SEMINAIRE D'ANALYSE NUMERIQUE
Année universitaire 2007-2008
Jeudi 29 novembre 2007 (salle 016) :
Sarah ENGLEDER
(TU Graz, Autriche)
Modified Boundary Integral Equations for the Helmholtz Equation
Although the exterior boundary value problems for the Helmholtz equation
with either Dirichlet or Neumann boundary conditions are uniquely
solvable, related boundary integral equations may not be solvable, or
the solutions are not unique. In particular, the boundary integral
operators are not injective when the wave number is an eigenvalue of the
interior Dirichlet or Neumann eigenvalue problem, respectively.
Considering linear combinations of different boundary integral
formulations, this results in combined boundary integral equations which
are unique solvable for all wave numbers. The most known formulations
are those of Brakhage-Werner and Burton-Miller.
However, since the
combined boundary integral equation involves boundary integral operators
of both first and second kind, the analytical framework offers different
settings. The classical combined boundary integral equations are
considered in $L_2(\Gamma)$, where the uniqueness results are based on
Gaarding's inequality and Fredholm's alternative. To ensure the
compactness of certain boundary integral operators, sufficient
smoothness of the surface is required. Here I will discuss different
regularized formulations, which ensure the unique solvability and
stability of the corresponding Galerkin method even for Lipschitz surfaces.