## Grants and Contracts Details

### Description

This proposal involves the development and analysis of
adaptive finite element methods (AFEM) for the reliable and efficient
computation of pointwise quantities and level sets in elliptic
problems. The proposer has previously proved a posteriori error
estimates for controlling pointwise and local pointwise gradient
errors in second-order scalar elliptic PDE using AFEM. This project
consists of three main components. First, we will further develop
fundamental theory for a posteriori control of pointwise and local
errors in scalar elliptic problems. The second component is the
extension of the proposer's theory for a posteriori control of
pointwise and local pointwise gradient errors in elliptic scalar
problems to elliptic systems arising in important applications, in
particular the stationary Stokes system and the equations of linear
elasticity. The latter application is of particular practical
interest in engineering computations as it seeks to address the
pointwise control of errors in the computation of stresses. The
third main goal is to develop adaptive algorithms for computations in
which the desired output is a subset of the computational domain
(e.g., a level set) instead of some norm of the error. In the
context of static problems, examples include computing the location
where the maximum temperature is taken on in a body at thermal
equilibrium, or where the stresses in a body under a particular load
exceed a given threshold. Answering these questions rigorously via
AFEM involves control of pointwise errors, and answering them with
maximum efficiency necessitates controlling errors locally. Thus we
will develop an algorithm based on our previously established local
pointwise a posteriori estimates.

Status | Finished |
---|---|

Effective start/end date | 7/1/07 → 6/30/11 |

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