Decay for solutions of the exterior problem for the wave equation

Morawetz, Cathleen S.

doi:10.1002/cpa.3160280204

Cited by 128 publications

(124 citation statements)

References 8 publications

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“…In this paper, however, we show that this task can be tackled using identities for solutions of the Helmholtz equation originally introduced by Morawetz. (This builds on the earlier work of two of the authors and their collaborators in [58].) Recall that Morawetz showed in [46] that bounding the solution of the exterior Dirichlet problem could be converted (via her identities) into constructing an appropriate vector field in the exterior of the obstacle, and then such a vector field was constructed by Morawetz, Ralston, and Strauss for two-dimensional nontrapping domains in [48, §4]. (This bound on the solution is equivalent to bounding the exterior Dirichlet-to-Neumann map, and can also be used to show local energy decay of solutions of the wave equation.)…”

Section: Introductionmentioning

confidence: 59%

“…Here we convert the problem of proving that A 0 k;Á is coercive into that of constructing a suitable vector field in both the exterior and the interior of the obstacle. In addition to needing a vector field in the interior as well as the exterior, the conditions that the vector field must satisfy for coercivity are stronger than those in [46] and [48]. Indeed, we prove that the conditions for coercivity cannot be satisfied if the obstacle is nonconvex, and then we construct a vector field satisfying these conditions for smooth, convex obstacles with strictly positive curvature in both two and three dimensions.…”

Section: Introductionmentioning

confidence: 90%

“…essentially introduced by Morawetz in [46], where Z.x/ is a vector field, andˇ.x/ and˛.x/ are scalar fields. Replacing the identity (1.31), coming from the multiplier rMu, by the more general identity coming from the multiplier Zu, and repeating the argument that led to (1.32), we find in Section 3 that, if Z is continuous across , <.…”

Section: Modifying the Classical Methods Using Morawetz's Identitiesmentioning

confidence: 99%

“…In Section 2 we recall the identities introduced for solutions of the Helmholtz equation by Morawetz in [46]. In Section 3 we translate the problem of proving that A 0 k;Á is coercive into that of constructing an appropriate vector field Z in the multiplier Zu.…”

Section: Outline Of the Papermentioning

confidence: 99%

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Coercivity of Combined Boundary Integral Equations in High‐Frequency Scattering

Spence

Kamotski

Smyshlyaev

2014

Comm Pure Appl Math

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We prove that the standard second-kind integral equation formulation of the exterior Dirichlet problem for the Helmholtz equation is coercive (i.e., sign-definite) for all smooth convex domains when the wavenumber k is sufficiently large. (This integral equation involves the so-called combined potential, or combined field, operator.) This coercivity result yields k-explicit error estimates when the integral equation is solved using the Galerkin method, regardless of the particular approximation space used (and thus these error estimates apply to several hybrid numerical-asymptotic methods developed recently). Coercivity also gives k-explicit bounds on the number of GMRES iterations needed to achieve a prescribed accuracy when the integral equation is solved using the Galerkin method with standard piecewise-polynomial subspaces. The coercivity result is obtained by using identities for the Helmholtz equation originally introduced by Morawetz in her work on the local energy decay of solutions to the wave equation.

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Section: Introductionmentioning

confidence: 59%

Section: Introductionmentioning

confidence: 90%

Section: Modifying the Classical Methods Using Morawetz's Identitiesmentioning

confidence: 99%

Section: Outline Of the Papermentioning

confidence: 99%

See 2 more Smart Citations

Coercivity of Combined Boundary Integral Equations in High‐Frequency Scattering

Spence

Kamotski

Smyshlyaev

2014

Comm Pure Appl Math

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“…The application of more sophisticated identities, such as those appearing in [51,53], to these type of problems (in particular for more general "nontrapping" scattering geometries) is under way.…”

Section: Resultsmentioning

confidence: 99%

A new frequency‐uniform coercive boundary integral equation for acoustic scattering

Spence

Chandler-Wilde

Graham

et al. 2011

Comm Pure Appl Math

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A new boundary integral operator is introduced for the solution of the soundsoft acoustic scattering problem, i.e., for the exterior problem for the Helmholtz equation with Dirichlet boundary conditions. We prove that this integral operator is coercive in L 2 ./ (where is the surface of the scatterer) for all Lipschitz star-shaped domains. Moreover, the coercivity is uniform in the wavenumber k D !=c, where ! is the frequency and c is the speed of sound. The new boundary integral operator, which we call the "star-combined" potential operator, is a slight modification of the standard combined potential operator, and is shown to be as easy to implement as the standard one. Additionally, to the authors' knowledge, it is the only second-kind integral operator for which convergence of the Galerkin method in L 2 ./ is proved without smoothness assumptions on except that it is Lipschitz. The coercivity of the star-combined operator implies frequency-explicit error bounds for the Galerkin method for any approximation space. In particular, these error estimates apply to several hybrid asymptoticnumerical methods developed recently that provide robust approximations in the high-frequency case. The proof of coercivity of the star-combined operator critically relies on an identity first introduced by Morawetz and Ludwig in 1968, supplemented further by more recent harmonic analysis techniques for Lipschitz domains.

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On the Low-frequency Asymptotic Expansion for some Second-order Elliptic Systems in a Two-dimensional Exterior Domain

Dan

1996

Math. Meth. Appl. Sci.

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Decay for solutions of the exterior problem for the wave equation

Cited by 128 publications

References 8 publications

Coercivity of Combined Boundary Integral Equations in High‐Frequency Scattering

Coercivity of Combined Boundary Integral Equations in High‐Frequency Scattering

A new frequency‐uniform coercive boundary integral equation for acoustic scattering

On the Low-frequency Asymptotic Expansion for some Second-order Elliptic Systems in a Two-dimensional Exterior Domain

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