Ph.D. (2004) Cornell University
Abstract: We study least-squares finite element methods for second-order elliptic partial differential equations.
Bramble, Lazarov and Pasciak  proposed a least-squares method which could be thought of as a stabilized Galerkin method. We modify their method by weakening the strength of the stabilization terms and present various error estimates. The modified method has all the desirable properties of the original method. At the same time, our numerical experiments show an improvement of the method due to the modification.
We also study a first-order least-squares method proposed by Cai, Lazarov, Manteuffel and McCormick . The method does not require the inf – sup condition. We bound the error in the primary function u and the flux p separately and obtain various error estimates with different approximating spaces.
(Note: The p above should have a bar under it.)