[Back]

A. Bespalov, T. Betcke, A. Haberl, D. Praetorius:
"Adaptive BEM for the Helmholtz equation";
Talk: IABEM 2018 - Symposium of the International Association for Boundary Element Methods, Paris (invited); 2018-06-26 - 2018-06-29.

We consider the weakly-singular integral equation $V_k \, \phi = f$ associated with the Helmholtz equation
for arbitrary but fixed wavenumber $k>0$.
For a standard conforming BEM discretization with piecewise polynomials,
usual duality arguments show that the underlying triangulation has to be sufficiently fine to ensure
the existence and uniqueness of the Galerkin solution.


Extending existing approaches, we prove in that adaptive mesh-refinement is capable of overcoming this preasymptotic behavior and eventually leads to convergence with optimal algebraic rates. Unlike previous works, one does not have to deal with the {\sl a~priori} assumption that the initial mesh is sufficiently fine.


By generalizing existing inverse-type estimates for the Laplace equation to arbitrary wavenumber $k>0$,
we prove that ABEM with the weighted-residual error estimator fits in
the abstract setting. Thus, we show that ABEM does not only lead to
linear convergence, but also guarantees optimal algebraic convergence behavior of the underlying sl a~posteriori error estimator.
The overall conclusion of our results thus is that adaptivity has stabilizing effects and can, in particular, overcome preasymptotic and possibly pessimistic restrictions on the meshes.