Basic Algorithms

BPX preconditioner
BramblePasciakXu
(1990) introduced what is now known as the BPXpreconditioner. This
method, originally described in Xu’s PhD thesis (1989), is one of the
two most powerful multigrid approaches for solving largescale
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
1990.


HX Preconditioner
Xu
(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 HiptmairXu
(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.





Finite Element Method for Highorder PDEs
WangXu
(2007, 2011) gives the only
canonical and universal construction of a class of convergent finite
element spaces for any order
of elliptic and parabolic equations in any spatial dimension. This method
allows the use of piecewise polynomials of the lowestorder degree for
constructing convergent, stable, and practical finite element
discretization methods for higher order partial differential equations,
including those that occur in magnetohydrodynamics modeled in plasma
physics and shell and those that in occur in plate models in structural
mechanics. 
Recently, XuZhang
(2012) applied order reduction on the solver level and developed a
robust and efficient preconditioner for biharmonic problems on
unstructured grids using the auxiliary space method framework.
The new preconditioner is formed by smoothers and Poisson
solvers. If the domain is convex, the
condition number is bounded by a constant. If the domain is nonconvex,
there are just a few bad eigenvalues (no more than the number of
concave corner points of the domain), such that the preconditoned
Krylov method will still be efficient.


Complex Fluids
LeeXu
(2006) presented an original approach to the numerical modeling of
complex fluids. Their provably stable numerical scheme for nonNewtonian
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, LeeXuZhang
(2010) developed numerical methods for nonNewtonian 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.

NonNewtonian 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
ZhangXu
(2012) proposed the first mathematically provable O(N log N)
algorithm for linear systems arising from the direct finite element
discretization of fourthorder problems on an unstructured grid of an
arbitrary domain. This onegrid multilevel method presents a new
approach to applying the divideandconquer strategy. It shows that
some mixedform discretizations of the fourthorder problems, which of
themselves lead to nondesirable—i.e., either nonoptimal 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 highorder problems. 
Eigenvalue distribution of a
preconditioned system: Only a few “bad” eigenvalues are evident, which
is favorable for the PCG method.
