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Friday, October 5, 2007 ASTRACT The inverse problem of heat conduction provides a model for many
applications governed by diffusion equations, for which interior
data are available. We wish to find the single, unknown, thermal
conductivity field by direct (i.e. non-iterative) computation.
This approach requires at least two interior temperature fields.
The strong form of the problem is governed by two partial differential
equations of pure advective transport. The given temperature fields
must satisfy a compatibility condition for the problem to have
a solution. The standard weak (variational) form does not provide
a suitable basis for finite element computation. A conventional
least-squares variational approach yields a well-posed problem, but
does not perform well numerically.
We introduce a novel variational formulation, the Adjoint Weighted
Equation (AWE), for solving the two-field problem. In this case, the
gradients of two given temperature fields must be linearly independent
in the entire domain, a weaker condition than the compatibility required
by the strong form. The solution of the AWE formulation is equivalent
to that of the strong form when both are well posed. The Galerkin
discretization of the AWE formulation converges with optimal rates.
Computational examples confirm these optimal rates, and demonstrate
superior performance to conventional numerical methods on problems
with both smooth and rough coefficients and solutions.
This work was done in collaboration with Assad Oberai of RPI and Paul
Barbone of BU. We are currently extending these ideas to problems of
incompressible elasticity.
Friday, November 9, 2007
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Problems Center at Rensselaer
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