35 research outputs found
On the convergence of local expansions of layer potentials
In a recently developed quadrature method (quadrature by expansion or QBX),
it was demonstrated that weakly singular or singular layer potentials can be
evaluated rapidly and accurately on surface by making use of local expansions
about carefully chosen off-surface points. In this paper, we derive estimates
for the rate of convergence of these local expansions, providing the analytic
foundation for the QBX method. The estimates may also be of mathematical
interest, particularly for microlocal or asymptotic analysis in potential
theory
Sweeping Preconditioner for the Helmholtz Equation: Moving Perfectly Matched Layers
This paper introduces a new sweeping preconditioner for the iterative
solution of the variable coefficient Helmholtz equation in two and three
dimensions. The algorithms follow the general structure of constructing an
approximate factorization by eliminating the unknowns layer by layer
starting from an absorbing layer or boundary condition. The central idea of
this paper is to approximate the Schur complement matrices of the factorization
using moving perfectly matched layers (PMLs) introduced in the interior of the
domain. Applying each Schur complement matrix is equivalent to solving a
quasi-1D problem with a banded LU factorization in the 2D case and to solving a
quasi-2D problem with a multifrontal method in the 3D case. The resulting
preconditioner has linear application cost and the preconditioned iterative
solver converges in a number of iterations that is essentially indefinite of
the number of unknowns or the frequency. Numerical results are presented in
both two and three dimensions to demonstrate the efficiency of this new
preconditioner.Comment: 25 page
Additive Sweeping Preconditioner for the Helmholtz Equation
We introduce a new additive sweeping preconditioner for the Helmholtz
equation based on the perfect matched layer (PML). This method divides the
domain of interest into thin layers and proposes a new transmission condition
between the subdomains where the emphasis is on the boundary values of the
intermediate waves. This approach can be viewed as an effective approximation
of an additive decomposition of the solution operator. When combined with the
standard GMRES solver, the iteration number is essentially independent of the
frequency. Several numerical examples are tested to show the efficiency of this
new approach.Comment: 27 page
Efficient coupling of inhomogeneous current spreading and dynamic electro-optical models for broad-area edge-emitting semiconductor devices
We extend a 2 (space) + 1 (time)-dimensional traveling wave model for broad-area edge-emitting semiconductor lasers by a model for inhomogeneous current spreading from the contact to the active zone of the laser. To speedup the performance of the device simulations, we suggest and discuss several approximations of the inhomogeneous current density in the active zone