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Fortino Garcia: WaveHoltz: Iterative solution of the Helmholtz equation via the wave equation

Time: Tue 2019-11-19 11.00 - 11.45

Location: KTH, 3418

Participating: Fortino Garcia, University of Colorado Boulder

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Designing efficient iterative solvers for the Helmholtz equation is notoriously difficult, with the two main difficulties being the resolution requirements and the highly indefinite character of the discretized system of equations. The dependence of the number of degrees of freedom on the frequency of the problem require high frequency Helmholtz solvers to be: (1) parallel, memory lean, and scalable, (2) high order accurate to overcome the penalty due to pollution/dispersion errors.

This talk will introduce a novel idea, the WaveHoltz iteration (WHI), for the solution of Helmholtz problems inspired by recent work on exact controllability (EC) methods. As in EC methods the iteration presented makes use of time domain methods for wave equations to design frequency domain Helmholtz solvers, but unlike EC methods no adjoint solves are needed. We show that the WaveHoltz iteration is symmetric and positive definite in the continuous setting. Numerical examples will be presented using various discretization techniques that demonstrate the method can be used to solve problems with rather high wave numbers in a memory lean and scalable fashion. This is joint work with Daniel Appelö (University of Colorado, Boulder) and Olof Runborg (KTH).