449 research outputs found
Convergence of infinite element methods for scalar waveguide problems
We consider the numerical solution of scalar wave equations in domains which
are the union of a bounded domain and a finite number of infinite cylindrical
waveguides. The aim of this paper is to provide a new convergence analysis of
both the Perfectly Matched Layer (PML) method and the Hardy space infinite
element method in a unified framework. We treat both diffraction and resonance
problems. The theoretical error bounds are compared with errors in numerical
experiments
Transparent boundary conditions based on the Pole Condition for time-dependent, two-dimensional problems
The pole condition approach for deriving transparent boundary conditions is
extended to the time-dependent, two-dimensional case. Non-physical modes of the
solution are identified by the position of poles of the solution's spatial
Laplace transform in the complex plane. By requiring the Laplace transform to
be analytic on some problem dependent complex half-plane, these modes can be
suppressed. The resulting algorithm computes a finite number of coefficients of
a series expansion of the Laplace transform, thereby providing an approximation
to the exact boundary condition. The resulting error decays super-algebraically
with the number of coefficients, so relatively few additional degrees of
freedom are sufficient to reduce the error to the level of the discretization
error in the interior of the computational domain. The approach shows good
results for the Schr\"odinger and the drift-diffusion equation but, in contrast
to the one-dimensional case, exhibits instabilities for the wave and
Klein-Gordon equation. Numerical examples are shown that demonstrate the good
performance in the former and the instabilities in the latter case
Limiting absorption principle and perfectly matched layer method for Dirichlet Laplacians in quasi-cylindrical domains
We establish a limiting absorption principle for Dirichlet Laplacians in
quasi-cylindrical domains. Outside a bounded set these domains can be
transformed onto a semi-cylinder by suitable diffeomorphisms. Dirichlet
Laplacians model quantum or acoustically-soft waveguides associated with
quasi-cylindrical domains. We construct a uniquely solvable problem with
perfectly matched layers of finite length. We prove that solutions of the
latter problem approximate outgoing or incoming solutions with an error that
exponentially tends to zero as the length of layers tends to infinity. Outgoing
and incoming solutions are characterized by means of the limiting absorption
principle.Comment: to appear in SIAM Journal on Mathematical Analysi
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Mini-Workshop: Efficient and Robust Approximation of the Helmholtz Equation
The accurate and efficient treatment of wave propogation phenomena is still a challenging problem. A prototypical equation is the Helmholtz equation at high wavenumbers. For this equation, Babuška & Sauter showed in 2000 in their seminal SIAM Review paper that standard discretizations must fail in the sense that the ratio of true error and best approximation error has to grow with the frequency. This has spurred the development of alternative, non-standard discretization techniques. This workshop focused on evaluating and comparing these different approaches also with a view to their applicability to more general wave propagation problems
Transparent Boundary Conditions for Time-Dependent Problems
A new approach to derive transparent boundary conditions (TBCs) for dispersive wave, Schrödinger, heat, and drift-diffusion equations is presented. It relies on the pole condition and distinguishes between physically reasonable and unreasonable solutions by the location of the singularities of the Laplace transform of the exterior solution. Here the Laplace transform is taken with respect to a generalized radial variable. To obtain a numerical algorithm, a Möbius transform is applied to map the Laplace transform onto the unit disc. In the transformed coordinate the solution is expanded into a power series. Finally, equations for the coefficients of the power series are derived. These are coupled to the equation in the interior and yield transparent boundary conditions. Numerical results are presented in the last section, showing that the error introduced by the new approximate TBCs decays exponentially in the number of coefficients
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