| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | July 8, 2013 |
| Latest Amendment Date: | July 8, 2013 |
| Award Number: | 1319172 |
| Award Instrument: | Standard Grant |
| Program Manager: |
matthias gobbert
DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | July 15, 2013 |
| End Date: | June 30, 2017 (Estimated) |
| Total Intended Award Amount: | $244,770.00 |
| Total Awarded Amount to Date: | $244,770.00 |
| Funds Obligated to Date: |
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| History of Investigator: |
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| Recipient Sponsored Research Office: |
202 HIMES HALL BATON ROUGE LA US 70803-0001 (225)578-2760 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
Baton Rouge LA US 70803-0100 |
| 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 |
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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
Elliptic variational inequalities are fundamental mathematical tools for modeling phenomena that involve elliptic partial differential operators and constrained optimization. This project will develop and analyze finite element methods for fourth and higher order elliptic variational inequalities, which arise naturally for example in mechanics and elliptic optimal control problems. A recent theoretical advance by the PIs demonstrates that, for the displacement obstacle problems of Kirchhoff plates, the heart of the error analysis involves only problems at the continuous level and therefore any finite element method that works for fourth order boundary value problems can also be adapted for obstacle problems. This new approach will be extended to other fourth and higher order variational inequalities with different types of constraints, which will bring well-developed finite element methodologies for boundary value problems (conforming and nonconforming methods, discontinuous Galerkin methods, generalized finite element methods, isoparametric finite element methods, local mesh refinement, singular function method, etc.) into the study of numerical solution of higher order variational inequalities. Fast solvers for higher order variational inequalities, such as multigrid methods, domain decomposition methods and adaptive methods, will also be developed. In particular this project will lead to new algorithms for second order elliptic distributed optimal control problems with pointwise state and/or control constraints that are fundamentally different from existing algorithms.
The results from this project will provide new insights to the numerical solution of higher order variational inequalities, an area that is becoming increasingly important as more and more complex phenomena in science, engineering and finance are being modeled by higher order differential equations. The outcomes of this project will impact diverse areas that require reliable and efficient numerical algorithms for the solution of such inequalities.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
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PROJECT OUTCOMES REPORT
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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.
Elliptic and parabolic variational inequalities are fundamental tools for modeling phenomena that involve partial differential operators, constrained optimization and free boundaries. They appear for example in physics (jet flows, Stefan problems), engineering (obstacle problems, Signorini problems, filtration through porous media), optimal control (control and/or state constrained problems) and finance (optimal stopping for American options).
However, most of the existing numerical methods are only for second order variational inequalities. This is due to a fundamental difference in the nature of the solutions of these problems. In the case of second order problems, the solutions of the constrained problems are as smooth as the solutions of the unconstrained problems and hence the well-developed numerical methods for unconstrained problems can also be applied to the variational inequalities. On the other hand, in the case of higher order problems, the solutions of the variational inequalities are never as smooth as the solutions of the unconstrained problems and therefore a fundamentally new approach is required.
The algorithms and analyses resulting from this project have demonstrated that indeed many of the numerical methods originally designed for unconstrained fourth order elliptic problems can also be applied to fourth order elliptic variational inequalities after appropriate modifications, and the reliability of these methods can be guaranteed mathematically. The outcomes of this project make it practical to use fourth order variational inequalities in mathematical models, which will significantly enlarge the mathematical toolbox for many applications in diverse areas.
Last Modified: 09/27/2017
Modified by: Susanne C Brenner
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