Convolution quadrature for the wave equation with a nonlinear impedance boundary condition

Lehel Banjai, Alexander Rieder

Research output: Contribution to journalArticle

Abstract

A rarely exploited advantage of time-domain boundary integral equations compared to their frequency counterparts is that they can be used to treat certain nonlinear problems. In this work we investigate the scattering of acoustic waves by a bounded obstacle with a nonlinear impedance boundary condition. We describe a boundary integral formulation of the problem and prove without any smoothness assumptions on the solution the convergence of a full discretization: Galerkin in space and convolution quadrature in time. If the solution is sufficiently regular, we prove that the discrete method converges at optimal rates. Numerical evidence in 3D supports the theory.
Original languageEnglish
Pages (from-to)1783-1819
Number of pages37
JournalMathematics of Computation
Volume87
Issue number312
Early online date4 Oct 2017
DOIs
Publication statusPublished - 2018

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Optimal Rates
Boundary Integral
Boundary Integral Equations
Acoustic Waves
Quadrature
Galerkin
Impedance
Nonlinear Problem
Time Domain
Wave equation
Convolution
Smoothness
Discretization
Scattering
Converge
Boundary conditions
Formulation
Evidence

Keywords

  • math.NA
  • 65M38, 65M12, 65R20

Cite this

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Convolution quadrature for the wave equation with a nonlinear impedance boundary condition. / Banjai, Lehel; Rieder, Alexander.

In: Mathematics of Computation, Vol. 87, No. 312, 2018, p. 1783-1819 .

Research output: Contribution to journalArticle

TY - JOUR

T1 - Convolution quadrature for the wave equation with a nonlinear impedance boundary condition

AU - Banjai, Lehel

AU - Rieder, Alexander

PY - 2018

Y1 - 2018

N2 - A rarely exploited advantage of time-domain boundary integral equations compared to their frequency counterparts is that they can be used to treat certain nonlinear problems. In this work we investigate the scattering of acoustic waves by a bounded obstacle with a nonlinear impedance boundary condition. We describe a boundary integral formulation of the problem and prove without any smoothness assumptions on the solution the convergence of a full discretization: Galerkin in space and convolution quadrature in time. If the solution is sufficiently regular, we prove that the discrete method converges at optimal rates. Numerical evidence in 3D supports the theory.

AB - A rarely exploited advantage of time-domain boundary integral equations compared to their frequency counterparts is that they can be used to treat certain nonlinear problems. In this work we investigate the scattering of acoustic waves by a bounded obstacle with a nonlinear impedance boundary condition. We describe a boundary integral formulation of the problem and prove without any smoothness assumptions on the solution the convergence of a full discretization: Galerkin in space and convolution quadrature in time. If the solution is sufficiently regular, we prove that the discrete method converges at optimal rates. Numerical evidence in 3D supports the theory.

KW - math.NA

KW - 65M38, 65M12, 65R20

U2 - 10.1090/mcom/3279

DO - 10.1090/mcom/3279

M3 - Article

VL - 87

SP - 1783

EP - 1819

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 312

ER -