CERFACS CERFACS Scientific Activity Report Jan. 2010 – Dec. 2011

ELECTROMAGNETISM AND ACOUSTICS TEAM
Influence of the frequency
GCSIE formulation
λ
k
o
h
n iter
4.715 14 23
9.43 14 60
18.85 14 115
PMCHWT formulation with SPAI preconditioner
λ o
k h
n iter
4.715 14 40
9.43 14 115
18.85 14 300(0.002)
[5] B.H. Jung and T.K. Sarkar and Y.S. Chung, A survey of various frequency domain integral equations
for the analysis of scattering from three-dimensional dielectric objets, PIER, 36 :193-246,2002.
2.7.2 A posteriori error analysis for integral equations
In the field of electromagnetism, acoustics or elastodynamics, the ability of the methods of integral
equations to solve large problems raised by the application has been widely proven. They are both less
costly in degrees of freedom and less dispersive than methods based on the discretization of the entire
domain (finite element method, finite difference method or discontinuous Galerkin method). In comparison
with the finite element methods, integral methods remain insufficiently popularized and they are generally
used by experienced specialists. We believe that one of the obstacles to wider use of these methods is the
lack of automatic tools, to ensure the accuracy of the computed solution. Indeed, the techniques of an a
posteriori error analysis and the adaptive methods are almost non-existent in the field of integral equations.
Nevertheless, there are some theoretical results [6, 7] showing the possibility of constructing a posteriori
error indicators for the integral formulations but these estimators do not seem to have been tried in practice.
This is especially true in the fields of electromagnetism and acoustics for which we did not have find any
result in the literature. The main technical difficulties to derive an a posteriori error indicator for integral
equations are the nonlocal character of operators and the singularity of the Green kernel.
In 2011, we began this subject by a first trainee period [EMA27] in which we studied the possibility to
use an a posteriori error indicator based on averaging techniques for the 2D acoustic problem. This first
results are promising (see Fig. 2.15). In particular, the adaptive mesh refinement algorithm obtained allows
to obtain the best rate of convergence of the numerical method when a singularity occurs. Moreover, the
CPU time spent to obtain an accurate solution decreases.
As perspective, a project in collaboration with CNRS, EADS-IW, IMACS and THALES was submitted to
the call for project MN of the ANR.
[6] C. CARSTENSEN and D. PRAETORIUS. Averaging techniques for the a posteriori bem error control
for a hypersingular integral equation in two dimensions. SIAM J.SCI. COMPUT.,29 :782, 2007.
[7] C. CARSTENSEN and D. PRAETORIUS. Averaging techniques for the a posteriori error control in
nite element and boundary element analysis. SIAM J.SCI. COMPUT., 2006.
CERFACS ACTIVITY REPORT 53