Graduation Semester and Year




Document Type


Degree Name

Master of Science in Mechanical Engineering


Mechanical and Aerospace Engineering

First Advisor

Brian Dennis


The Least-Squares Finite Element Method (LSFEM) is a numerical method for solving partial differential equations (PDE) approximately by minimizing the L-2 norm of the PDE residuals. Unlike the more common Galerkin approach, this method does not employ integration by parts to reduce the continuity requirements of the basis functions. Instead, the PDE is cast as a first-order system of differential equations, which allows for the solution of primary and secondary variables with the same order of accuracy. In addition, this approach always leads to a direct minimization problem and therefore not subject to the restrictive inf-sup condition and does not result in an indefinite system of equations like the Galerkin method. Despite these advantages, it has been noted by other researchers that the choice of mesh and numerical integration scheme results in an implicit weighting of the functionals. This leads solutions that are very mesh sensitive and linear systems of equations that are ill-conditioned. This first part of this research is focused on creating a perfectly determined problem whose solution is independent of any implicit weighting. This is accomplished in 2-D by selectively subdividing quadrilateral mesh elements into triangles and employing reduced integration. This results in a discrete system with exactly the same number of equations as unknowns. A sensitivity analysis is used on the whole domain and elements which are most sensitive to a weight factor are split until the desired number of equations is reached. A 2-D LSFEM solver was developed for hybrid quadrilateral/triangle meshes to demonstrate the method for elliptic and hyperbolic-elliptic equations. The conjugate gradient method was used as a solver for the resulting system of equations. Results show the solution to the resulting well-determined system is independent of any user defined weights applied to the functionals. An optimized set of weights was then obtained to minimize the number of conjugate gradient iterations required to solve the linear system of equations.


Least squares finite element, Weight, Condition number


Aerospace Engineering | Engineering | Mechanical Engineering


Degree granted by The University of Texas at Arlington