November 26, 2024
Clément Richefort will defend his PhD, titled Development of an algebraic multigrid solver for the indefinite Helmholtz equation on Wednesday 27th of November at 2pm in the Ada Lovelace room at INRIA. The presentation will be in English.
Abstract: The numerical simulation of complex physical phenomena generally requires to solve systems of linear equations. The solver should benefit from modern computing machines and scale on highly parallel architectures. In particular, multigrid methods are scalable methods that enable the resolution of a wide range of problems where the discretization matrix is symmetric positive definite and the near-kernel space geometrically smooth. In this thesis, our target is to extend multigrid methods to the oscillatory and indefinite Helmholtz equation. Each multigrid operator should be adapted to these challenging properties. In particular, the smoother should capture large eigenvalues independently of the sign. Moreover, the range of interpolation should now approximate the oscillatory near-kernel space that is unknown at the set-up phase of multigrid. Last, indefinite matrices do not generate a norm. As a consequence, the coarse correction has no minimization properties and can amplify the error. In this thesis, we present an alternative coarse correction that contracts the error properly. We target a multi-level method that converges in a constant number of iterations independently of the matrix size.