High-frequency behaviour of corner singularities in Helmholtz problems

T. Chaumont-Frelet, S. Nicaise
2018 Mathematical Modelling and Numerical Analysis  
We analyze the singular behaviour of the Helmholtz equation set in a nonconvex polygon. Classically, the solution of the problem is split into a regular part and one singular function for each re-entrant corner. The originality of our work is that the "amplitude" of the singular parts is bounded explicitly in terms of frequency. We show that for high frequency problem, the "dominant" part of the solution is the regular part. As an application, we derive sharp error estimates for finite-element
more » ... for finite-element discretizations. These error estimates show that the "pollution effect" is not changed by the presence of singularities. Furthermore, a consequence of our theory is that locally refined meshes are not needed for high-frequency problems, unless a very accurate solution is required. These results are illustrated with numerical examples, that are in accordance with the developed theory. 2010 Mathematics Subject Classification. Primary 35J05 35J75 65N30; Secondary 78A45. Key words and phrases. Helmholtz problems; Corner singularities; Finite elements; Pollution effect. T. Chaumont-Frelet has received funding from the Projects of the Spanish Ministry of Economy and Competitiveness with reference MTM2016-76329-R, and the BCAM "Severo Ochoa" accreditation of excellence SEV-2013-0323. 1 1 as usual, for s > 0, a function g belongs toH s (Γ 0 ) ifg, its extension by zero outside Γ 0 , belongs to H s (∂D ω,R )
doi:10.1051/m2an/2018031 fatcat:abbqwmmbcfb65cedrijefuka4a