Wavenumber-explicit stability and convergence analysis of ℎ𝑝 finite element discretizations of Helmholtz problems in piecewise smooth media
Maximilian Bernkopf, T. Chaumont-Frelet, Jens Markus Melenk
Abstract
We present a wavenumber-explicit convergence analysis of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h p"> <mml:semantics> <mml:mrow> <mml:mi>h</mml:mi> <mml:mi>p</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">hp</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Finite Element Method applied to a class of heterogeneous Helmholtz problems with piecewise analytic coefficients at large wavenumber <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Our analysis covers the heterogeneous Helmholtz equation with Robin, exact Dirichlet-to-Neumann, and second order absorbing boundary conditions, as well as perfectly matched layers.