mach-number compressible flows, two-fluid flows, convective flows, and scattering waves.
Here is a comprehensive introduction to the least-squares finite element method (LSFEM) for numerical solution of PDEs. It covers the theory for first-order systems, particularly the div-curl and the div-curl-grad system. Then LSFEM is applied systematically to permissible boundary conditions for the incompressible Navier-Stokes equations, to show that the divergence equations in the Maxwell equations are not redundant, and to derive equivalent second-order versions of the Navier-Stokes equations and the Maxwell equations. LSFEM is simple, efficient and robust, and can solve a wide range of problems in fluid dynamics and electromagnetics, including incompressible viscous flows, rotational inviscid flows, low-Mach-number compressible flows, two-fluid and convective flows, scattering waves, etc.
This is the first monograph on the subject, providing a comprehensive introduction to the LSFEM method for numerical solution of PDEs. LSFEM is simple, efficient and robust, and can solve a wide range of problems in fluid dynamics and electromagnetics.
Table of Contents
Contents (preliminary): I. The Basic Concept of the LSFEM.- 1. Introduction.- 2. The first-order Scalar Differential Equation in One-Dimension.- 3. The First-Order System in One-Dimension.- II. Fundamentals of the LSFEM.- 4. Fundamentals of the LSFEM.- 5. The Div-Curl System.- 6. The Div-Curl-Grad System.- III. The LSFEM in Fluid Dynamics.- 7. Inviscid Irrotational Flows.- 8. Incompressible Viscous Flows.- 9. Convective Transport.- 10. Rotational Inviscid Flows.- 11. Two-face Flows.- 12. Compressible Viscous Flows.- 13. High-Speed Compressible Flows.- 14. P-Version Least-Squares Finite Element Method.- IV. The LSFEM in Electromagnetics.- 15. Electromagnetics.- V. Solution of the Discrete Equations 16. Iterative Methods for Solving Linear Systems of Equations.- Appendix A: Operation on Vectors.- B. Green's Formula.- C. Finite Element Interpolation.- D. The Lax-Milgram Theory.- References.- Index.