(1990) introduced what is now known as the BPX-preconditioner. This
method, originally described in Xu’s PhD thesis (1989), is one of the
two most powerful multigrid approaches for solving large-scale
algebraic systems that arise from the discretization of models in
science and engineering described by partial differential equations.
The method has been widely used by researchers and practitioners since
(1996) is a pioneering work on the auxiliary space method, a
technique that uses a more structured space to construct an efficient
subspace correction method for less structured problems. A
generalization of this idea when used in concert with the BPX
preconditioner led to the optimal preconditioner of Hiptmair-Xu
(2007) for Maxwell equations, and it was recently identified by the
U.S. Department of Energy as one of the top
ten breakthroughs in computational science in recent years.
Researchers from Sandia, Los Alamos, and Lawrence Livermore National
Labs use this algorithm for modeling fusion with magnetohydrodynamic
equations. Moreover, this approach will also be instrumental in
developing optimal iterative methods in structural mechanics,
electrodynamics, and modeling of complex flows.
|Lee-Xu (2006) presented an original approach to
the numerical modeling of complex fluids.
Their provably stable numerical scheme for non-Newtonian
flows to simulate rheological
phenomena leads to
numerical techniques that preserve the positive
definiteness of the conformation tensor for any
range of Weissenberg numbers. This
represents a significant advance toward the
solution of the famous High Weissenberg Number Problem.
In addition, Lee-Xu-Zhang
(2010) developed numerical methods for
non-Newtonian fluids that guarantee the
discrete system has a unique solution and there
exists an iterative algorithm that converges
uniformly with respect to the Weissenberg number
and Reynolds number.
Non-Newtonian flow passing through a cylinder. Left: velocity and pressure at t=0.5; Right: stress at t=0.5.
Left: Drag coefficent (different Weissenberg number); Right: History of drag coefficient.
An Optimal Preconditioner for the Biharmonic Problem
|Zhang-Xu (2012) proposed the first mathematically
provable O(N log N) algorithm for linear systems
arising from the direct finite element
discretization of fourth-order problems on an
unstructured grid of an arbitrary domain.
This one-grid multilevel method presents a new
approach to applying the divide-and-conquer
strategy. It shows that some mixed-form
discretizations of the fourth-order problems,
which of themselves lead to non-desirable—i.e., either
non-optimal or nonconvergent—approximations of
the original solution, can provide optimal
preconditioners for direct finite element
discretizations. It is rigorously shown
that the preconditioners can be reduced to the
solution of a fixed number of discrete Poisson
equations. This approach will also be
instrumental in the development of optimal
iterative methods in high-order problems.
Eigenvalue distribution of a preconditioned system: Only a few “bad” eigenvalues are evident, which is favorable for the PCG method.