17 research outputs found
Sharp high-frequency estimates for the Helmholtz equation and applications to boundary integral equations
We consider three problems for the Helmholtz equation in interior and
exterior domains in R^d (d=2,3): the exterior Dirichlet-to-Neumann and
Neumann-to-Dirichlet problems for outgoing solutions, and the interior
impedance problem. We derive sharp estimates for solutions to these problems
that, in combination, give bounds on the inverses of the combined-field
boundary integral operators for exterior Helmholtz problems.Comment: Version 3: 42 pages; improved exposition in response to referee
comments and added several reference
Unique continuation for the Helmholtz equation using stabilized finite element methods
In this work we consider the computational approximation of a unique
continuation problem for the Helmholtz equation using a stabilized finite
element method. First conditional stability estimates are derived for which,
under a convexity assumption on the geometry, the constants grow at most
linearly in the wave number. Then these estimates are used to obtain error
bounds for the finite element method that are explicit with respect to the wave
number. Some numerical illustrations are given.Comment: corrected typos; included suggestions from reviewer
A first order system least squares method for the Helmholtz equation
We present a first order system least squares (FOSLS) method for the
Helmholtz equation at high wave number k, which always deduces Hermitian
positive definite algebraic system. By utilizing a non-trivial solution
decomposition to the dual FOSLS problem which is quite different from that of
standard finite element method, we give error analysis to the hp-version of the
FOSLS method where the dependence on the mesh size h, the approximation order
p, and the wave number k is given explicitly. In particular, under some
assumption of the boundary of the domain, the L2 norm error estimate of the
scalar solution from the FOSLS method is shown to be quasi optimal under the
condition that kh/p is sufficiently small and the polynomial degree p is at
least O(\log k). Numerical experiments are given to verify the theoretical
results
Stability estimate for the Helmholtz equation with rapidly jumping coefficients
The goal of this paper is to investigate the stability of the Helmholtz
equation in the high- frequency regime with non-smooth and rapidly oscillating
coefficients on bounded domains. Existence and uniqueness of the problem can be
proved using the unique continuation principle in Fredholm's alternative.
However, this approach does not give directly a coefficient-explicit energy
estimate. We present a new theoretical approach for the one-dimensional problem
and find that for a new class of coefficients, including coefficients with an
arbitrary number of discontinuities, the stability constant (i.e., the norm of
the solution operator) is bounded by a term independent of the number of jumps.
We emphasize that no periodicity of the coefficients is required. By selecting
the wave speed function in a certain \resonant" way, we construct a class of
oscillatory configurations, such that the stability constant grows
exponentially in the frequency. This shows that our estimates are sharp.Comment: a) Added references, b) rewritten the introduction with a summary of
the results/techniques of the paper, c) Corrected typo