
NSF Org: |
DMS Division Of Mathematical Sciences |
Recipient: |
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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 2014 = $82,134.00 FY 2015 = $60,987.00 |
History of Investigator: |
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Recipient Sponsored Research Office: |
1 PROSPECT ST PROVIDENCE RI US 02912-9100 (401)863-2777 |
Sponsor Congressional District: |
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Primary Place of Performance: |
Office of Sponsored Projects Providence RI US 02912-9093 |
Primary Place of
Performance Congressional District: |
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Unique Entity Identifier (UEI): |
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Parent UEI: |
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NSF Program(s): | COMPUTATIONAL MATHEMATICS |
Primary Program Source: |
01001415DB NSF RESEARCH & RELATED ACTIVIT 01001516DB NSF RESEARCH & RELATED ACTIVIT |
Program Reference Code(s): |
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Program Element Code(s): |
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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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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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