Abstract
We present super-algebraic compatible Nyström discretizations for the four Helmholtz boundary operators of Calderón’s calculus on smooth closed curves in 2D. These discretizations are based on appropriate splitting of the kernels combined with very accurate product-quadrature rules for the different singularities that such kernels present. A Fourier based analysis shows that the four discrete operators converge to the continuous ones in appropriate Sobolev norms. This proves that Nyström discretizations of many popular integral equation formulations for Helmholtz equations are stable and convergent. The convergence is actually super-algebraic for smooth solutions.
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Acknowledgements
Catalin Turc gratefully acknowledge support from NSF through contract DMS-1312169. Víctor Domínguez is partially supported by Ministerio de Economía y Competitividad, through the grant MTM2014-52859.
This research was partially supported by Spanish MINECO grants MTM2011-22741 and MTM2014-54388.
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Domínguez, V., Turc, C. (2016). High Order Nyström Methods for Transmission Problems for Helmholtz Equation. In: Ortegón Gallego, F., Redondo Neble, M., Rodríguez Galván, J. (eds) Trends in Differential Equations and Applications. SEMA SIMAI Springer Series, vol 8. Springer, Cham. https://doi.org/10.1007/978-3-319-32013-7_15
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DOI: https://doi.org/10.1007/978-3-319-32013-7_15
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