A. Delabourdonnaye, HIGH-FREQUENCY APPROXIMATION OF INTEGRAL-EQUATIONS MODELING SCATTERING PHENOMENA, Modelisation mathematique et analyse numerique, 28(2), 1994, pp. 223-241
In this paper, we present a new way of discretizing integral equations
coming from high frequency wave propagation. Indeed, using the eikona
l equation, we will write that the solution is locally the product of
an amplitude by an oscillating function whose phase gradient modulus i
s the wave number. Discretizing in order to keep this relation, we wil
l show that, is the limit of high frequencies, the matrices we obtain
are sparse (as sparse as volumic finite-element methods, in fact), whi
ch is not the case with the classical way of discretizing for example
with P1-Lagrange or H(div) (see [11] or [13]) finite elements. More pr
ecisely, if N is the number of degrees of freedom, we lower the comple
xity from O(N2) to O(N).