Applied Math Seminar
Monday, December 6, 1999, 4:00pm, 120 Physics
Peter K. Moore (Tulane University)
An Adaptive H-Refinement Finite Element For Solving Systems of Parabolic Partial Differential Equations in Three Space Dimensions
Abstract:- Adaptive methods for solving systems of partial differential equations
have become widespread. Robust
adaptive software for solving parabolic systems in one and two space
dimensions is now widely available.
Three spatial adaptive strategies and combinations thereof are
frequently employed: mesh refinement
(h-refinement); mesh motion (r-refinement); and order variation
(p-refinement). These adaptive
strategies are driven by a priori and a posteriori error estimates.
I will present an adaptive h-refinement finite element code in three
dimensions on structured grids.
These structured grids contain irregular nodes. Solution values at these
nodes are determined by
continuity requirements across element boundaries rather than by the
differential equations. The
differential-algebraic system resulting from the spatial discretization
is integrated using Linda
Petzold's multistep DAE code DASPK. The large linear systems resulting
from Newton's method applied to
nonlinear system of differential algebraic equations is solved using
preconditioned GMRES. In DASPK the
matrix-vector products needed by GMRES are approximated by a
``directional derivative''. Thus, the
Jacobian matrix need not be assembled. However, this approach is
inefficient. I have modified DASPK to
compute the matrix-vector product using stored Jacobian matrix. As in
the earlier version of DASPK,
DASSL, this matrix is kept for several time steps before being updated.
I will discuss appropriate
preconditioning strategies, including fast-banded preconditioners. In
three dimensions when using
multistep methods for time integration it is crucial to use a ``warm
restart'', that is, to restart the
dae solver at the current time step and order. This requires
interpolation of the history information.
The interpolation must be done in such a way that mode irregularity is
enforced on the new grid.
A posteriori error estimates on uniform grids can easily be generalized
from two-dimensional results
(Babuska and Yu showed that in the case of odd order elements, jumps
across elemental boundaries give
accurate estimates, and in the case of even order elements, local
parabolic systems must be solved to
obtain accurate estimates). Babuska's work can even be generalized to
meshes with irregular modes but
now they no longer converge to the true error (in the case of odd order
elements). I have developed a
new set of estimates that extend the work of Babuska to irregular meshes
and finite difference methods.
These estimates provide a posteriori error indicators in the finite
element context.
Several examples that demonstrate the effectiveness of the code will be
given.
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