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Improving the Performance of Perfectly Matched Layers by Means of Hp-Adaptivity C Improving the Performance of Perfectly Matched Layers by Means of hp-Adaptivity C. Michler, L. Demkowicz, J. Kurtz, D. Pardo Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin, Austin, Texas 78712 Received 10 February 2006; accepted 31 January 2007 Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.20252 We improve the performance of the Perfectly Matched Layer (PML) by using an automatic hp-adaptive discretization. By means of hp-adaptivity, we obtain a sequence of discrete solutions that converges expo- nentially to the continuum solution. Asymptotically, we thus recover the property of the PML of having a zero reflection coefficient for all angles of incidence and all frequencies on the continuum level. This allows us to minimize the reflections from the discrete PML to an arbitrary level of accuracy while retaining optimal computational efficiency. Since our hp-adaptive scheme is automatic, no interaction with the user is required. This renders tedious parameter tuning of the PML obsolete. We demonstrate the improvement of the PML performance by hp-adaptivity through numerical results for acoustic, elastodynamic, and electromagnetic wave-propagation problems in the frequency domain and in different systems of coordinates. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 832–858, 2007 Keywords: Perfectly Matched Layer; hp-adaptivity; exterior boundary-value problem; acoustic scattering; elastodynamic wave propagation; electromagnetic scattering I. INTRODUCTION Most wave-propagation problems arising in acoustics, elastodynamics, and electromagnetics are posed on a spatially unbounded domain. Since domain-based discretization methods such as finite elements can handle only bounded domains, a truncation of the unbounded physical domain to a bounded computational domain is necessary. Any artificial boundary condition that is applied on the truncated domain boundary must not essentially alter the original problem, i.e., it must allow for outgoing waves only and it has to be essentially reflectionless. Many techniques have been developed for this purpose such as Infinite Elements (see e.g. Refs. [1,2]), the Dirichlet-to- Neumann map (see e.g. Ref. [3]), exact nonreflecting boundary conditions (see e.g. Ref. [4]), the continued-fraction absorbing boundary condition [5], and the Perfectly Matched Layer [6]. We refer to Refs. [7–9] for an overview over these so-called absorbing boundary conditions, all of which have their specific strengths and weaknesses. Correspondence to: C. Michler, Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin, Austin, TX 78712 (email: [email protected]) © 2007 Wiley Periodicals, Inc. IMPROVING THE PERFORMANCE OF PERFECTLY MATCHED LAYERS 833 Among the various absorbing boundary conditions, the Perfectly Matched Layer (PML) in particular has become very popular due to its performance, conceptual simplicity, and ease of implementation. The PML is based on the concept of analytic continuation of a real function into the complex plane (see e.g. Ref. [10]), typically also referred to as complex-coordinate stretch- ing [11]. This concept enables straightforward derivation of the PML equations for different types of wave propagation and different systems of coordinates; see e.g. Refs. [12–15]. The PML concept has been extensively studied and analyzed regarding the so-called split and unsplit formu- lations and well-posedness (see e.g. Refs. [16,17]), causality [18,19], long-time behavior [16,20], termination of the PML [21, 22], and finite-element dispersion and convergence analysis in the frequency domain [23, 24]. A recommendable review of the PML in the context of Maxwell’s equations is presented in Ref. [14]. The PML has the remarkable property of having a zero reflection coefficient for all angles of incidence and all frequencies on the continuum level. However, under discretization, this property is compromised and spurious reflections typically do occur at the discrete PML. Much effort has thus been spent on optimizing the PML parameters to minimize the error resulting from discretiza- tion and thus obtain optimal performance of the PML; see e.g. Refs. [21,25,26]. In this work, we advocate the opposite approach, i.e. we optimize the discretization for any given PML parameters, rather than resorting to tedious parameter tuning. A highly effective methodology to construct an optimal discretization is the hp-adaptive finite-element method that allows for locally adapting the element size h and polynomial approximation order p to the local resolution requirements of the solution. Such adaptivity can be accomplished in an automatic way, i.e. no interaction with the user is required. This is achieved by a so-called two-grid strategy [27] that guides the adaptive refinements. By solving the problem on a sequence of coarse and corresponding fine grids, an approximation of the error function is constructed at each iteration, which serves as an error estimate and based on which it is determined where to refine and how to refine, i.e. in h or in p. This yields optimal accuracy for the least computational expense. For details on the automatic hp-adaptive procedure, we refer to Ref. [27] and to the recent book by Demkowicz [28]. The immense benefit that hp-adaptivity can offer for increasing the performance of the PML is due to the property of the PML of having a zero reflection coefficient on the continuum level. Although this property is lost under discretization, it can be recovered asymptotically, i.e. upon convergence to the continuum solution. This implies that upon convergence of the hp-adaptive discretization, the discretization error and the resulting reflections can be minimized to an arbitrary level of accuracy. The PML is designed such that traveling waves that leave the actual domain of interest and enter the encompassing PML are converted into evanescent waves that are made to decay suffi- ciently fast, i.e. they vanish before they can get reflected at the domain boundary. The rapid decay that is enforced on the solution in the PML implies strong solution gradients which need to be resolved to prevent reflections and an accompanying loss in accuracy. Although the significance of resolving the PML has long been recognized, conventional discretization methods generally face difficulties in resolving strong solution gradients and, therefore, typically revert to adjusting the PML parameters and the associated damping profile to a given discretization. Conversely, our adaptive strategy automatically adapts the discretization to resolve any damping profile that is imposed on the solution. This allows us to choose the geometry of the PML independent of the geometry of the initial grid, i.e. the respective geometries need not coincide (a cylindrical PML and a Cartesian grid, for instance)—a truly remarkable feature. In this paper, we show that by means of hp-adaptivity PML-induced solution gradients can be effectively resolved and the performance of the PML can be greatly improved. This renders tedious parameter tuning of the PML obsolete. We demonstrate the performance and the versatility of our approach by Numerical Methods for Partial Differential Equations DOI 10.1002/num 834 MICHLER ET AL. numerical results for acoustic, elastodynamic, and electromagnetic wave-propagation problems in the frequency domain and in different systems of coordinates. The contents of this paper are organized as follows: Sections II, III, and IV present the PML formulation of the Helmholtz, linear elasticity, and Maxwell’s equations, respectively, and discuss the particularities of the respective PML formulation. Each of these sections presents numerical results that demonstrate the potential of hp-adaptivity for improving the performance of the PML. Finally, Section V contains concluding remarks. II. HELMHOLTZ EQUATION In this section, we study the formulation of the PML for an exterior boundary-value problem governed by the Helmholtz equation. In Section A, we provide a problem statement in coordinate- invariant form. In Section B, we derive the PML formulation in Cartesian coordinates. Based on the formalism introduced in Section B, in Sections C and D, we state the PML formulation in 2D polar and 3D spherical coordinates, respectively. In Section E, we present numerical results for exterior Helmholtz problems that demonstrate the improvement in performance of the PML that can be obtained by hp-adaptive finite elements. A. The Helmholtz Equation in Coordinate-Invariant Form We consider the unbounded exterior domain = Rn − int with int ⊂ Rn, n = 1, 2, 3, a given bounded interior domain and n the number of space dimensions. We seek a solution for the pressure p(x), x ∈ ¯ that satisfies the Helmholtz equation in , −p − k2p = 0in, (2.1) where denotes the Laplacian operator, and k := ω/c is the wavenumber with ω and c the angular frequency and the sound speed, respectively. Throughout this work, we shall assume a time-harmonic dependence of the form eiωt, where i is the imaginary unit and t denotes time and, moreover, we consider the governing equations in their non-dimensional form. Equation (2.1) is supplemented with a Dirichlet boundary condition on the interior-domain boundary int p = pD on = ∂ , (2.2) and the Sommerfeld radiation condition at infinity ∂p + ikp ∈ L2(), (2.3) ∂r where r denotes the radius. Note that this particular form of the Sommerfeld radiation condition has the advantage of being
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