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Preprints, Working Papers, ... Year : 2024

Stability of Perfectly Matched Layers for Maxwell's Equations in Rectangular Solids

Abstract

Perfectly matched layers from the nineties are extensively used to compute approximate solutions of Maxwell's equations in $\RR^{1+3}$ using a bounded computational domain, usually a rectangular solid. A smaller domain of interest is surrounded by layers designed to absorb outgoing waves in perfectly reflectionless manner. On the external boundary of the computational domain imperfect absorbing conditions are imposed. The method replaces the Maxwell equations by a larger system, with absorption coefficients nonzero in the layers. Well posedness of the resulting initial boundary value problem is proved here for the first time. The Laplace transform of the resulting Helmholtz system is studied. For real values of the transform variable $\tau$ it is classical that the system has $H^1$ solutions. To prove that they yield solutions of Maxwell's equations requires more regularity. For ${\rm Im}\,\tau\ne 0$ the problem is much more difficult. It is is complex, its Dirichlet form loses its positivity as do the boundary terms expressing dissipativity. We smooth the domain and construct $H^2$ solutions with uniform $H^1$ estimates. Maxwell's equations are recovered and the smoothing is removed. Need to carefully choose boundary conditions at the smoothed boundaries, the estimates require subtle multipliers, and, a method of Jerison-Kenig-Mitrea is extended to help overcome the nonpositivity of the flux.
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Dates and versions

hal-04049577 , version 1 (28-03-2023)
hal-04049577 , version 2 (23-06-2024)

Identifiers

  • HAL Id : hal-04049577 , version 2

Cite

Laurence Halpern, Jeffrey Rauch. Stability of Perfectly Matched Layers for Maxwell's Equations in Rectangular Solids. 2024. ⟨hal-04049577v2⟩
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