Pure Theories

  • Subspace Correction

In Xu (1992), a general theoretical framework on iterative methods based on space decompositions and subspace correction is proposed. This framework changed the understanding of iterative methods and, together with subsequent research on subspace correction methods, it is considered a milestone in the development of multilevel iterative methods. Special meetings are devoted to the method proposed in this paper.

  • X-Z identity

Xu-Zikatanov (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 X-Z 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”

Xu-Zikatanov (2003) made a key observation on abstract error estimates for Galerkin approximations based on Babuska-Brezzi 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}. $$

  • Wang-Xu Elements for High-order PDEs in High Dimensions

Wang-Xu (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 spatial-dimensions. This method allows piecewise polynomials of the lowest-order degree to be used in 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 plate models in structural mechanics.


  • Lower Bounds of the Discretization Error for Piecewise Polynomials

Lin-Xie-Xu (2014) proved the sharp lower-bound 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: $$
\|u-u_h\|_{j,p,h} \geq Ch^{r-j}, \quad 0\leq j \leq r, \quad \forall u \in W^{r+\delta,p}(\Omega), $$
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 lower-bound eigenvalue approximations.