Research

Research Interests

My research interests lie in the fields of Numerical Analysis and Scientific Computing, both with an emphasis on using domain decomposition, adaptivity, and multigrid/multilevel methods to solve partial differential equations. I am interested in designing, analyzing and implementing provably robust and efficient numerical methods for large-scale real-life problems. Particular applications of my research include simulations of surface-subsurface hydrology, oil reservoirs, multiphase flows in porous media, and phase-field models in material science.

Projects (not up-to-date)

Spectral Domain Decomposition Preconditioning for PDEs with Highly Varying/Multiscale Coefficients

Formulating Optimized Schwarz and 2-Lagrange domain decomposition preconditioners that are robust for this special class of problems.

Collaborators: Sébastien Loisel, Rob Scheichl
Publication:

1. S. Loisel, H. Nguyen and R. Scheichl, Optimized Schwarz and 2-Lagrange Methods for Multiscale PDEs, submitted to SISC.
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Domain Decomposition and Adaptivity for Phase-Field Models

• Developing domain decomposition and adaptivity for phase-field models (Cahn-Hilliard systems, Navier-Stokes-Cahn-Hilliard systems) to simulate solidification in material science and multiphase flows in porous media (using deal.II)

• Analyzing preconditioners based on Additive Schwarz methods for related obstacle problems and saddle point problems
  

Collaborators: Lubomir Banas, David Neil Geer, Anastasios Karangelis

Publication:

Domain Decomposition Preconditioners for Parallel Adaptive Finite Elements

Formulated and analyzed preconditioners to use when combining domain decomposition and adaptivity.

Collaborator: Sébastien Loisel
Publications:

1. S. Loisel, H. Nguyen, On the Convergence of Additive Schwarz Preconditioners for Parallel Adaptive Finite Elements, in preparation.
   

2. S. Loisel, H. Nguyen, A Comparison of Additive Schwarz Preconditioners for Parallel Adaptive Finite Elements, in Domain Decomposition Methods in Science and Engineering XXII, Lecture Notes in Computational Science and Engineering, to appear.
   [PDF]

3. S. Loisel, H. Nguyen, An Optimal Schwarz Preconditioner for Parallel Adaptive Finite Elements, submitted to SINUM.
   [PDF]

Nonlinear/Linear Solvers for Multiscale Integrated Hydrologic Models

Formulated and implemented advanced techniques to improve the convergence and accuracy of nonlinear/linear solvers for IWFM, a water resources management and planning model developed at Bay-Delta Office, Department of Water Resources, California, USA.

Publications:

1. H. Nguyen and Z. Bai, E. Dogrul, T. Kadir, C. Brush, F. Chung, On Using the Newton-PGMRES Method for Multiscale Integrated Hydrologic Models.
   [PDF]

2. H. Nguyen and Z. Bai, E. Dogrul, T. Kadir, C. Brush, F. Chung, Adaptive Accuracy Control of Nonlinear Newton-Krylov Methods for Multiscale Integrated Hydrologic Models, in XIX International Conference on Computational Methods in Water Resources, 2012.
   [link|PDF]

hp-Adaptive Finite Elements and Bank-Holst Parallel Adaptive Meshing

Formulated and implemented a posteriori error estimates for elements of arbitrarily high order, special nodal basis functions for transition elements. Investigated the combination of high-order elements and domain decomposition using Bank-Holst paradigm.

Publications:

1. R. E. Bank and H. Nguyen, A parallel hp-adaptive finite element method, in Recent Advances in Scientific Computing and Applications, vol. 586 of Contemporary Mathematics, Amer. Math. Soc., Providence, RI, 2013, pp. 23–33.
   [link|PDF]

2. R. E. Bank and H. Nguyen, Mesh regularization in Bank-Holst parallel hp-adaptive meshing, in Domain Decomposition Methods in Science and Engineering XX, R. Bank, M. Holst

3. R. E. Bank and H. Nguyen, hp Adaptive Finite Elements Based on Derivative Recovery and Superconvergence, Computing and Visualization in Science, 14 (2011), pp 287-299.
   [link|PDF]

4. R. E. Bank and H. Nguyen, Domain decomposition and hp-adaptive finite elements, in Domain Decomposition Methods in Science and Engineering XIX, Y. Huang, R. Kornhuber, O. Widlund, and J. Xu, eds., vol. 78 of Lecture Notes in Computational Science and Engineering, 2011, Springer, pp. 3–13.
   [link|PDF]

5. Hieu Trung Nguyen, p-adaptive and automatic hp-adaptive finite element methods for elliptic partial differential equations, PhD Thesis, UC SanDiego, 2010.
   [link|PDF]

Code Analysis and Optimization

Analyzed and optimized AWP-ODC, an earthquake simulation application developed at San Diego Supercomputer Center.

Publication:

1. H. Nguyen, Yifeng Cui, Kim Olsen, Kwangyoon Lee , Single CPU optimizations of SCEC AWP-Olsen application, Poster, Southern California Earthquake Center Annual Meeting, 2009.
   [PPT|PDF]

Shooting methods for two-point boundary-value problems (ODEs)

Publication:

1. Nguyen Trung Hieu, Remark on the shooting methods for nonlinear two-point boundary-value problems, Journal of Science, Vietnam National University, T. XIX, No 3, 2003.
   [PDF]