Abstract
We examine the use of the Dirichlet-to-Neumann coarse space within an additive Schwarz method to solve the Helmholtz equation in 2D. In particular, we focus on the selection of how many eigenfunctions should go into the coarse space. We find that wave number independent convergence of a preconditioned iterative method can be achieved in certain special cases with an appropriate and novel choice of threshold in the selection criteria. However, this property is lost in a more general setting, including the heterogeneous problem. Nonetheless, the approach converges in a small number of iterations for the homogeneous problem even for relatively large wave numbers and is robust to the number of subdomains used.
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Notes
- 1.
Note that if ΓR = ∅ then the problem will be ill-posed for certain choices of k corresponding to Dirichlet eigenvalues of the corresponding Laplace problem.
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Bootland, N., Dolean, V. (2021). On the Dirichlet-to-Neumann Coarse Space for Solving the Helmholtz Problem Using Domain Decomposition. In: Vermolen, F.J., Vuik, C. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2019. Lecture Notes in Computational Science and Engineering, vol 139. Springer, Cham. https://doi.org/10.1007/978-3-030-55874-1_16
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DOI: https://doi.org/10.1007/978-3-030-55874-1_16
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