Conference abstracts

Session B6 - Multiresolution and Adaptivity in Numerical PDEs

July 13, 17:30 ~ 17:55 - Room T1

Adaptive Finite Element Methods for Unresolved Diffusion Coefficients

Andrea Bonito

Texas A&M, USA   -   bonito@math.tamu.edu

Elliptic partial differential equations (PDEs) with discontinuous diffusion coefficients occur in application domains such as diffusions through porous media, electro-magnetic field propagation on heterogeneous media, and diffusion processes on rough surfaces.

In most applications, the discontinuities do not lie on the boundaries of the cells in the initial triangulation. Rather, the discontinuities occur on curves, surfaces, or manifolds, and could even be unknown beforehand. One of the obstacles to treating such discontinuity problems is that the usual perturbation theory for elliptic PDEs assumes bounds for the distortion of the coefficients in the $L_\infty$ norm and this in turn requires that the discontinuities are matched exactly when the coefficients are approximated.

We present a new approach based on distortion of the coefficients in an $L_q$ norm with $q<\infty$, which therefore does not require the exact matching of the discontinuities. We then use this new distortion theory to formulate new adaptive finite element methods (AFEMs) for such discontinuity problems. We show that such AFEMs are optimal in the sense of distortion versus number of computations, and report insightful numerical results supporting our analysis.

If time permits, we will proceed further and discuss how this perturbation theory can be used in turn to derive stability estimates for parameter recovery processes in diffusion problems.

Refs:

A. Bonito, R.A. DeVore, and R.H. Nochetto: Adaptive Finite Element Methods for Elliptic Problems with Discontinuous Coefficients.

A. Bonito, A. Cohen, R.A. DeVore, G. Petrova, and G. Welper: Diffusion Coefficients Estimation for Elliptic Partial Differential Equations.

Joint work with Ronald A. DeVore (Texas A&M University, USA) and Ricardo H. Nochetto (University of Maryland, USA).

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