RESUMEN
In this paper we consider
the numerical solution of discrete
boundary integral equations
on polyhedral surfaces in three dimensions. When the solution contains
typical edge singularities, highly stretched meshes are preferred to uniform
meshes, since they reduce the number of degrees
of freedom needed to obtain
a fixed accuracy. The classical panel-clustering method can still be applied in the presence of
such highly stretched meshes. However, we will
show that the savings in computation time and storage become
suboptimal because the nearfield matrix
arising in the panel-clustering algorithm is no longer as sparse as it is
in the case of uniform meshes.
Hence, a natural question arises as to whether
a new enhanced panel-clustering algorithm can be designed which performs efficiently even in the presence
of highly stretched meshes. The main result
of this paper
is to formulate
such an enhanced
version of the panel-clustering algorithm. The key features of
the algorithm are (i) the employment of partial analytic
integration in the direction of stretching,
yielding a new kernel function on a one-dimensional manifold where the influence of
high aspect ratios in the stretched elements
is removed, and (ii) the introduction
of a generalized admissibility condition with respect to
the partially integrated kernel, which ensures that
certain stretched clusters which are inadmissible in the classical sense
now become admissible. In the context of a model
problem, we prove that our
algorithm yields an accurate (up to the discretization
error) matrix-vector multiplication
which requires O(NlogkN) operations, where N is the
number of degrees of freedom
and k is small and independent of the aspect ratio. The generalized admissibility condition can be viewed as an addition
to the classical
method which may be useful in general when stretched meshes are present. We also
have performed a numerical experiment which shows that
the sparsity of the nearfield
matrix for the enhanced panel-clustering method is not negatively
affected by stretched elements, and the
method will perform optimally.