Nellie Rajarova, (2009-05). A Comparison of Least-Squares Finite Element Models with the Conventional Finite Element Models of Problems in Heat Transfer and Fluid Mechanics. Master's Thesis.
Thesis
In this thesis, least-squares based finite element models (LSFEM) for the Poisson equation and Navier-Stokes equation are presented. The least-squares method is simple, general and reliable. Least-squares formulations offer several computational and theoretical advantages. The resulting coefficient matrix is symmetric and positive-definite. Using these formulations, the choice of approximating space is not subject to any compatibility condition. The Poisson equation is cast as a set of first order equations involving gradient of the primary variable as auxiliary variables for the mixed least-square finite element model. Equal order C0 continuous approximation functions is used for primary and auxiliary variables. Least-squares principle was directly applied to develop another model which requires C1continous approximation functions for the primary variable. Each developed model is compared with the conventional model to verify its performance. Penalty based least-squares formulation was implemented to develop a finite element for the Navier Stokes equations. The continuity equation is treated as a constraint on the velocity field and the constraint is enforced using the penalty method. Velocity gradients are introduced as auxiliary variables to get the first order equivalent system. Both the primary and auxiliary variables are interpolated using equal order C0 continuous, p-version approximation functions. Numerical examples are presented to demonstrate the convergence characteristics and accuracy of the method.
In this thesis, least-squares based finite element models (LSFEM) for the Poisson equation and Navier-Stokes equation are presented. The least-squares method is simple, general and reliable. Least-squares formulations offer several computational and theoretical advantages. The resulting coefficient matrix is symmetric and positive-definite. Using these formulations, the choice of approximating space is not subject to any compatibility condition. The Poisson equation is cast as a set of first order equations involving gradient of the primary variable as auxiliary variables for the mixed least-square finite element model. Equal order C0 continuous approximation functions is used for primary and auxiliary variables. Least-squares principle was directly applied to develop another model which requires C1continous approximation functions for the primary variable. Each developed model is compared with the conventional model to verify its performance. Penalty based least-squares formulation was implemented to develop a finite element for the Navier Stokes equations. The continuity equation is treated as a constraint on the velocity field and the constraint is enforced using the penalty method. Velocity gradients are introduced as auxiliary variables to get the first order equivalent system. Both the primary and auxiliary variables are interpolated using equal order C0 continuous, p-version approximation functions. Numerical examples are presented to demonstrate the convergence characteristics and accuracy of the method.