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Applied
Math Seminar |
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Spectral projections enjoy high order convergence for globally
smooth functions. However, a single discontinuity introduces O(1) spurious
oscillations, Gibbs' Phenomena, and reduces the high order convergence
rate to first order. We will show how adaptive mollifiers can be used
to recover the high order convergence rate as well as remove the spurious
oscillations found near a discontinuity. In addition, when these adaptive
mollifiers are applied to an equidistant sampling of piecewise smooth
functions we obtain an exponentially accurate "interpolation" scheme.
This is a powerful new tool for equidistant data with applications in
image processing and non-linear conservation laws. Preliminary applications
for time dependent problems will be shown. |