Award Abstract # 1318108
Topics in the analysis of finite elements

NSF Org: DMS
Division Of Mathematical Sciences
Recipient: BROWN UNIVERSITY
Initial Amendment Date: July 10, 2013
Latest Amendment Date: July 14, 2015
Award Number: 1318108
Award Instrument: Continuing Grant
Program Manager: Leland Jameson
DMS
 Division Of Mathematical Sciences
MPS
 Directorate for Mathematical and Physical Sciences
Start Date: July 1, 2013
End Date: June 30, 2017 (Estimated)
Total Intended Award Amount: $210,000.00
Total Awarded Amount to Date: $210,000.00
Funds Obligated to Date: FY 2013 = $66,879.00
FY 2014 = $82,134.00

FY 2015 = $60,987.00
History of Investigator:
  • Johnny Guzman (Principal Investigator)
    Johnny_Guzman@brown.edu
Recipient Sponsored Research Office: Brown University
1 PROSPECT ST
PROVIDENCE
RI  US  02912-9100
(401)863-2777
Sponsor Congressional District: 01
Primary Place of Performance: Brown University
Office of Sponsored Projects
Providence
RI  US  02912-9093
Primary Place of Performance
Congressional District:
01
Unique Entity Identifier (UEI): E3FDXZ6TBHW3
Parent UEI: E3FDXZ6TBHW3
NSF Program(s): COMPUTATIONAL MATHEMATICS
Primary Program Source: 01001314DB NSF RESEARCH & RELATED ACTIVIT
01001415DB NSF RESEARCH & RELATED ACTIVIT

01001516DB NSF RESEARCH & RELATED ACTIVIT
Program Reference Code(s): 9263
Program Element Code(s): 127100
Award Agency Code: 4900
Fund Agency Code: 4900
Assistance Listing Number(s): 47.049

ABSTRACT

Three distinct projects that study the behavior of finite element methods (FEM) will be considered. The first project is studying the pollution effects of immersed boundaries in the immersed boundary finite element method. A sharp error analysis will be given that measures how far one has to be from the immersed boundary to obtain optimal convergence. The second project will involve adaptive Discontinuous Galerkin (DG) methods. Contraction properties of weakly penalized DG methods will be proved. The final project is max-norm stability analysis of inf-sup stable finite element methods for the Stokes problem. A Fortin projection that is exponentially decaying will be constructed for the lowest-order Taylor-Hood element in three dimensions. Exponentially decaying projections will be an important tool to prove max-norm stability estimates.

FEM are widely used to simulate a variety of problems in engineering and science. Users of these methods rely on theoretical results that give them some guarantee of their reliability. The P.I. will use mathematical analysis to describe the behavior of FEM for three important FE methods. In particular, the P.I. will mathematically study the behavior of the immeresed boundary FEM which is a method especially suited for fluid-solid interactions. For example, these methods have been used to simulate blood flow and animal locomotion, to name a few. The results of this investigation will give users theoretical guidance on where to put more computational effort which in turn will make their simulations more accurate for imporant applications.

PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH

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J. Guzman, M. Sanchez-Uribe and M. Sarkis "On the accuracy of finite element approximations to a class of interface problems," Mathematics of Computation , v.85 , 2016
Johnny Guzman, Manuel Sanchez Uribe, Marcus Sarkis "Higher-order finite element methods for elliptic problems with interfaces" ESAIM: Mathematical Modelling and Numerical Analysis , v.50 , 2016

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.

Immersed boundary and immersed finite element methods are standard method to solve interface problems. In this project we mostly considered time independent second order elliptic interface  problems. The standard methods are  known to suffer from loss of accuracy near the interface. However, it was not clear how far from the interface is there error pollution. One of our first task was to answer this question. In particular, we showed that if, roughly, one is a square root of h (where h is the mesh size) away from the interface then one recovers optimal accuracy. The next task was to alter the immersed finite element method  to and design a family of higher order methods that are optimal up to the interface.  The above results were for a problem were the diffusion coefficients were the same on either side of the interface.  We, therefore, also considered  an interface problem with a jump on the diffusion coefficients. We in fact, defined a method and proved that the estimates are independent of the contrast. In other words, the jump in coefficients can be arbitrarly large. 

A graduate student was trained in this project. Several talks in conferences and seminars were given discussing the results. The PI attendend several conferences with a large presence of historically underrepresented students and discussed the research project with them.

 

 


Last Modified: 08/02/2017
Modified by: Johnny Guzman

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