Perfectly Matched Layers
This is a survey of some recent developments on the so called “Perfectly Matched Layer” (PML) method. We take as model the scattering problems in linear acoustics. First, the Cartesian PML equations are described in the time domain for the split Berenger
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Departamento de Matem´atica Aplicada, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain [email protected] Departamento de Matem´aticas, Universidade da Coru˜na, 15707 A Coru˜na, Spain [email protected] Departamento de Matem´atica Aplicada, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain [email protected] GI2 MA, Departamento de Ingenier´ıa Matem´atica, Universidad de Concepci´on, Casilla 160–C, Concepci´on, Chile [email protected]
Summary. This is a survey of some recent developments on the so called “Perfectly Matched Layer” (PML) method. We take as model the scattering problems in linear acoustics. First, the Cartesian PML equations are described in the time domain for the split Berenger and the unsplit Zhao–Cangellaris formulations. The energy estimates existing in the literature are revised, and the coupled fluid/PML problem is introduced. Next, the pressure formulation of the Cartesian PML model is derived in the frequency domain. We show that a PML method based on a non–integrable absorbing function allows recovering the exact solution in the physical domain of interest, in the framework of plane waves with oblique incidence. We revise the theoretical results that state the well–posedness of the continuous model for the acoustic scattering problem. Finally, we illustrate with some numerical results the efficiency and accuracy of the Cartesian PML approach and compare different absorbing profiles. Finally, we introduce the pressure formulation of the radial PML model in the frequency domain and revise the theoretical results that assess the accuracy of this technique in the continuous model. Under convenient assumptions, we show its convergence when the thickness of the PML becomes larger and its exactness when a non–integrable absorbing function is used. The numerical accuracy of this approach is also illustrated.
6.1 Introduction One problem to be tackled for the numerical solution of any scattering problem in an unbounded domain is truncating the computational domain without perturbing too much the solution of the original problem. In an ideal framework, this truncation should satisfy, at least, three properties: efficiency, easiness of implementation,
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A Berm´udez, L Hervella–Nieto, A Prieto, R Rodr´ıguez
and robustness. In fact, the typical first step for the numerical solution by either finite elements or finite differences is to choose boundary conditions to replace the Sommerfeld radiation condition at infinity, see, for instance, [26]. Several numerical techniques have been developed with this purpose: boundary element methods, infinite element methods, Dirichlet–to–Neumann operators based on truncating Fourier expansions, absorbing boundary conditions, etc. The potential advantages of each of them have been widely studied in the literature, see, for instance, [3, 21, 31, 40], and [26] for a classical review on this subject. The last mentioned technique, absorbing boundary conditions (ABCs), can be used to preserve the computational effic
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