My research ranges from pure theoretical analysis, basic algorithmic development to practical applications I mainly work on numerical methods for partial differential equations (PDEs), such as finite element, multi-grid (MG) and domain decomposition (DD) methods. Theoretical elegance and practical usefulness can go together, and the design and analysis of algorithms can be beautiful. Theory is the soul of what I do, and practical needs are what motivate me.  One thing that I very much enjoy doing these days is to help people to speed up their simulation codes from a few times to a few orders of magnigudes (for very large scale problems).  In all my work, I try to strike a balance between rigor versus practicality.

Examples of my better known works include the Bramble-Pasciak-Xu preconditioner (a basic algorithm for solving elliptic PDEs) and the Hiptmair-Xu preconditioner (an effective Maxwell solver which was featured in a 2008 report by the U.S. Department of Energy as one of the top 10 breakthroughs in computational science in recent years). I developed the framework and theory of the Method of Subspace Corrections that have been widely used in the literature for the design and analysis of iterative methods and later established the Xu-Zikatanov identity, giving the optimal theory for these methods. Other examples include: