In the Galerkin Asymptotic Wave Expansion (GAWE) the finite-element matrix equation
is expanded about a specified frequency (expansion point
*f*_{0}) in terms of a power series in
*f* – *f*_{0}, where the
initial value of *f*_{0} is chosen adaptively by
the solver. The solution at each *f*_{i} near the
expansion point is then approximated by a sum over basis vectors that are determined
from the matrix equation expansion. Generally the accuracy of the expansion degrades as
|*f*_{i} –
*f*_{0}| gets larger, causing the process to
yield acceptable answers only within a range
*f*_{min} <
*f*_{0} <
*f*_{max}, where the bounds are determined by
checking the residual of the original matrix equation. Converged frequencies are
archived, and new expansion points are then picked in unconverted regions of the band,
with the process continuing until all frequency points are converged. The following
discussion assumes all materials are independent of frequency; the terms in the
expansion change for frequency-dependent materials.

We start with

where the coefficients in the first three terms are given by

and

The term in parentheses to the left of the equal sign is the port mode expansion, given by

The term in parentheses to the right of the equal sign is the source expansion, given by

In these equations you have

and

The combined expression can be written in the following form:

and you can create a vector basis as follows:

and

You can very efficiently obtain the solution at a particular frequency near the expansion point via projection:

where *V* is a matrix whose columns are the
*v*_{j}. This approach provides very accurate
results for frequencies "near enough" to the expansion point (note that
*V* and the * A _{i}
* are functions of the expansion frequency, and that only one matrix
factorization is required per expansion point). In practice, the solutions obtained from
this procedure are inserted into the original matrix equation and their residuals are
checked to determine when to generate a new expansion point.

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