Xu Zhang's Homepage

Research Grants

My research has been supported by the following grants and awards.

NSF Grant DMS-2224003 (PI)
NSF Grant DMS-2110833 (PI)
NSF Grant DMS-2005272 (PI)
NSF Grant DMS-1720425 (PI)
ORAU 2021 Powe Junior Faculty Enhancement Award (PI)

Research Interests

Numerical analysis and scientific computing
Numerical methods for partial differential equations
Finite element and discontinuous Galerkin methods
Immersed finite element methods for interface problems
A posteriori error estimation and adaptive finite element methods
Superconvergence of finite element methods

Research Description

My research focuses on immersed finite element (IFE) methods for interface problems and their applications. IFE methods are designed to solve partial differential equations with interfaces, including both static and moving interface problems, while allowing the use of meshes that do not necessarily conform to the interface geometry.

Static Interface Problems

In science and engineering, many simulations are carried out on domains consisting of multiple materials separated by curves or surfaces. This often leads to interface problems for partial differential equations, where the coefficients, solutions, or fluxes may be discontinuous across the material interface. Such problems can be solved by several classes of finite element methods, depending on how the mesh represents the interface.

$Body-fitting triangular mesh$ $Cartesian mesh cut by an interface$

Conventional finite element methods can be used to solve interface problems when the computational mesh is tailored to fit the interface geometry. Such a body-fitting mesh typically requires each element to lie essentially on one side of the interface.

Immersed finite element methods can use non-body-fitting meshes, such as Cartesian meshes, to solve interface problems. IFE basis functions are interface-dependent and are constructed to incorporate the physical jump conditions across the interface. My research in this direction includes the development, analysis, and implementation of IFE methods for elliptic interface problems, elasticity interface problems, and related adaptive algorithms.

Selected related publications:

Cuiyu He, Xu Zhang.
Residual based a posteriori error estimation for immersed finite element methods.
Journal of Scientific Computing, 81 (2019), no. 3, 2051–2079.
Tao Lin, Dongwoo Sheen, Xu Zhang.
A nonconforming immersed finite element method for elliptic interface problems.
Journal of Scientific Computing, 79 (2019), no. 1, 442–463.
Waixiang Cao, Xu Zhang, Zhimin Zhang.
Superconvergence of immersed finite element methods for interface problems.
Advances in Computational Mathematics, 43 (2017), no. 4, 795–821.
Tao Lin, Yanping Lin, Xu Zhang.
Partially penalized immersed finite element methods for elliptic interface problems.
SIAM Journal on Numerical Analysis, 53 (2015), no. 2, 1121–1144.
Highly cited paper in Mathematics in Web of Science.
Tao Lin, Qing Yang, Xu Zhang.
A priori error estimates for some discontinuous Galerkin immersed finite element methods.
Journal of Scientific Computing, 65 (2015), no. 3, 875–894.
Tao Lin, Dongwoo Sheen, Xu Zhang.
A locking-free immersed finite element method for planar elasticity interface problems.
Journal of Computational Physics, 247 (2013), 228–247.

Moving Interface Problems

Many simulations involve moving interfaces, such as phase-transition problems and free-boundary problems. A major advantage of immersed finite element methods is that they avoid repeated mesh generation as the interface evolves. Moreover, the number and locations of the degrees of freedom remain fixed, which makes it natural to combine IFE methods with the method of lines for efficient simulations on Cartesian meshes.