Research

Research Area

• • Computational and applied mathematics (CAM)

Research Interests

• • Numerical analysis and scientific computing

  • Numerical solutions of deterministic and stochastic partial differential equations: finite element methods, discontinuous and spectral Galerkin methods, multi-grid methods, domain decomposition methods.

  • Nonlinear partial differential equations: geometric evolution PDEs and fully nonlinear PDEs.

  • Homogenization and multi-scale methods

  • Uncertainty quantification (UQ) 

  • Fractional calculus, fractional/nonlocal DEs, and their numerical methods

  • Deep neural networks, AI and machine learning algorithms, and high-dimensional computation

Application Areas

• • Computer vision and image processing: PDE and variational methods for imaging process.

  • Materials science: Phase transitions and related geometric problems of moving surfaces and curves, multi-scale modeling, and simulations. Superconductivity and soft matters. 

  • Fluid mechanics: porous media flow, reservoir simulation and groundwater contaminant simulation, geostrophic flow, viscoelastic fluids (liquid crystals, polymers).

  • Solid mechanics: linear elasticity, plates and shells, poro-elasticity (gels, soft matters)

  • Wave propagation: acoustic waves, elastic waves, electromagnetic waves, inverse scattering, fluid-solid interactions, quantum waves.

  • General relativity: numerical solutions of wave maps, mean curvature and inverse mean curvature flow, and Einstein equations.

  • Mathematical biology: systems biology, gene regulatory network identification, and functional prediction. Modeling and simulating drug-eluting stents in arteries. 

Books, Book Chapter, and Special Issue

•  Feng, X; Schulze, T. (Editors) Recent Advances in Numerical Methods for Partial Differential Equations and Applications, Contemporary Mathematics, vol. 306, American Mathematical Society, 2002.

• Alexiades, V.; Feng, X.;  Schulze, T. (Editors) Multi-scale Modeling and Simulation in Materials Sciences, Special Issue of Journal of Scientific Computing, vol. 37, no. 1, 2008.

• Feng, X; Karakashian, O. A.;  Xing, Y. (Editors) Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations, vol.157, IMA book series, Springer, 2013.

• Du, Q; Feng, X. The Phase Field Method for Geometric Moving Interfaces and Their Approximations, Chapter 5 of  Handbook of Numerical Analysis, vol. 21, 425--508, 2020.  

Selected Refereed Journal Papers  ​​(below are10 most cited papers, a complete list of 100+ publications can be found at Google Scholar,  ResearchGate, MathSciNet (requires to login), arXiv, and in CV) 

• Feng, X.;  Prohl, A. Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows.  Numerische Mathematik 94 (1), 33--65, 2003.

• Feng, X. Fully discrete finite element approximations of the Navier-Stokes-Cahn-Hilliard diffuse interface model for two-phase fluid flows. SIAM Journal on Numerical Analysis 44 (3), 1049--1072, 2006.

• Feng, X.; Wu, H. Discontinuous Galerkin methods for the Helmholtz equation with large wave number. SIAM Journal on Numerical Analysis 47 (4), 2872--2896, 2009.

• Feng, X.;  Prohl, A. Error analysis of a mixed finite element method for the Cahn-Hilliard equation. Numerische Mathematik 99 (1), 47--84, 2004.

• Feng, X.; Karakashian, O. A. Two-level additive Schwarz methods for a discontinuous Galerkin approximation of second order elliptic problems. SIAM Journal on Numerical Analysis 39 (4), 1343--1365, 2001.

• Cummings, P.; Feng, X. Sharp regularity coefficient estimates for complex-valued acoustic and elastic Helmholtz equations. Mathematical Models and Methods in Applied Sciences 16 (01), 139--160, 2006.

• Feng, X.; Wu, H. ℎ𝑝-discontinuous Galerkin methods for the Helmholtz equation with large wave number. Mathematics of Computation 80 (276), 1997--2024, 2011.

• Feng, X; Glowinski, R.; Neilan, M. Recent developments in numerical methods for fully nonlinear second order partial differential equations. SIAM Review 55 (2), 205--267, 2013.

• Feng, X; Neilan, M. Vanishing moment method and moment solutions for fully nonlinear second order partial differential equations. Journal of Scientific Computing 38 (1), 74--98, 2009.

• Feng, X.; He, Y.; Liu, C. Analysis of finite element approximations of a phase field model for two-phase fluids. Mathematics of Computation 76 (258), 539--571, 2007. 

Grants (below is a list of the NSF and DOE grants since 2010, a complete list can be found in CV)

• 2023--2026, NSF (DMS-2309626,  PI) 

• 2023--2024, ORNL - UT-Battelle - Oak Ridge National Laboratory (PR136014) 

• 2020--2023, NSF (DMS-2012414,  PI) 

• 2018, DOE - ORNL - UT-Battelle - Oak Ridge National Laboratory (PI) 

• 2016--2020, NSF (DMS-1620168, PI) 

• 2013--2016, NSF (DMS-1216416, PI) 

• 2010--2013, NSF (DMS-1016173, PI)  

Editorial Boards

• Associate Editor,  Journal of Computational Mathematics, 12/2005-12/2022. 

• Associate Editor,  MDPI, Mathematics. 8/2022-present.