## Research Interests

- Broad Areas of Scientific Computing and Numerical Analysis
- Finite Element Method, Discontinuous Galerkin Method, Virtual Element Method
- Machine Learning Methods for Partial Differential Equations
- Interface problems and Coupling Mathematical Models Arising from Applications
- Design, Analysis and Applications of Immersed Finite Element Method for interface problems

## Research Description

**
** Interface Problem

In scientific and engineering, simulations based on partial differential equations (PDE) are widely used among mechanics, physics and portfolio investment. PDEs with discontinuous coefficients on several sub-domains separated by curve 'interface' are getting attention in real world application, e.g. modeling displacement of an elastic plane with a circular inclusion made by dissimilar materials. To handle these so called 'interface' problems, Immersed Finite Element Method (Z.Li, 1998) was proposed.

As shown in the below figure, the whole domain \(\Omega\) is separated into \(\Omega^+\) and \(\Omega^-\) by red interface. Classical Finite Element method constructs 'body-fitting' mesh (left) to fit the interface. Under such treatment, every element lies in one of the subdomains. However, Immersed Finite Element method just lets the interface pass through elements and constructs basis functions directly on standard Cartesian mesh. This is so-called non-body-fitting mesh (right) which saves computational resource of classical method. To adapt to interface, local space on elements with interface are 'immersed'. Notably this construction ensures that the degree of freedom is inmdependent of interface locations.

My research focuses on design, analysis and application of Immersed Finite Element method. A branch of my work deals with interfaces with geometric singularities. Examples of these singularities include triple point in multi-domain interface problem (K.Xia et al, 2011), sharp edge interface problem (S.Hou et al, 2010), etc (Also see (b) in the following diagram for a heart shape interface with one singularity point). Recently, I focus on interface problems under the context of Computational Fluid Dynamics. Sepecificly, we follow spirits of Immersed Finite Element method to develop efficient unfitted numerical methods for Stokes and Navier-Stokes interface problem. One famous application of such modeling problem is the simulation of multi-phase flow. See (a) in the following diagram for an exemplified illustration of two-phase flow: rising bubbles.

**Related publications:**

**Y.Chen**, S.Hou, X.Zhang, An Immersed Finite Element Method for Elliptic Interface Problems with Multi-Domain and Triple Junction Points.**AAMM**, 11 (2019), no. 5, 1005-1021.**Y.Chen**, S.Hou, X.Zhang, A Bilinear Partially Penalized Immersed Finite Element Method for Elliptic Interface Problems with Multi-domains and Triple Junction Points.**RINAM**, 8 (2020), 100100.**Y.Chen**, X.Zhang, A \(\mathcal{P}_2\)-\(\mathcal{P}_1\) Partially Penalized Immersed Finite Element Method for Stokes Interface Problem.**IJNAM**, 18 (2021), no. 1, 120-141.