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Award Abstract # 1418934
Multigrid Methods for a Class of Saddle Point Problems

NSF Org: DMS
Division Of Mathematical Sciences
Recipient: UNIVERSITY OF CALIFORNIA IRVINE
Initial Amendment Date: August 7, 2014
Latest Amendment Date: July 6, 2016
Award Number: 1418934
Award Instrument: Continuing Grant
Program Manager: Leland Jameson
DMS  Division Of Mathematical Sciences
MPS  Directorate for Mathematical and Physical Sciences
Start Date: August 15, 2014
End Date: July 31, 2018 (Estimated)
Total Intended Award Amount: $204,996.00
Total Awarded Amount to Date: $204,996.00
Funds Obligated to Date: FY 2014 = $141,432.00
FY 2016 = $63,564.00
History of Investigator:
  • Long Chen (Principal Investigator)
     chenlong@math.uci.edu
Recipient Sponsored Research Office: University of California-Irvine
 160 ALDRICH HALL
 IRVINE
 CA  US  92697-0001
(949)824-7295
Sponsor Congressional District: 47
Primary Place of Performance: University of California-Irvine
 Rowland Hall room 510F
 Irvine
 CA  US  92697-3875
Primary Place of Performance
Congressional District:		 47
Unique Entity Identifier (UEI): MJC5FCYQTPE6
Parent UEI: MJC5FCYQTPE6
NSF Program(s): COMPUTATIONAL MATHEMATICS
Primary Program Source: 01001415DB NSF RESEARCH & RELATED ACTIVIT
01001617DB NSF RESEARCH & RELATED ACTIVIT
Program Reference Code(s): 8396, 8609, 9263
Program Element Code(s): 127100
Award Agency Code: 4900
Fund Agency Code: 4900
Assistance Listing Number(s): 47.049

ABSTRACT

The fast multigrid methods developed and studied in this work are expected to have a broader impact on the numerical solutions of a large class of practical problems. Important applications include: vector (Hodge) Laplacian, Maxwell equations, Stokes equations, Oseen and Navier-Stokes equations, and Magnetohydrodynamics (MHD) etc. MHD, in particular, has important applications in the development of fusion technology and casting processes. In these applications, since no experimentation is nowadays possible, the numerical simulation of the corresponding partial differential equations is indispensable. These simulations are very challenging, requiring large computational resources. The multigrid solvers developed in this project offer the potential for increasingly accurate models to be solved. In addition, our improvements in algorithm developments will have impact on many other areas, such as image processing, and computer graphics.

This project is divided into two parts: algorithmic development and theoretical analysis. For the algorithmic development, multigrid solvers will be developed for mixed finite element discretization based on Finite Element Exterior Calculus (FEEC). In our study, effective smoothers, which are the key of multigrid methods, will be developed by using existing effective preconditioners or splitting schemes. One such example is a distributive smoother proposed in this project which is highly related to the well-known projection methods used in computational fluid dynamic. In addition to the algorithmic development, a more completed convergence theory of multigrid methods for saddle point problems will be developed. This theory aims to relax the strong regularity assumption in existing work. Consequently our theory can be applied to more realistic problems especially for solutions with singularities. Our theoretical investigation will also provide insight for the algorithmic development, e.g., the construction of approximated distributive smoothers and Schwarz smoothers, and optimal choice of relaxation parameters used in several smoothers.

PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH

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(Showing: 1 - 10 of 14)
B. Zheng, L.P. Chen, X. Hu, L. Chen, R.H. Nochetto, J. Xu "Fast multilevel solvers for a class of discrete fourth order parabolic problems" Journal of Scientific Computing , 2016 10.1007/s10915-016-0189-6
J. Huang, L. Chen and H. Rui "Multigrid Methods for A Mixed Finite Element Method of The Darcy-Forchheimer Model" Journal of Scientific Computing , 2017 https://doi.org/10.1007/s109
L. Chen. "Multi-Grid Methods for Saddle Point Systems using Constrained Smoothers" Computers and Mathematics with Applications , v.70 , 2015 , p.2854 doi:10.1016/j.camwa.2015.09.020
L. Chen, J. Hu, X. Huang "Fast Auxiliary Space Preconditioner for Linear Elasticity in Mixed Form" Mathematics of Computation , 2017 https://doi.org/10.1090/mcom/3285
L. Chen, J. Hu, X. Huang "Multigrid Methods for Hellan-Herrmann-Jonson Mixed Method of Kirchhoff Plate Bending Problems" Journal of Scientific Computing , v.76 , 2018 , p.673 https://doi.org/10.1007/s10915-017-0636-z
L. Chen, J. Hu, X. Huang "Stabilized mixed finite element methods for linear elasticity on simplicial grids in R^n" Numerische Mathematik , 2016 10.1515/cmam-2016-0035
L. Chen, J. Hu, X. Huang, and H. Man "Residual-Based A Posteriori Error Estimates for Symmetric Conforming Mixed Finite Elements for Linear Elasticity Problems" Science China Mathematics , v.61 , 2017 , p.973 https://doi.org/10.1007/s11425-017-9181-2
L. Chen, J. Wang, Y. Wang and X. Ye "Au Auxiliary Space Multigrid Preconditioner for the Weak Galerkin Method" Computers and Mathematics with Applications , v.70 , 2015 , p.330 doi:10.1016/j.camwa.2015.04.016
L. Chen, R.H. Nochetto, E. Otarola, A.J. Salgado "Multilevel methods for nonuniformly elliptic operators and Fractional Diffusion" Mathematics of Computation , 2016 10.1090/mcom/3089
L. Chen, Y. Wu, L. Zhong and J. Zhou "Multigrid Preconditioners for Mixed Finite Element Methods of Vector Laplacian" Journal of Scientific Computing , 2018 https://doi.org/10.1007/s10915-018-0697-7
Long Chen and Jianguo Huang "Some Error Analysis on Virtual Element Methods" Calcolo , v.55 , 2018 , p.5 https://doi.org/10.1007/s10092-018-0249-4
(Showing: 1 - 10 of 14)

PROJECT OUTCOMES REPORT

Disclaimer

This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.

In the numerical simulation of many complex phenomena in physics and engineering, it is always an active research topic to efficiently and effectively solve a set of partial differential equations (PDEs) arising from the mathematical models of practical problems concerned. Such equations encapsulate the most intrinsic features of these phenomena accurately, and are used in our everyday life, from testing the structural stability of a building, to forecasting the impact of the global climate change. However, these equations rarely can be solved in simple formulas. Hence, part of this project is on the development, analysis, and application of multigrid methods, one of the most efficient solvers, to a class of saddle point problems, which arise from the mixed finite element methods for these PDEs. The finite element method is one of the best tools for over 60 years in the computer-aided approximation of these PDEs, and is crucial in many areas of science and engineering like electromagnetics, structural mechanics, and fluid dynamics.

The project is divided into algorithmic development and theoretical analysis parts. In the algorithmic development, multigrid solvers or preconditioners have been developed for a class of saddle point problemsarising from mixed finite element methods discretization of PDEs. One such example is the block smoother for the plate blending problem by revealing the underlying differential complex, which further motivates a theoretical investigation on the deep structure of a large class of similar problems.

Besides the algorithmic development, we have developed a more complete convergence theory of multigrid methods for saddle point problems. Our new approach can be applied to more realistic problems especially for solutions with singularity. Our theoretical investigation have also provided insight on the algorithmic development, e.g., the construction of approximated distributive smoothers and block smoothers.

A list of work related to fast solvers
The fast multigrid methods developed and studied in this work are expected to have a broader impact on the numerical solutions of a large class of practical problems. Important applications include: vector (Hodge) Laplacian, Stokes equations, Oseen and Navier-Stokes equations, and Maxwell equations etc. The numerical simulations of these equations are very challenging, requiring large computational resources. The fast solvers developed in this project offer the potential for increasingly accurate models to be simulated.
Besides the main focus on the fast solvers, we have also developed or analyzed numerical methods for solving Cahn-Hillard equation, linear elasticity, elliptic interface problems, and Hodge Laplacian equations etc.

A list of work on numerical analysis
Another impact of this project is the education and training of the next generation of computational mathematicians. By further developing the open-source software package iFEM, the PI has improved a project-oriented course for better training of students.  The developed software is also very beneficial to and recognized by the research community. 
Supervised research topics related to this project have been designed for junior and senior graduates and young scholars. With this support, the PI has supervised 1 postdocs, 7 Ph.D students (2 graduated), and 6 visiting students and scholars. 

Standard methods of peer-reviewed journal articles and presentations at colloquia, seminars, workshops, and conferences is also used to further disseminate our results. During the awarding period, the PI has published or accepted 17 articles and has submitted 5 manuscripts.

Last Modified: 09/02/2018
Modified by: Long Chen

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