# Shuhao Cao

## Contact Information

Email: scao (at) math DOT purdue DOT edu
Office: 435 math building

## Backgrounds

### Education

B.S, Sichuan University, 2002-2006
M.S, Purdue University, 2006-2008
Ph.D, Purdue University, 2008-present

### Research Interests

1. Adaptive finite element method for partial differential equations with discontinuous coefficients from physics in 3D
- The design and analysis of the a posteriori error estimation technique
2. Construction of finite element spaces and efficient matrix assembly
- Hierarchical differential form basis on simplex (Nédélec first and second kinds, Raviart-Thomas, Brezzi-Douglas-Marini)
- Finite element exterior calculus in 4D spacetime mesh
- Canonical construction of nonconforming finite element spaces based on exterior calculus
- Symmetric composite element for stress tensor
3. Spacetime Discontinuous Galerkin finite element methods for hyperbolic conservation laws
The work of my Ph.D. thesis mainly focused on the designing various a posteriori error estimators, proving the reliability and efficiency bounds, and implementing the adaptive finite element method for the following problem in a bounded Lipschitz domain $$\Omega\subset \mathbb{R}^3$$, in which $$\mu$$ and $$\beta$$ can have severe discontinuities across the interfaces: \left\{ \begin{aligned} \nabla\times (\mu^{-1} \nabla\times \boldsymbol{u}) + \beta \boldsymbol{u} &= \boldsymbol{f} &\text{ in }& \Omega, \\[1mm] \boldsymbol{u} \times \boldsymbol{n} &= \boldsymbol{g}_D &\text{ on }& \Gamma_D, \\ (\mu^{-1} \nabla\times \boldsymbol{u}) \times \boldsymbol{n} & = \boldsymbol{g}_N &\text{ on }& \Gamma_N. \end{aligned} \right.
I am starting to work on two near future projects:
1. The a posteriori error analysis for the Helmholtz type problem arising from time-harmonic Maxwell's equations $$\nabla\times (\mu^{-1} \nabla\times \boldsymbol{u}) + (i\omega \sigma- \omega^2 \epsilon) \boldsymbol{u} = \boldsymbol{f}$$, and the more difficult case while $$\sigma=0$$.
2. The well-posedness, the finite element method, and error analysis for the model problem arising from Magnetohydrodynamics (MHD), while magneto-convection (a first order convective term) is present: $$\nabla\times (\alpha \nabla\times \boldsymbol{B}) + \nabla\times(\boldsymbol{v}\times \boldsymbol{B}) + \gamma \boldsymbol{B} = \boldsymbol{f}$$.
In my spare time, I participate the Q&A sites Mathematics/Scientific Computing StackExchange mostly to answer questions and have discussions with people on partial differential equations:

### Awards

2013. 1st Place in the graduate talks of SIAM@Purdue Computational Science and Engineering Student Conference
2010-2011. Meritorious contributor to Problem of the Week column at Purdue Mathematics
2009-2010. Excellence in Teaching Award as a graduate instructor, Department of Mathematics, Purdue University
2006. Meritorious winners ranking 19th/835 in US Mathematical Contest in Modeling (COMAP)

### Papers and Notes

• (with Zhiqiang Cai) Recovery-Based A Posteriori Error Estimators for $$\boldsymbol{H}(\mathbf{curl})$$-Interface Problems, submitted
• (with Zhiqiang Cai) Two New A Posteriori Error Estimators For Nédélec finite elements in $$\boldsymbol{H}(\mathbf{curl})$$-Problem, to be submitted
• (with Zhiqiang Cai, and Rob Falgout) A Posteriori Error Estimation for Finite Element Approximation to $$\boldsymbol{H}(\mathbf{curl})$$-Problems, to be submitted
• A Discrete de Rham Complex for Discontinuous Functions on Stars, unpublished
• A Canonical Construction of Nonconforming Vector Finite Elements based on de Rham Complex, unpublished

### Teaching

Spring 2014: Instructor, MA224 Business Calculus II, Distant Learning Online section
Fall 2013: Instructor, MA224 Business Calculus II, Distant Learning Online section
Summer 2013: Instructor/Course coordinator, MA266 Ordinary Differential Equations
Summer 2012: Instructor/Course coordinator, MA266 Ordinary Differential Equations
Fall 2011: Instructor, MA223 Business Calculus I
Summer 2010: Instructor, MA266 Ordinary Differential Equations
Spring 2010: Instructor, MA223 Business Calculus I
Fall 2009: Instructor, MA153 Algebra And Trigonometry I
Summer 2009: Instructor, MA162 Calculus II
Fall 2008: Instructor, MA153 Algebra And Trigonometry I
Summer 2008: Instructor, MA162 Calculus II
Spring 2008: Recitation TA, MA161 Calculus I
Fall 2007: Recitation TA, MA262 Linear Algebra and Differential Equations I
Spring 2007: Recitation TA, MA173 Honor Calculus II

### Running

I became a runner in 2010. While running, my mind enjoys the total relaxation and the overwhelming intensity of the so-called endorphins release. I could think freely on math, physics, and philosophy (just to rephrase Kant held the view that freedom is thinkable, yet unknown to be achievable or not, such an agnostic rationalist). I have run two full-marathons, and many half-marathons. My ultimate goal is to qualify for Boston Marathon, as well as to finish an Ironman Triathlon. Here are my personal bests:
 5K 22m47s 10K 48m46s Half-marathon 1h55m Marathon 4h18m