In this research, least-squares based finite element formulations and their applications
in fluid mechanics are presented. Least-squares formulations offer several computational
and theoretical advantages for Newtonian as well as non-Newtonian fluid flows. Most
notably, these formulations circumvent the inf-sup condition of Ladyzhenskaya-Babuska-
Brezzi (LBB) such that the choice of approximating space is not subject to any compatibility
condition. Also, the resulting coefficient matrix is symmetric and positive-definite. It
has been observed that pressure and velocities are not strongly coupled in traditional leastsquares
based finite element formulations. Penalty based least-squares formulations that
fix the pressure-velocity coupling problem are proposed, implemented in a computational
scheme, and evaluated in this study. The continuity equation is treated as a constraint on
the velocity field and the constraint is enforced using the penalty method. These penalty
based formulations produce accurate results for even low penalty parameters (in the range
of 10-50 penalty parameter). A stress based least-squares formulation is also being proposed
to couple pressure and velocities. Stress components are introduced as independent
variables to make the system first order. The continuity equation is eliminated from the
system with suitable modifications. Least-squares formulations are also developed for viscoelastic
flows and moving boundary flows. All the formulations developed in this study
are tested using several benchmark problems. All of the finite element models developed
in this study performed well in all cases.
A method to exploit orthogonality of modal bases to avoid numerical integration and have a fast computation is also developed during this study. The entries of the coefficient
matrix are calculated analytically. The properties of Jacobi polynomials are used and most
of the entries of the coefficient matrix are recast so that they can be evaluated analytically
