## Research Grants

**My research is partially supported by**

**the NSF Grant DMS-2110833 (PI)****the NSF Grant DMS-2005272 (PI)****the NSF Grant DMS-1720425 (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 the Immersed Finite Element (IFE) methods for interface problems and their applications. IFE methods can be used to solve both static interface problems and moving interface problems.

### Static Interface Problems

In science and engineering, many simulations are carried out over domains consisting of multiple materials which are separated by curves or surfaces. This usually leads to the so-called interface problems of partial differential equations whose coefficients are discontinuous across the material interface. These interface problems can be solved by different types of finite element methods.

**Conventional Finite Element Method**
can be used to solve the interface problems provided that the solution mesh is tailored to fit the interface geometry (body-fitting mesh). Geometrically, such mesh requires each element to be placed essentially on one side of the interface.

**Immersed Finite Element (IFE) Method**
can use non-body-fitting meshes, such as Cartesian meshes to solve interface problem. IFE basis functions are interface-dependent and they are constructed to fit physical interface jump conditions. We focus on IFE methods for the second order elliptic interface problems and elasticity interface problems. Our research includes both developing efficient IFE methods and carrying out related error estimation.

**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. An immediate benefit of using **Immersed Finite Element Method** is the avoidance of regenerating meshes. More importantly, numbers and locations of degrees of freedom remain unchanged even though interfaces evolve. These features enable us to combine IFEs with the method of lines to efficiently solve moving interface problems on a Cartesian mesh.

**Related publications:**

- Wenqiang Feng, Xiaoming He, Yanping Lin, Xu Zhang

Immersed finite element method for interface problems with algebraic multigrid solver.

**Communications in Computational Physics**15 (2014), no. 4, 1045-1067. - Tao Lin, Yanping Lin, Xu Zhang

A method of lines based on immersed finite elements for parabolic moving interface problems.

**Advances in Applied Mathematics and Mechanics**5 (2013), no. 4, 548-568. - Tao Lin, Yanping Lin, Xu Zhang

Immersed finite element method of lines for moving interface problems with nonhomogeneous flux jump.

**Contemporary Mathematics**586 (2013), 257-265. - Xiaoming He, Tao Lin, Yanping Lin, Xu Zhang

Immersed finite element methods for parabolic equations with moving interface.

**Numerical Methods for Partial Differential Equations**29 (2013), no. 2, 619-646.