Accurate evaluation of is a rapid procedure
because it only involves integration over the boundary
.
It requires
basis function evaluations,
where the number of discretization points on the boundary is
(see Appendix G).
However the evaluation of the norm matrix
at first sight seems like a
tremendous bottleneck.
Eq.(5.13) for
involves computing
overlaps
each of which is a
dimensional
integral over the domain.
Since the
are oscillatory at scale
, each integral
would require
basis function evaluations, and the construction
of
would scale as
.
If this were performed literally, then much of the advantage of
a surface method would be lost.
In fact, in any implementation of the
PWDM as originally described,
almost all the time is spent performing normalizations
over the domain, which is wasteful.
The aim of this section is to describe more rapid evaluations
of , both approximate and exact, which scale like the boundary.
(In Section 5.5 we will see that the accuracy of
needed is
actually quite low).
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