Dissertation
The effect of boundary correctors on scattering by a periodic obstacle
Doctor of Philosophy (Ph.D.), Drexel University
Jun 2020
DOI:
https://doi.org/10.17918/00001010
Abstract
We present results in the homogenization of a transmission problem arising in the scattering theory for bounded inhomogeneities with periodic coefficient in the lower-order term of the Helmholtz equation. The squared index of refraction is assumed to be a periodic function of the fast variable, specified over the unit cell with characteristic size epsilon. We obtain improved convergence results that assume lower regularity than previous estimates (which also allow for periodicity in the second-order operator). In particular we show that, in contrast to Dirichlet problems, the O([epsilon]) boundary corrector is nontrivial and can be observed in the far field. We further demonstrate the latter far-field effect is larger than that of the "bulk'' corrector - the so-called periodic drift, which is found to emerge only at O([epsilon]²). We illustrate the analysis with examples in one and two spatial dimensions. Because the convergence of the boundary correctors to their limits is in general slow, we explore in detail their use in a second order approximation and show a new convergence estimate for the second order boundary corrector on a square. We show numerical examples of the higher order forward approximation in one and two dimensions. Furthermore, we continue our study by using the first order boundary correction as an asymptotic model for inversion and show numerical examples of inversion in two dimensions. Our work provides new results in an area of scattering theory where the boundary effect has not been previously studied.
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Details
- Title
- The effect of boundary correctors on scattering by a periodic obstacle
- Creators
- Tayler Anne Pangburn
- Contributors
- Shari Moskow (Advisor)
- Awarding Institution
- Drexel University
- Degree Awarded
- Doctor of Philosophy (Ph.D.)
- Publisher
- Drexel University; Philadelphia, Pennsylvania
- Number of pages
- xi, 124 pages
- Resource Type
- Dissertation
- Language
- English
- Academic Unit
- College of Arts and Sciences; Drexel University; Mathematics
- Other Identifier
- 991014695238904721