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MAT 767, Topics in Advanced Mathematics

Numerical PDEs for Multiphysics Problems

Fall 2026

(08/24/2026-12/05/2026)

COURSE DESCRIPTION:

This course will focus on modeling studies (classical PDEs, coupled PDEs in multiphysics, interface PDEs, PDEs with moving domain/interfaces, etc.), numerical methodologies (finite element/volume/difference methods, deep neural network method, etc.), and numerical analyses (well-posedness, stability, convergence, etc.) for multiphysics problems arising from interdisciplinary areas (fluid dynamics, electrohydrodynamics, fuel cell dynamics, solid mechanics, hemodynamics, fluid-structure interactions, etc.). The following topics are on the lecture list, tentatively:

1. Elliptic equations:
   • Linear convection-diffusion-reaction case
     • Strong and weak form
     • Finite element discretization
     • Stability analysis
     • H¹-norm error estimate
     • L²-norm error estimate
   • Nonlinear diffusion-reaction case
     • Strong and weak form
     • Finite element discretization
     • Stability analysis
     • H¹-norm error estimate
     • L²-norm error estimate
     • Picard's linearization
     • Newton's linearization
   • Nonlinear convection-diffusion-reaction case
2. Parabolic equations:
   • Linear convection-diffusion case
     • Semi discretization
       • Stability analysis
       • H¹-norm error estimate
       • L²-norm error estimate
     • Full discretization
       • Backward Euler scheme
       • Crank-Nicolson scheme
       • Energy-stable scheme
   • Nonlinear convection-diffusion-reaction case
     • Semi discretization
     • Full discretization
       • Backward Euler scheme
       • Crank-Nicolson scheme
3. Hyperbolic/wave equations:
   • Nonlinear convection-diffusion-reaction case
     • Semi discretization
     • Full discretization
   • Energy-preserving scheme
   • Newmark scheme
4. Stokes equations:
   • Well-posedness
   • Mixed finite element method
     • Stability
     • Error estimates
5. Navier-Stokes equations:
   • Well-posedness
   • Mixed finite element method
     • Stability
     • Error estimates
6. Charge carrier transport problems
   • Poisson-Nernst-Planck (PNP) equations
   • PNP/Navier-Stokes coupling system - electrohydrodynamics
7. Two-phase flow problems
   • Modeling study
   • Mixed finite element method
   • Stokes-Darcy coupling problem
   • Phase Field Model
8. Fluid-structure interaction problems
   • ALE finite element method
   • Fictitious domain method
9. Deep Neural Networks/meshfree approach
   • Deep neural networks (DNN)
   • Physics-informed neural networks (PINN)
   • PINN for solving PDEs

In addition, we may also pay attention to the finite element and deep neural network algorithm abstraction and software development by virtue of existing packages for the implementation of the above multiphysics problems.

PREREQUISITE: MAT 665/666 and/or MAT729/730 and/or MAT765/766 or equivalent.

COURSE OBJECTIVES/LEARNING OUTCOME: Upon completion of this course, the graduate student will be able to

• Develop and pursue a unique study question through substantial, legitimate research that fosters focus and flexibility.
• Maintain a research note documenting work and sources.
• Gain a thorough understanding of the topic through investigation and discussion of modeling and numerical studies, such as conservation laws, monolithic fluid-structure interaction model, Poisson-Nernst-Planck equations, and the associated scientific and engineering computing.
• Contribute original scholarship of the topic, including developing a summary of existing work, writing a report of investigation and analysis, and implementing a numerical algorithm by developing a source code.

TEST AND GRADING POLICY: Homework are assigned on the weekly basis,  and the final grade is based on the performance of the routine homework.

Final course letter grades will be assigned as follows:

NOTE: Your grade will be based EXCLUSIVELY on the homework.

CLASS POLICIES:

HOMEWORK:

The homework problems are given in the class, and collected on every Monday. You are required to do the homework outside of class.  Homework are reviewed and returned to you in the following Monday.

 

ATTENDANCE:

You are required to attend class regularly. If you miss a class for any reason, you are responsible for all missed lectures, materials and announcements made in class.

ACADEMIC INTEGRITY:

Academic integrity is the pursuit of scholarly activity in an open, honest and responsible manner. Academic integrity is a basic guiding principle for all academic activity at University of Nevada, Las Vegas, and all members of the University community are expected to act in accordance with this principle. Consistent with this expectation, the University's Code of Conduct states that all students should act with personal integrity, respect other students' dignity, rights and property, and help create and maintain an environment in which all can succeed through the fruits of their efforts.

Academic integrity includes a commitment not to engage in or tolerate acts of falsification, misrepresentation or deception. Such acts of dishonesty violate the fundamental ethical principles of the University community and compromise the worth of work completed by others. Based on the University's Student Academic Misconduct Policy, a range of academic sanctions may be taken against a student who engages in academic dishonesty.