TITLE:  Spectrally Matched Grids for Anisotropic Problems

SPEAKER:  Shari Moskow

AFFILIATION:  Drexel University

 
Spectrally Matched Grids (SMG) allow one to obtain spectral (exponential) 
super-convergence of standard second order finite difference schemes at 
targeted subsets (manifolds) of computational domains. They have been 
applied very successfully to domain decomposition, truncation of exterior 
computational domains, and remote sensing, where the solutions is needed 
only at receiver locations. 

 

Originally the SMG were designed for isotropic problems. However, spectral 
convergence of the Neumann to Dirichlet map was also observed for anisotropic
 problems, despite the fact that the grids were designed for isotropic ones. 
 We explain why this happens. For elliptic problems, the gridding algorithm 
is reduced to a Stieltjes rational approximation on an interval of a line 
in the complex plane instead of the real axis as in the isotropic case. We 
show rigorously why this occurs for a semi-infinite and bounded interval. 
We then extend the gridding algorithm to hyperbolic problems.  For the 
propagative modes, the problem is reduced to a rational approximation on 
an interval of the negative real semiaxis, similarly to in the isotropic 
case. We present numerical examples for truncation of unbounded domains 
for acoustic well logging applications in anisotropic media, where 
traditional absorbing boundary conditions and Perfectly Matched Layer (PML) 
approaches fail.  Collaborators: S. Asvadurov  V. Druskin, V. Lisitsa.