35J05 Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation [See also 31Axx, 31Bxx]
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- carrier and current densities (2)
- density matrices (2)
- dissipative Schroedinger-Poisson systems (2)
- dissipative Schroedinger-type operators (2)
- Asymptotic analysis (1)
- Helmholtz equation (1)
- Pade approximation (1)
- Periodic surface homogenization (1)
- Singular asymptotic expansions (1)
- a priori estimates (1)
- discrete transparent boundary conditions (1)
- finite difference method (1)
- one-way Helmholtz equation (1)
- pole condition (1)
- resonance problems (1)
- scattering theory (1)
- smooth operators (1)
- split-step method (1)
- spurious solutions (1)
- transparent boundary condition (1)
The present work deals with the resolution of the Poisson equation in a bounded domain made of a thin and periodic layer of finite length placed into a homogeneous medium.
We provide and justify a high order asymptotic expansion which takes into account the boundary layer effect occurring in the vicinity of the periodic layer as well as the corner singularities appearing in the neighborhood of the extremities of the layer. Our approach combines mixes
the method of matched asymptotic expansions and the method of periodic surface homogenization.
When simulating isolated resonators, the application of transparent boundary conditions causes the approximated spectrum to be polluted with spurious solutions. Distinguishing these artificial solutions from solutions with a physical meaning is often difficult and requires a priori knowledge of the spectrum or the expected field distribution of resonant states. We present an implementation of the pole condition that distinguishes between incoming and outgoing waves by the location of the poles of their Laplace transform as transparent boundary condition. This implementation depends on one tuning parameter. We will use the sensitivity of the computed solutions to perturbations of this parameter as a means to identify spurious solutions. To obtain global statements, we will combine this technique with a convergence monitor for the boundary condition.
This work deals with the efficient numerical solution of
the two-dimensional one-way Helmholtz equation
posed on an unbounded domain.
In this case one has to introduce
artificial boundary conditions to confine the computational domain.
Here we construct with the Z-transformation
so-called discrete transparent
boundary conditions
for higher-order parabolic equations schemes.
These methods are Pade ``Parabolic'' approximations of
the one-way Helmholtz equation
and frequently used in integrated optics and (underwater) acoustics.