Pure Theories

Subspace Correction


XZ identity
XuZikatanov
(2002), published in
the Journal of
American Mathematical Society (JAMS) (a top journal
in all areas of pure and applied mathematics), made
substantial progress in regard to the theory of the
subspace correction method and obtained the
sharpest possible estimate for the convergence
of the subspace correction method. Most of the
existing estimates (which have been studied in
hundreds of papers) can be easily derived from
this identity. Consider the linear system $A u = f$. Let operator $B$ be defined by the Sucessive Subspace Correction (SSC) method. Then, the following XZ identity holds when each subspace problem is solved exactly: $$ \ I  BA \^{2}_{A} = 1  \frac{1}{1+c_{0}}, $$ where $c_0 = \sup_{\v\_A = 1} \inf_{\sum v_i = v} \sum_{i=0}^{J} \ P_i \sum_{j>i} v_j \_{A_i}^2$. 

Removal of “1”
XuZikatanov
(2003) made a key observation on
abstract
error estimates for Galerkin approximations
based on BabuskaBrezzi conditions. A
basic error estimate $$ \ u  u_h \_{U} \le (1 + C) \inf_{w_h} \ u  w_h \_{U} $$ is sharpened by means of an identity whereby $\ P \ = \ I  P \$ for any nontrivial idempotent operator $P$. The constant “1” is removed in general, i.e., $$ \ u  u_h \_{U} \le C \inf_{w_h} \ u  w_h \_{U}. $$ 

WangXu Elements for Highorder PDEs in High Dimensions
WangXu
(2013) 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
spatialdimensions. This method allows
piecewise polynomials of the lowestorder degree
to be used in constructing convergent, stable,
and practical finite element discretization
methods for higherorder partial differential
equations, including those that occur in
magnetohydrodynamics modeled in plasma physics
and shell and plate models in structural
mechanics. 

Lower Bounds of the Discretization Error for Piecewise Polynomials
LinXieXu
(2014)
proved the sharp lowerbound error estimate of
the approximation error by piecewise
polynomials function spaces. Precisely, the
following lower error bounds are valid for
finite element (consisting of piecewise
polynomials of a degree less than $r$)
approximation to 2$m$th order elliptic
boundary value problems:
$$ \uu_h\_{j,p,h} \geq Ch^{rj}, \quad 0\leq j \leq r, \quad \forall u \in W^{r+\delta,p}(\Omega), $$ where the positive constant $C$ is independent of the mesh size $h$. This result is further extended to various situations including general shape regular grids and many different types of finite element spaces. It plays a fundamental role in the numerical analysis, especially in the analysis lowerbound eigenvalue approximations. 