The Helmholtz equation in heterogeneous media: a priori bounds, well-posedness, and resonances
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In this paper, the exterior Dirichlet problem for the heterogeneous Helmholtz equation is considered and new a priori bounds on the solution under conditions on A, n, and the domain that ensure nontrapping of rays are proved.About:
This article is published in Journal of Differential Equations.The article was published on 2019-03-05 and is currently open access. It has received 54 citations till now. The article focuses on the topics: Dirichlet problem & Helmholtz equation.read more
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Stability and finite element error analysis for the Helmholtz equation with variable coefficients
Ivan G. Graham,Stefan A. Sauter +1 more
TL;DR: In this article, the authors discuss the stability theory and numerical analysis of the Helmholtz equation with variable and possibly nonsmooth or oscillatory coefficients, and give an existence-uniqueness result for this problem, which holds under rather general conditions on the coefficients and on the domain.
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Wavenumber explicit convergence analysis for finite element discretizations of general wave propagation problems
TL;DR: In this article, a general methodology to derive stability conditions and error estimates that are explicit with respect to the wavenumber is proposed. But the method is not suitable for the case of convected sound waves.
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Domain decomposition preconditioners for high-order discretisations of the heterogeneous Helmholtz equation
TL;DR: These estimates prove rigorously that, with enough absorption and for $k$ large enough, GMRES is guaranteed to converge in a number of iterations that is independent of $k,p,$ and the coefficients.
Posted Content
A sharp relative-error bound for the Helmholtz $h$-FEM at high frequency
TL;DR: A key ingredient in the proofs is a result describing the oscillatory behaviour of the solution of the plane-wave scattering problem, which is proved using semiclassical defect measures and is sufficient for the relative error of the FEM solution in 2 or 3 dimensions to be controllably small.
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For most frequencies, strong trapping has a weak effect in frequency-domain scattering
TL;DR: In this paper, it was shown that even in the presence of the strongest-possible trapping, if a set of frequencies of arbitrarily small measure is excluded, the Helmholtz solution operator grows at most polynomially as the frequency tends to infinity.
References
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TL;DR: Second-order boundary value problems in polygons have been studied in this article for convex domains, where the second order boundary value problem can be solved in the Sobolev spaces of Holder functions.
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Fourier Integral Operators. I
TL;DR: In this paper, a more general class of pseudo-differential operators for non-elliptic problems is discussed. But their value is rather limited in genuinely nonelliptical problems.
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Strongly Elliptic Systems and Boundary Integral Equations
TL;DR: In this article, the Laplace equation, the Helmholtz equation, and the Sobolev spaces of strongly elliptic systems have been studied and further properties of spherical harmonics have been discussed.