Simon Tavener
Department of Mathematics, Colorado State University
Visiting fellow, Oxford Centre for Collaborative Applied Mathematics
"A posteriori analysis and adaptive error control for operator decomposition
approaches to coupled physics problems"
Abstract
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*A posteriori* error estimation and adaptive error control based on the
formulation and solution of an adjoint problem are well established for
problems involving a single type of physics. For systems that exhibit
sufficiently complicated physics or a range of scales so that they severely
challenge standard solution techniques, operator decomposition
provides an attractive way to decompose the problem into components with
relatively simple physics or into behaviors that occur over a modest range
of scales.
Operator decomposition creates additional challenges for adaptive error
control since errors in one component may limit the accuracy in another
component, yet a global adjoint solution is typically not available. In
this talk, I will describe an *a posteriori* analysis of operator
decomposition methods for coupled elliptic systems. This analysis takes
into account the accuracy with which individual components are solved as
well as the global effects of operator decomposition. The estimates
obtained provide the means for adaptive error control.
Extensions of these ideas that are currently under investigation include
coupled elliptic-parabolic and coupled parabolic systems arising in
studies of cardiac physiology. Time-dependent problems present additional
challenges and I will describe a one approach we have developed which we
call "block adaptivity".