Escapil-Inchauspe, P., & Jerez-Hanckes, C. (2021). Bi-parametric operator preconditioning. Comput. Math. Appl., 102, 220–232.
Abstract: We extend the operator preconditioning framework Hiptmair (2006) [10] to Petrov-Galerkin methods while accounting for parameter-dependent perturbations of both variational forms and their preconditioners, as occurs when performing numerical approximations. By considering different perturbation parameters for the original form and its preconditioner, our bi-parametric abstract setting leads to robust and controlled schemes. For Hilbert spaces, we derive exhaustive linear and super-linear convergence estimates for iterative solvers, such as h-independent convergence bounds, when preconditioning with low-accuracy or, equivalently, with highly compressed approximations.
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Fierro, I., & Jerez-Hanckes, C. (2020). Fast Calderon preconditioning for Helmholtz boundary integral equations. J. Comput. Phys., 409, 22 pp.
Abstract: Calderon multiplicative preconditioners are an effective way to improve the condition number of first kind boundary integral equations yielding provable mesh independent bounds. However, when discretizing by local low-order basis functions as in standard Galerkin boundary element methods, their computational performance worsens as meshes are refined. This stems from the barycentric mesh refinement used to construct dual basis functions that guarantee the discrete stability of L-2-pairings. Based on coarser quadrature rules over dual cells and H-matrix compression, we propose a family of fast preconditioners that significantly reduce assembly and computation times when compared to standard versions of Calderon preconditioning for the three-dimensional Helmholtz weakly and hyper-singular boundary integral operators. Several numerical experiments validate our claims and point towards further enhancements. (C) 2020 Elsevier Inc. All rights reserved.
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