The solution process of problems on unbounded domains usually require a domain truncation and therefore artificial boundary conditions, leading to techniques such as perfectly matched layers (PML) or absorbing boundary conditions (ABC), see [1, 2] for references. To be concrete, taking \(\Omega \subset \mathbb {R}^2\) as an infinite strip (sometimes called a waveguide), then the original problem (or its discretization)

\begin{equation*} \begin{aligned} \mathcal {L}u &= f \quad \mathrm {in} \; \Omega , \\ \mathcal {B}u &= g \quad \mathrm {on} \; \partial \Omega , \end {aligned} \qquad \left (\mathrm {or} \quad L^{\infty } \mathbf {u} = \mathbf {f}^{\infty } \right ) \end{equation*}

is truncated to

\begin{equation*} \begin{aligned} \mathcal {L}v &= f \quad \mathrm {in} \; \Omega ^{\mathrm {trunc}} \\ \mathcal {B}^{\mathrm {trunc}}v &= g \quad \mathrm {on} \; \partial \Omega ^{\mathrm {trunc}}, \end {aligned} \qquad \left (\mathrm {or} \quad L \mathbf {v} = \mathbf {f} \right ) \end{equation*}

where \(\hat { \Omega }\) is the region in which we want to approximately compute \(u\), \(\Omega ^{\mathrm {ABC}}\) is a the bounded region with which we replace the (originally unbounded) \(\Omega \backslash \hat { \Omega }\) and \(\Omega ^{\mathrm {trunc}} = \hat { \Omega } \cup \Omega ^{\mathrm {ABC}}\) is bounded. We have \(\mathcal {B}^{\mathrm {trunc}} = \mathcal {B}\) wherever \(\partial \Omega ^{\mathrm {trunc}}\) coincide with \(\partial \Omega \) and usually introduce a simple boundary condition along the remainder of \(\partial \Omega ^{\mathrm {trunc}}\), e.g., Dirichlet. Naturally, this is also reflected at the discrete level where the infinite matrix \(L^{\infty }\) is replaced by a finite matrix \(L\), which is identical with \(L^{\infty }\) for the unknowns of the interior of \(\Omega ^{\mathrm {trunc}}\) and those where \(\partial \Omega ^{\mathrm {trunc}}\) coincide with \(\partial \Omega \). Domain truncation is also important in domain decomposition methods where a given computational domain is decomposed into many smaller subdomains, and then subdomain solutions are computed independently in parallel. The solutions on the smaller subdomains can naturally be interpreted as solutions on truncated domains, and thus it is of interest to use ABC or PML techniques at the interfaces between the subdomains. The classical Schwarz method uses Dirichlet transmission conditions between subdomains and an overlap to achieve convergence [2]. The overlap coupled with the Dirichlet boundary condition can be thus interpreted as a specific ABC once the unknowns of the overlap are folded onto the interface – an idea that inspired number of iterative solvers, see [1] and the references therein.

An interesting question of a discrete optimized ABC/PML for problems with finite difference grids has been discussed in [3] for \(\mathcal {L}\) being the Laplacian and then extended to the Helmholtz equation in [4] – in both of these, the authors answer the question:   

Here, we are interested in the complementary question:   

We also start with \(\mathcal {L}\) being the Laplacian and, after discertization, start with the known correspondence of the discrete ABC and the Schur complement, see [1, Remark 14 and below]. We use its eigendecomposition, which is closely linked with its Fourier analysis (sometimes also called the frequency domain analysis), and show that in the spectral domain the ABC is naturally represented as the \(i\)-th convergent of a particular continued fraction, namely

\begin{equation*} \mathrm {ABC}(z) \; \sim \; 2+z - \frac {1}{2+z-\frac {1}{2+z-\frac {\ddots }{2+z-\frac {1}{2+z}}}}, \end{equation*}

where the fraction has “\(i\) levels” and \(z\) corresponds to the Fourier frequency. After relating \(i\) to the prolongation of \(\Omega ^{\mathrm {ABC}}\), as posed in our question, we also show that the infinite continued fraction (i.e., without stopping after \(i\) levels) gives a natural representation of the optimal ABC for the infinite problem \(L\mathbf {u} = \mathbf {f}\), hence obtaining the first part of the answer:   

Thanks to the deep results connecting continued fractions and approximation theory, namely Padé approximation (see [6]), we expand on this by concluding   

This suggest that for \(i\) not too large the approximation quality is rather poor around the left endpoint of the spectrum of \(L\), showing us some room for improvement. One such improvement corresponds to considering different boundary conditions where we can, i.e., along what we above called “the remainder of \(\partial \Omega ^{\mathrm {trunc}}\)”, e.g., Robin boundary condition. Using the free parameters well, e.g., the Robin parameter, we can decrease the approximation error. Another, different, to improve on the above ABC is to change the Padé expansion point, hence introducing a new ABC/PML technique. Notice that in such case, we still obtain a different PML to these in [3, 4] as we do not change the discretization. Both of these improvements can be optimized so as to decrease the approximation error \(v|_{\hat {\Omega }} \approx u|_{\hat {\Omega }}\). We demonstrate all of our results also numerically and then comment on possible generalizations. Some of these results have been published in [5].

References

• [1]  M. J. Gander, H. Zhang. A class of iterative solvers for the Helmholtz equation: factorizations, sweeping preconditioners, source transfer, single layer potentials, polarized traces, and optimized Schwarz methods. SIAM Review, volume (61): 3–76, 2019

• [2]  M. M. Rana, V. E. Howle, K. Long, A. Meek, W. Milestone. A New Block Preconditioner for Implicit Runge-Kutta Methods for Parabolic PDE Problems. SIAM Journal on Scientific Computing, volume (43): S475–S495, 2021

• [3]  D. Ingerman, V. Druskin, L. Knizhnerman.Optimal finite difference grids and rational approximations of the square root : I. Elliptic problems. Communications on Pure and Applied Mathematics, volume (53): 1039–1066, 2000

• [4]  V. Druskin, S. Güttel, L. Knizhnerman. Near-optimal perfectly matched layers for indefinite Helmholtz problems. SIAM Review, volume (58): 90–116, 2016

• [6]  L. Lorentzen, H. Waadeland. Continued Fractions with Applications. Elsevier, North Holland, in series Studies in Computational Mathematics 3, 1992

• [5]  M. J. Gander, L. Jakabčin, M. Outrata. Domain truncation, absorbing boundary conditions, Schur Complements, and Padé approximation. Electronic Transactions on Numerical Analysis, volume (59): 319–341, 2024