3.5. Finite-Difference and Finite-Element Methods
The PE approximation, ignoring backscattered waves, is not appropriate for a medium with strong range dependence. As an alternative to coupled modes, general numerical discretization methods of FD or FE type can be applied to solve the Helmholtz equation. With the FDM, derivatives are approximated point-wise by difference quotients involving a discrete computational grid. With the FEM, the physical space is divided into non-overlapping subsets called elements and the sought function is approximated on each element by a polynomial. There are several text books where FDM and FEM are described in detail, and this section merely illustrates the basic ideas for a few wave propagation applications.
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