  The discretization of surface intrinsic elliptic partial differential
equations (PDEs) poses interesting challenges not seen in flat space. The
discretization of these PDEs typically proceeds by either parametrizing the
surface, triangulating the surface, or embedding the surface in a higher
dimensional flat space. The closest point method (CPM) is an embedding method
that represents surfaces using a function that maps points in the embedding
space to their closest points on the surface. In the CPM, this mapping also
serves as an extension operator that brings surface intrinsic data onto the
embedding space, allowing PDEs to be numerically approximated by standard
methods in a narrow tubular neighborhood of the surface. We focus here on
numerically approximating the positive Helmholtz equation,
$\left(c-\Delta_\mathcal{S}\right)u=f,~c\in\mathbb{R}^+$ by the CPM paired with
finite differences. This yields a large, sparse, and non-symmetric system to
solve. Herein, we develop restricted additive Schwarz (RAS) and optimized
restricted additive Schwarz (ORAS) solvers and preconditioners for this
discrete system. In particular, we develop a general strategy for computing
overlapping partitions of the computational domain, as well as defining the
corresponding Dirichlet and Robin transmission conditions. We demonstrate that
the convergence of the ORAS solvers and preconditioners can be improved by
using a modified transmission condition where more than two overlapping
subdomains meet. Numerical experiments are provided for a variety of analytical
and triangulated surfaces. We find that ORAS solvers and preconditioners
outperform their RAS counterparts, and that using domain decomposition as a
preconditioner gives faster convergence over using it as a solver, as expected.
The methods exhibit good parallel scalability over the range of process counts
tested.

Numerical Analysis;Numerical Analysis