TY - JOUR
T1 - Analysis of the [$L^2$, $L^2$, $L^2$] Least-Squares Finite Element Method for Incompressible Oseen-Type Problems
AU - C. L. Chang & S.-Y. Yang
JO - International Journal of Numerical Analysis and Modeling
VL - 3-4
SP - 402
EP - 424
PY - 2007
DA - 2007/04
SN - 4
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/869.html
KW - Navier-Stokes equations, Oseen-type equations, finite element methods, least squares.
AB - <p style="text-align: justify;">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 $H^1 \times L^2 \times L^2$ norm for the velocity, vorticity, and pressure, but only
continuous in the $H^1 \times H^1 \times H^1$ 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.</p>