# The Helmholtz equation in heterogeneous media: a priori bounds, well-posedness, and resonances

@article{Graham2018TheHE, title={The Helmholtz equation in heterogeneous media: a priori bounds, well-posedness, and resonances}, author={I. Graham and O. R. Pembery and E. Spence}, journal={arXiv: Analysis of PDEs}, year={2018} }

We consider a variety of boundary value problems (BVPs) for the heterogeneous Helmholtz equation, i.e. the equation $\nabla\cdot(A \nabla u ) + k^2 n u =-f$ where both $A$ and $n$ are functions of position. We prove new a priori bounds on the solution under conditions on $A$, $n$, and the domain that ensure nontrapping of rays; the novelty is that these bounds are explicit in $k$, $A$, $n$, and geometric parameters of the domain. We then show that these a priori bounds hold when $A$ and $n$ are… CONTINUE READING

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- ArXiv
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- ArXiv
- 2019

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- J. Comput. Appl. Math.
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- Mathematics, Computer Science
- ArXiv
- 2020

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- Mathematics, Computer Science
- ArXiv
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#### References

SHOWING 1-10 OF 98 REFERENCES

The Helmholtz Equation in Random Media: Well-Posedness and A Priori Bounds

- Mathematics, Computer Science
- SIAM/ASA J. Uncertain. Quantification
- 2020

5- PDF

Acoustic transmission problems: wavenumber-explicit bounds and resonance-free regions

- Mathematics, Physics
- 2017

25- PDF

Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions

- Computer Science, Mathematics
- Math. Comput.
- 2010

139- PDF

Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: what is the largest shift for which wavenumber-independent convergence is guaranteed?

- Mathematics, Computer Science
- Numerische Mathematik
- 2015

68- PDF

Wavenumber-Explicit Bounds in Time-Harmonic Acoustic Scattering

- Computer Science, Mathematics
- SIAM J. Math. Anal.
- 2014

39- PDF

A new frequency-uniform coercive boundary integral equation for acoustic scattering

- Mathematics
- 2011

43- PDF

An inequality for the reduced wave operator and the justification of geometrical optics

- Mathematics
- 1968

107 Highly Influential

A Sign-Definite Preconditioned High-Order FEM, Part I: Formulation and Simulation for Bounded Homogeneous Media Wave Propagation

- Computer Science, Mathematics
- SIAM J. Sci. Comput.
- 2017

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