|dc.description.abstract||We study a class of numerical schemes for Navier-Stokes equations
(NSE) or Stokes equations (SE) for incompressible fluids in a
bounded domain with given boundary value of velocity.
The incompressibility constraint and non-slip boundary condition
have made this problem very challenging. Their treatment by finite
element method leads to the well-known inf-sup compatibility
condition. Their treatment by finite difference method leads to
the very popular projection method, which suffers from low
resolution near the boundary.
In [LLP], the authors propose an unconstrained formulation of NSE
or SE, which replace the divergence-free constraint by a pressure
equation with an appropriate boundary condition. All of the
schemes in this thesis are based on this new formulation. In
contrast to traditional methods, these schemes do not need to
fulfill the traditional inf-sup compatibility condition between
velocity space and pressure space. More importantly, they can
achieve high-order accuracy very easily and are very efficient due
to the decoupling of the update of velocity and pressure. They can
even be proved to be unconditionally stable.
There are two ways to analyze the schemes that we propose. The
first is based upon the sharp estimate of the pressure in [LLP].
The second relies on a nice identity.
Using the pressure estimates, we propose and study a $C^1$ finite
element (FE) scheme for the steady-state SE as well as for the
time-dependent NSE. For steady-state SE, we can either use an
iterative scheme or solve velocity and pressure together.
Using the nice identity, we prove that the semi-discrete iterative
scheme for the steady-state SE converges ("semi-discrete" means
that the spatial variable are kept continuous). This identity will
also be crucial for our proofs of the stability and error
estimates of the time-dependent $C^0$ FE schemes.
Associated numerical computations demonstrate stability and
accuracy of these schemes.
We also present the numerical results of yet another $C^0$ FE
scheme ([JL]) for the time-dependent NSE for which the theory of
the fully discrete case is yet lacking.||en_US
|dc.title||A Class of Stable, Efficient Navier-Stokes Solvers||en_US
|dc.contributor.publisher||Digital Repository at the University of Maryland||en_US
|dc.contributor.publisher||University of Maryland (College Park, Md.)||en_US
|dc.subject.pquncontrolled||incompressible Navier-Stokes equations||en_US