Least squares finite element methods for the Stokes and Navier-Stokes equations
Bochev, Pavel B.
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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 analyzed 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 Navier-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 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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