Analysis of the [L^2, L^2, L^2] least squares finite element method for incompressible Oseen-type problems
Ching L. Chang 1, Suh-Yuh Yang 21 Department of Mathematics, Cleveland State University, Cleveland, Ohio 44115, USA
2 Department of Mathematics, National Central University, Jhongli City 32001, Taiwan
Received by the editors February 9, 2006, and, in revised form, February 27, 2006
In this paper we analyze several first-order systems of Oseen-type equations that are obtained from the time-dependent incompressible Navier-Stokes equations after introducing the additional vorticity and possibly total pressure variables, time-discretizing the time derivative and linearizing the non-linear terms. We apply the [L-2, L-2, L-2] least-squares finite element scheme to approximate the solutions of these Oseen-type equations assuming homogeneous velocity boundary conditions. All of the associated least-squares energy functionals are defined to be the sum of squared L-2 norms of the residual equations over an appropriate products space. We first prove that the homogeneous least-squares functionals are coercive in the H-1 x L-2 x L-2 norm for the velocity, vorticity, and pressure, but only continuous in the H-1 x H-1 x H-1 norm for these variables. although equivalence between the homogeneous least-squares functionals and one of the above two product norms is not achieved, by using these a priori estimates and additional finite element analysis we are nevertheless able to prove that the least-squares method produces an optimal rate of convergence in the H-1 norm for velocity and suboptimal rate of convergence in the L-2 norm for vorticity and pressure. numerical experiments with various Reynolds numbers that support the theoretical error estimates are presented. In addition, numerical solutions to the time-dependent incompressible Navier-Stokes problem are given to demonstrate the accuracy of the semi-discrete [L-2, L-2, L-2] least-squares finite element approach.
AMS subject classifications: 65N15, 65N30, 76M10
Key words: Navier-Stokes equations; Oseen-type equations; finite element methods; least squares
Email: firstname.lastname@example.org (C. L. Chang), email@example.com (S.-Y. Yang)