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\(T\)-coercivity: application to the discretization of Helmholtz-like problems. (English) Zbl 1252.35022

Summary: To solve variational indefinite problems, a celebrated tool is the Banach-Nečas-Babuška theory, which relies on the inf-sup condition. Here, we choose an alternate theory, \(T\)-coercivity. This theory relies on explicit inf-sup operators, both at the continuous and discrete levels. It is applied to solve Helmholtz-like problems in acoustics and electromagnetics. We provide simple proofs to solve the exact and discrete problems, and to show convergence under fairly general assumptions. We also establish sharp estimates on the convergence rates.

MSC:

35A35 Theoretical approximation in context of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
46N99 Miscellaneous applications of functional analysis
65J05 General theory of numerical analysis in abstract spaces
76Q05 Hydro- and aero-acoustics
78A10 Physical optics
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