Iterative methods for time-harmonic waves

Towards accuracy and scalability

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The bottleneck in designing iterative solvers for the Helmholtz equation lies in balancing the trade-off between accuracy and scalability. Both the accuracy of the numerical solution and the number of iterations to reach convergence deteriorate in higher dimensions and increase with the wavenumber. To address these issues in this dissertation, we formulated three research pillars: accuracy, wavenumber independent convergence and linear complexity. Below, we summarize the core findings of this dissertation:

We develop the first preconditioning technique which leads to close to wavenumber independent convergence for very large wavenumbers in 1D, 2D and 3D. Building on a two-level deflation projection method, we incorporated Quadratic Rational Bezier curves to construct the deflation space and vectors (Chapter 7). As a result, the near-zero eigenvalues of the coarse grid operator remain aligned with the fine-grid operator, keeping the spectrum of the preconditioned system clustered, leading to superior convergence properties compared to previous methods.

For over 30 years, applied mathematicians have tried to make convergent (standard) multigrid solvers for the Helmholtz equation. Multigrid solvers use sequences of smaller problem sizes and are computationally cheap and easy to implement. Unfortunately, multigrid methods diverge for Helmholtz and solving this issue remained an open problem. Using standard smoothing techniques, combined with similar higher-order coarse spaces, we constructed a fully convergent V- and W-cycle algorithm (Chapter 9). The key features of the algorithm are the use of higher-order transfer operators (instead of deflation vectors in the previous application) and a complex shift in the smoothing operator. While the method converges and the preliminary results have been proven, much research can still be conducted in this area, as this could support a paradigm shift in solving the complexity issue for very large wavenumbers in 2D and 3D. In light of this, we extended the two-level deflation solver to a multi-level deflation solver to address both the issue of wavenumber and problem size dependence (Chapter 8). In this part, we show better convergence properties and provide numerical experiments on challenging 2D and 3D test problems to corroborate the theoretical results.

Finally, we developed an unprecedented way to study the accuracy of the numerical solutions by studying the eigenvalues of systems where the analytical solution is known (Chapter 5). Expressing the pollution error in terms of these eigenmodes, enabled theoretical accuracy studies and dispersion corrections in higher dimensions, irrespective of the wave propagation angles. Something which was previously impossible. We also studied the application of Isogeometric Analysis (IgA) to improve the accuracy and reduce the pollution error (Chapter 6). Our results showed that the use of IgA was able to significantly suppress the pollution error compared to Finite Elements Discretizations of the same order.