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SUMMARY:Numerical analysis of high frequency wave scattering via semiclass
 ical analysis: a case study with non-uniform meshes - Jeffrey Galkowski (U
 CL)
DTSTART:20250206T150000Z
DTEND:20250206T160000Z
UID:TALK227986@talks.cam.ac.uk
CONTACT:Matthew Colbrook
DESCRIPTION:In recent years\, semiclassical analysis has significantly adv
 anced our understanding of numerical algorithms for high-frequency wave sc
 attering. This talk will begin with an overview of how semiclassical metho
 ds have influenced the theory of numerical methods for frequency-domain wa
 ve problems. As a case study\, we will then focus on the finite element me
 thod (FEM)\, a classical approach for approximating solutions to high-freq
 uency scattering problems. In FEM\, the solution is typically approximated
  using piecewise polynomials of degree p on a mesh of width h. A fundament
 al question is then: how should h be chosen (as a function of the frequenc
 y\, k) so that the error in the numerical solution is small? It has been k
 nown since the seminal work of Babuska and Ihlenberg that the natural conj
 ecture hk<<1 is not sufficient. Instead\, one must require that (hk)^{2p} 
 \\rho(k)<< 1 to maintain constant relative error\, where \\rho(k) is the n
 orm of the relevant solution operator. In this talk\, we will show that th
 is condition can be substantially weakened by using a non-uniform mesh whi
 ch strategically concentrates resolution in regions with apriori higher er
 rors.
LOCATION:Centre for Mathematical Sciences\, MR14
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