Improving the performance of perfectly matched layers by means of hp‐adaptivity

Abstract

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 exponentially 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.

Parametric finite elements, exact sequences and perfectly matched layers

“…We follow the choices made in the derivation of the original formulation, and relax or keep in the strong form the same equations, comp. [19,17]. When integrating over the stretched domain, we need to account for the Jacobian of the stretching,…”

A Comparison Between Pml, Infinite Elements and an Iterative Bem as Mesh Truncation Methods for Hp Self-Adaptive Procedures in Electromagnetics

“…Mapping (3) is introduced into (1) and the stretching yields to complex material constants (see [36] for details). Different mappings can be selected yielding to different PML "profiles".…”

“…However, their contributions to the error are not taken into account in the comparison. Although hp-adaptivity has demonstrated to accurately resolve (in contrast to classical FE discretizations) the electromagnetic field, independently of the PML profile [36,37], there is an optimum profile (in terms of the number of unknowns needed to achieve a given level of accuracy of the field solution), which is problem dependent.…”

“…Indeed, PML may damage the accuracy and increase the conditioning of FEM solutions (e.g., see [35] and references therein). However, for the case of hp-adaptivity, the PML seems a good candidate to truncate the computational domain, as shown in [36,37].…”

On spurious solutions in finite element approximations of resonances in open systems

“…(iii) Use hp-adaptivity, e.g. a technique similar to [41] for PML and [18] for DtN, to reduce the errors in the target eigenvalues. (iv) Check that the new eigenpars significantly reduce the residual in the Lippmann-Schwinger equation.…”