What could we gain, by using the locally divergence-free solution
spaces in discontinuous Galerkin methods? (Fengyan Li, University of
South Carolina)
Abstract:
There are many partial differential equations with solutions which
have divergence-free components. Examples include the incompressible
Euler and Navier-Stokes equations, the magnetohydrodynamics (MHD)
equations, and the Maxwell equations. For some of the problems,
such as the MHD equations and the Maxwell equations, the divergence
constraints seem to be redundant, as in the continuous model, the
solutions will automatically satisfy the divergence-free conditions if
the initial data is divergence-free. However, many works in
literature show that the actual negligence in dealing with the
divergence-free condition numerically can lead to serious defects of
the schemes.
We here try to explore the capabilities of a new type of methods, the
locally divergence-free discontinuous Galerkin Methods, through
intensive theoretical and numerical studies. In particular, we focus
on their applications in Maxwell equations and MHD equations in this presentation.
This is an abstract of a talk to be presented at the
2004 SEAMS Workshop in Charleston, SC. For more information, visit
the workshop's homepage at math.cofc.edu/SEAMS.