Parametric finite elements, exact sequences and perfectly matched layers

Pawel J. Matuszyk 0 1 2 Leszek F. Demkowicz 0 1 2 0 L. F. Demkowicz Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin , Austin, TX 78712, USA 1 P. J. Matuszyk Department of Applied Computer Science and Modeling, AGH University of Science and Technology , Krakw, Poland 2 Present Address: P. J. Matuszyk ( 3 More precisely, Piola-like transforms The paper establishes a relation between exact sequences, parametric finite elements, and perfectly matched layer (PML) techniques. We illuminate the analogy between the Piola-like maps used to define parametric H 1-, H(curl)-, H(div)-, and L2-conforming elements, and the corresponding PML complex coordinates stretching for the same energy spaces. We deliver a method for obtaining PML-stretched bilinear forms (constituting the new weak formulation for the original problem with PML absorbing boundary layers) directly from their classical counterparts. The purpose of this work is to illuminate the relation between the use of Piola transforms1 in the definition of pull-back maps defining parametric finite elements (FE), and the analogous use of the same concept in the construction of the cor- - responding PML via complex stretching. If nothing else, this helps in coding the PML for multiphysics problems involving a simultaneous discretization using H 1-, H(curl)-, H(div)-, and L2-conforming fields. The perfectly matched layer (PML) was introduced by Brenger [1] as a technique for reflectionless absorbing of electromagnetic (EM) waves at the boundary of the domain of interest. Absorption of the waves inside of the PML was achieved by matching material conductivities, and by splitting the EM field components into subcomponents. In [2], Chew and Weedon proposed an alternative formulation of the PML through a complex coordinate stretching of the spatial variables of the original differential equations posed in the frequency domain. The approach was later reinterpreted in terms of an analytic continuation [3,4], and motivated the extension of the PML to curvilinear coordinates, conformal mesh terminations, and more general media. Simultaneously, the new technique was successfully applied to acoustics [5], elastic wave propagation [6], and poroelasticity [7]. Sacks et al. [8] developed a non-split version of the PML for Maxwells equations, preserving the original form of the equations (the so called Maxwellian PML) at the expense of introducing anisotropic material properties. In consequence, the resulting field in the PML can be interpreted as a physical field in the aforementioned anisotropic medium. This is the manifestation of the metric invariance of Maxwells equations. For more general hyperbolic equations (e.g. elastodynamics), due to the presence or absence of specific symmetries in the physical fields in the PML introduced medium, the interpretation of the PML in terms of an anisotropic material seems to be impossible. A rigorous proof of stability for such systems remains open [9]. In the series of papers [1012], Teixeira and Chew showed that an appropriate change of field variables (analogous to Piola transforms for H(curl)- and H(div)-conforming fields) combined with complex coordinate stretching leads to Maxwellian PML for general curvilinear coordinates and anisotropic linear media. A similar approach was used for elastic wave propagation in [13,14], where the H(div)-conforming Piola transform was applied to the stress tensor. Finally, in [15], Teixeira and Chew proposed a generalized geometric viewpoint to the PML obtained via complex coordinate stretching in the Fourier domain. It appears that the analytic continuation is equivalent to a complexification of the space metric tensor. Since the spatial differential operators depend on the metric, the PML amounts to a modification of these operators due to the new complex metric. This enables the construction of more elaborate versions of PML, provided one can determine the transformation matrix relating original and complexified metric tensors, which in turn is used to define stretched differential operators or field variables. In this paper, we present a unified framework for a systematic derivation of complex stretchings for all fields and exterior differentiation operators present in the classical grad-curl-div exact sequence. We expose the algebraic structure of the construction for arbitrary curvilinear systems of coordinates, and establish practical rules for transformation of bilinear forms corresponding to classical weak formulations. As a result of the construction, the PML-stretched operators and spaces preserve the exact sequence property. The content of the paper is as follows. We begin in Sect. 2 with a review of the notion of the exact sequence and its connection with a proper definition of FE spaces for parametric elements through the use of Piola-like transforms. Section 3 describes the PML technique, and reinterprets +it in terms of the new Piola-like complex transforms. In Sect. 4, we illustrate the general technique with two specific examples of calculations of Jacobian matrices describing the PML transformations. Finally, in Sect. 5, we show how to derive the bilinear forms corresponding to various PMLs by a direct stretching of the classical bilinear forms for several standard variational formulations involving the exact sequence energy spaces. We conclude with a summary in Sect. 6. 2 Exact sequence and parametric elements The following energy spaces, used henceforth, are implied by physics: H(curl, ) := {E L2() : E L2()} , H(div, ) := {V L2() : V L2()} , Along with operators of grad, curl, and div, for a simply connected domain , the spaces form the well-known exact sequence shown below. The sequence can be reproduced at the level of a single element or a Finite Element (FE) mesh using various versions of Lagrange, Ndlec, Raviart-Thomas, and discontinuous elements, possibly with a variable order (the hp elements), see [16,17]. Figure 1 presents the lowest order elements on a tetrahedral element. The corresponding degrees-of-freedom for H(curl) elements are defined using tangential components along edges, and for H(div) spaces, using normal components over faces; hence, the frequently-used names of edge and face elements. R id H(curl) H(div) Qhp The de Rham complex (1) combines the continuous and discrete versions of the sequence with specially constructed Projection-Based Interpolation (PBI) operators that make the diagram commute [18]. Notation. We use superscripts for contravariant and subscripts for covariant components of vectors and tensors. The Latin alphabet is used for coordinates and indices in Cartesian coordinate systems, and the Greek alphabet for their curvilinear counterparts. 2.1 Cartesian coordinates We reproduce the reasoning from [17, p. 34]. Given a bijective smooth map Ty : y x (alternatively xi = xi (y j ), i, j = 1, 2, 3), transforming a master element (...truncated)