Title page for ETD etd-06062008-165910

Type of Document Dissertation
Author Bochev, Pavel B.
URN etd-06062008-165910
Title Least squares finite element methods for the Stokes and Navier-Stokes equations
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Gunzburger, Max D. Committee Chair
Lin, Tao Committee Member
Peterson, Janet S. Committee Member
Ribbens, Calvin J. Committee Member
Rogers, Robert C. Committee Member
  • Least squares
Date of Defense 1994-07-18
Availability unrestricted
The central goal of this work is to define and analyze least squares finite element methods

for the Stokes and Navier-Stokes equations that are practical and optimal in a systematic

and rigorous way. To accomplish this task we begin by developing the least squares theory

for the linear Stokes equations. We introduce least squares methods based on the minimization

of functionals that involve residuals of the equations of an equivalent first order

formulation for the Stokes problem. We show that for the Stokes equations there are two

general types of boundary conditions. For the first type, practical least squares methods

can be defined and analized in a fairly standard way, based on application of the Agmon,

Douglis and Nirenberg a priori estimates. For the second type of boundary conditions this

task is more difficult and involves mesh dependent (weighted) least squares functionals.

Among the main results are the optimal error estimates for the weighted least squares

method in two and three space dimensions. Then, we formulate two least squares methods

for the nonlinear Na vier-Stokes equations written as a first order system. We consider the

first method as a conforming discretization of an abstract nonlinear problem and the second

weighted one, which is more practical, as a nonconforming discretization of the same

abstract problem. As a result, the analysis of the first method fits into the framework of the

approximation theory of Brezzi, Rappaz and Raviart and the analysis of the second does

not. Thus, we develop an abstract approximation theory that is suitable for nonconforming

discretizations of the abstract problem. The central result is based on the application

of our abstract theory to the weighted least squares method. We prove that this method

results in optimally accurate approximations for the Navier-Stokes equations. We believe

that these error analyses of Chapter are the first treatment of a least squares formulation

for a nonlinear problem in the current literature. We then discuss various implementation

issues, including theoretical and numerical estimates of the condition numbers and the

presentation of numerical examples. In particular, we study the numerical convergence rates

of various implementations of least squares methods and demonstrate that the weights are

necessary for the optimal rates to hold. Finally, we compare numerical results for the driven

cavity flow problem with some benchmark results reported in the literature.

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