2017/08/28 by Simon N. Chandler‐Wilde, Euan A. Spence, Chandler-Wilde, Simon N. +5 · 2 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1708.08415
openalex publication_date 2017/08/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper is concerned with resolvent estimates on the real axis for the\nHelmholtz equation posed in the exterior of a bounded obstacle with Dirichlet\nboundary conditions when the obstacle is trapping. There are two resolvent\nestimates for this situation currently in the literature: (i) in the case of\nelliptic trapping the general "worst case" bound of exponential growth applies,\nand examples show that this growth can be realised through some sequence of\nwavenumbers, (ii) in the prototypical case of hyperbolic trapping where the\nHelmholtz equation is posed in the exterior of two strictly convex obstacles\n(or several obstacles with additional constraints) the nontrapping resolvent\nestimate holds with a logarithmic loss.\n This paper proves the first resolvent estimate for parabolic trapping by\nobstacles, studying a class of obstacles the prototypical example of which is\nthe exterior of two squares (in 2-d), or two cubes (in 3-d), whose sides are\nparallel. We show, via developments of the vector-field/multiplier argument of\nMorawetz and the first application of this methodology to trapping\nconfigurations, that a resolvent estimate holds with a polynomial loss over the\nnontrapping estimate. We use this bound, along with the other trapping\nresolvent estimates, to prove results about integral-equation formulations of\nthe boundary value problem in the case of trapping. Feeding these bounds into\nexisting frameworks for analysing finite and boundary element methods, we\nobtain the first wavenumber-explicit proofs of convergence for numerical\nmethods for solving the Helmholtz equation in the exterior of a trapping\nobstacle.\n