Computational Math Seminar: Tom Manteuffel

Hybrid FOSLS/FOSLL*  

Tom Manteuffel

Applied Mathematics, University of Colorado Boulder

Date and time: 

Tuesday, September 17, 2013 - 10:00am

Location: 

GRVW Conference Room

Abstract: 

Standard FOSLS formulation for a system of PDEs enjoys a simple a posteriori error measure that is both locally sharp and globally reliable. Of course, one would like an error measure that is also a local upper bound on the error. FOSLS local error measures provide an upper bound in a special semi-norm of the error, similar to an H1-semi-norm. This talk will include a discussion of the implications of this distinction.

The FOSLL* formulation, which can be described as a least-squares formulation on the adjoint first-order system, minimizes the L2-norm of the error over a nonstandard finite element space. However, this formulation does not easily admit a local a posteriori error measure.

A hybrid FOSLS/FOSLL* formulation will be introduced that attempts to combine the best aspects of both FOSLS and FOSLL*. Let L represent a first order system of PDEs. The hybrid method essentially optimizes the graph norm of the error, ||L(u-uh)||2 + ||u-uh||2, over the finite element space, with very mild coercivity and continuity bounds. It attains enhanced local L2 accuracy, retains the same local a posteriori error measure of FOSLS, and provides a sharp local L2 error measure. The strengths and weaknesses of this approach will be discussed. Numerical results will be presented that demonstrate the use of the a posteriori error measures to direct local adaptive refinement in a number of applications. This approach essentially eliminates loss of conservation in fluid flow problems.