Projects per year

### Abstract

We prove that the standard second-kind integral equation formulation of the exterior Dirichlet problem for the Helmholtz equation is coercive (i.e., sign-definite) for all smooth convex domains when the wavenumber k is sufficiently large. (This integral equation involves the so-called combined potential, or combined field, operator.) This coercivity result yields k-explicit error estimates when the integral equation is solved using the Galerkin method, regardless of the particular approximation space used (and thus these error estimates apply to several hybrid numerical-asymptotic methods developed recently). Coercivity also gives k-explicit bounds on the number of GMRES iterations needed to achieve a prescribed accuracy when the integral equation is solved using the Galerkin method with standard piecewise-polynomial subspaces. The coercivity result is obtained by using identities for the Helmholtz equation originally introduced by Morawetz in her work on the local energy decay of solutions to the wave equation

Original language | English |
---|---|

Pages (from-to) | 1587-1639 |

Number of pages | 53 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 68 |

Issue number | 9 |

Early online date | 6 Oct 2014 |

DOIs | |

Publication status | Published - 1 Sep 2015 |

## Fingerprint Dive into the research topics of 'Coercivity of combined boundary integral equations in high-frequency scattering'. Together they form a unique fingerprint.

## Projects

- 2 Finished

### Post Doc Fellowship - New Methods and Analysis for Wave Propagation Problems

Engineering and Physical Sciences Research Council

1/04/11 → 31/03/14

Project: Research council

### Boundary Integral Equation Methods for HF Scattering Problems

Graham, I. & Smyshlyaev, V. P.

Engineering and Physical Sciences Research Council

24/03/09 → 23/09/12

Project: Research council

## Profiles

### Euan Spence

- Department of Mathematical Sciences - Professor
- EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)

Person: Research & Teaching

## Cite this

Spence, E. A., Kamotski, I. V., & Smyshlyaev, V. P. (2015). Coercivity of combined boundary integral equations in high-frequency scattering.

*Communications on Pure and Applied Mathematics*,*68*(9), 1587-1639. https://doi.org/10.1002/cpa.21543