The Least-Squares Finite Element Method

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.