Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering*
- S. Chandler-Wilde, I. Graham, S. Langdon, E. Spence
- MathematicsActa Numerica
- 19 April 2012
Recent progress on the design, analysis and implementation of hybrid numerical-asymptotic boundary integral methods for boundary value problems for the Helmholtz equation that model time harmonic acoustic wave scattering in domains exterior to impenetrable obstacles is described.
Sharp High-Frequency Estimates for the Helmholtz Equation and Applications to Boundary Integral Equations
- Dean Baskin, E. Spence, J. Wunsch
- MathematicsSIAM Journal on Mathematical Analysis
- 4 April 2015
We consider three problems for the Helmholtz equation in interior and exterior domains in $\mathbb{R}^d$ ($d=2,3$): the exterior Dirichlet-to-Neumann and Neumann-to-Dirichlet problems for outgoing…
Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: what is the largest shift for which wavenumber-independent convergence is guaranteed?
The question of how large varepsilon can be can be so that the shifted problem provides a preconditioner that leads to k-independent convergence of GMRES is focused on, and the main result is a sufficient condition on £ε for this property to hold.
Wavenumber-Explicit Bounds in Time-Harmonic Acoustic Scattering
- E. Spence
- MathematicsSIAM Journal on Mathematical Analysis
- 28 August 2014
Borders on the Dirichlet-to-Neumann map for the Helmholtz equation in the exterior of a bounded obstacle are proved, which are the sharpest yet obtained (for their respective problems) in terms of their dependence on the wavenumber.
Boundary Value Problems for Linear Elliptic PDEs
- E. Spence
- Mathematics
- 2010
This thesis is concerned with new analytical and numerical methods for solving boundary value problems for the 2nd order linear elliptic PDEs of Poisson, Helmholtz, and modified Helmholtz in two…
Numerical Estimation of Coercivity Constants for Boundary Integral Operators in Acoustic Scattering
It is found that coercivity holds, uniformly in the wavenumber $k$, for a wide variety of domains, and convergence estimates for the numerical range of Galerkin projections of a general bounded linear operator on a Hilbert space are proved.
Is the Helmholtz Equation Really Sign-Indefinite?
New sign-definite formulations of the Helmholtz equation posed in either the interior of a star-shaped domain with impedance boundary conditions or the exterior of aStar-shape domain, with implications for both the analysis and the practical implementation of finite element methods are introduced.
A new frequency‐uniform coercive boundary integral equation for acoustic scattering
- E. Spence, S. Chandler-Wilde, I. Graham, V. Smyshlyaev
- Mathematics
- 1 October 2011
A new boundary integral operator is introduced for the solution of the soundsoft acoustic scattering problem, i.e., for the exterior problem for the Helmholtz equation with Dirichlet boundary…
A spectral collocation method for the Laplace and modified Helmholtz equations in a convex polygon
- S. Smitheman, E. Spence, A. Fokas
- Mathematics
- 1 October 2010
Integral representations for the solutions of the Laplace and modified Helmholtz equations can be obtained using Green's theorem. However, these representations involve both the solution and its…
A new transform method II: the global relation and boundary-value problems in polar coordinates
- E. Spence, A. S. Fokas
- MathematicsProceedings of the Royal Society A
- 8 August 2010
A new method for solving boundary-value problems (BVPs) for linear and certain nonlinear PDEs was introduced by one of the authors in the late 1990s. For linear PDEs, this method constructs novel…
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