Abstract
In d dimensions, accurately approximating an arbitrary function oscillating with frequency \(\lesssim k\) requires \(\sim k^{d}\) degrees of freedom. A numerical method for solving the Helmholtz equation (with wavenumber k) suffers from the pollution effect if, as \(k\rightarrow \infty \), the total number of degrees of freedom needed to maintain accuracy grows faster than this natural threshold. While the h-version of the finite element method (FEM) (where accuracy is increased by decreasing the meshwidth h and kee** the polynomial degree p fixed) suffers from the pollution effect, the hp-FEM (where accuracy is increased by decreasing the meshwidth h and increasing the polynomial degree p) does not suffer from the pollution effect. The heart of the proof of this result is a PDE result splitting the solution of the Helmholtz equation into “high” and “low” frequency components. This result for the constant-coefficient Helmholtz equation in full space (i.e. in \(\mathbb {R}^{d}\)) was originally proved in Melenk and Sauter (Math. Comp 79(272), 1871–1914, 2010). In this paper, we prove this result using only integration by parts and elementary properties of the Fourier transform. The proof in this paper is motivated by the recent proof in Lafontaine et al. (Comp. Math. Appl. 113, 59–69, 2022) of this splitting for the variable-coefficient Helmholtz equation in full space use the more-sophisticated tools of semiclassical pseudodifferential operators.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aziz, A.K., Kellogg, R.B., Stephens, A.B.: A two point boundary value problem with a rapidly oscillating solution. Numer. Math. 53(1), 107–121 (1988)
Babuška, I. M., Sauter, S.A.: Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? SIAM Rev., 451–484 (2000)
Barucq, H., Chaumont-Frelet, T., Gout, C.: Stability analysis of heterogeneous Helmholtz problems and finite element solution based on propagation media approximation. Math. Comp. 86(307), 2129–2157 (2017). https://doi.org/10.1090/mcom/3165
Bernkopf, M., Chaumont-Frelet, T., Melenk, J.M.: Wavenumber-explicit stability and convergence analysis of hp finite element discretizations of Helmholtz problems in piecewise smooth media. ar**v:2209.03601 (2022)
Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Texts in Applied Mathematics, vol. 15. Springer (2008)
Brown, D.L., Gallistl, D., Peterseim, D.: Multiscale Petrov-Galerkin method for high-frequency heterogeneous Helmholtz equations. In: Meshfree Methods for Partial Differential Equations VIII, pp 85–115. Springer (2017)
Cao, H., Wu, H.: IPCDGM and multiscale IPDPGM for the Helmholtz problem with large wave number. J. Comput. Appl. Math. 369, 112590 (2020)
Chandler-Wilde, S.N., Monk, P.: Wave-number-explicit bounds in time-harmonic scattering. SIAM J. Math. Anal. 39(5), 1428–1455 (2008)
Chaumont-Frelet, T.: On high order methods for the heterogeneous Helmholtz equation. Comput. Math. Appl. 72(9), 2203–2225 (2016)
Chaumont-Frelet, T., Valentin, F.: A multiscale hybrid-mixed method for the Helmholtz equation in heterogeneous domains. SIAM J. Num. Anal. 58 (2), 1029–1067 (2020)
Chaumont-Frelet, T., Nicaise, S.: Wavenumber explicit convergence analysis for finite element discretizations of general wave propagation problem. IMA J. Numer. Anal. 40(2), 1503–1543 (2020)
Ciarlet, P.G.: Basic error estimates for elliptic problems. In: Handbook of Numerical Analysis, Vol. II. Handb. Numer. Anal., II, pp 17–351, North-Holland (1991)
Colton, D.L., Kress, R.: Integral Equation Methods in Scattering Theory, p 271. Wiley, New York (1983)
Du, Y., Wu, H.: Preasymptotic error analysis of higher order FEM and CIP-FEM for Helmholtz equation with high wave number. SIAM J. Numer. Anal. 53(2), 782–804 (2015)
Dyatlov, S., Zworski, M.: Mathematical Theory of Scattering Resonances. AMS (2019)
Esterhazy, S., Melenk, J.M.: On stability of discretizations of the Helmholtz equation. In: Graham, I.G., Hou, T.Y., Lakkis, O., Scheichl, R. (eds.) Numerical Analysis of Multiscale Problems. Lecture Notes in Computational Science and Engineering, vol. 83, pp 285–324. Springer (2012)
Feng, X., Wu, H.: Discontinuous Galerkin methods for the Helmholtz equation with large wave number. SIAM J. Numer. Anal. 47(4), 2872–2896 (2009)
Feng, X., Wu, H.: hp-Discontinuous Galerkin methods for the Helmholtz equation with large wave number. Math. Comput. 80(276), 1997–2024 (2011)
Gallistl, D., Chaumont-Frelet, T., Nicaise, S., Tomezyk, J.: Wavenumber explicit convergence analysis for finite element discretizations of time-harmonic wave propagation problems with perfectly matched layers. Comm. Math. Sci. 20(1), 1–52 (2022)
Gallistl, D., Peterseim, D.: Stable multiscale Petrov–Galerkin finite element method for high frequency acoustic scattering. Comput. Methods Appl. Mech. Eng. 295, 1–17 (2015)
Galkowski, J., Lafontaine, D., Spence, E.A., Wunsch, J.: Decompositions of high-frequency Helmholtz solutions via functional calculus, and application to the finite element method. SIAM J. Math. Anal. to appear. ar** resolvent estimates. Pure Appl. Anal. 2(1), 157–202 (2020)
Galkowski, J., Spence, E.A.: Does the Helmholtz boundary element method suffer from the pollution effect? SIAM Review to appear (2023)
Graham, I.G., Sauter, S.: Stability and finite element error analysis for the Helmholtz equation with variable coefficients. Math. Comp. 89(321), 105–138 (2020)
Graham, I.G., Pembery, O.R., Spence, E.A.: The Helmholtz equation in heterogeneous media: a priori bounds, well-posedness, and resonances. J. Differ. Equ. 266(6), 2869–2923 (2019)
Ihlenburg, F., Babuška, I.: Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation. Int. J. Numer. Meth. Eng. 38(22), 3745–3774 (1995)
Ihlenburg, F., Babuska, I.: Finite element solution of the Helmholtz equation with high wave number part II: the hp version of the FEM. SIAM J. Numer. Anal. 34(1), 315–358 (1997)
Ihlenburg, F., Babuška, I.: Finite element solution of the Helmholtz equation with high wave number Part I: The h-version of the FEM. Comp. Math. Appl. 30(9), 9–37 (1995)
Lafontaine, D., Spence, E.A., Wunsch, J.: Wavenumber-explicit convergence of the hp-FEM for the full-space heterogeneous Helmholtz equation with smooth coefficients. Comp. Math. Appl. 113, 59–69 (2022)
Li, Y., Wu, H.: FEM and CIP-FEM for Helmholtz equation with high wave number and perfectly matched layer truncation. SIAM J. Numer. Anal. 57(1), 96–126 (2019)
Melenk, J.M., Sauter, S.: Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions. Math. Comp 79(272), 1871–1914 (2010)
Melenk, J.M.: On generalized finite element methods. PhD thesis, The University of Maryland (1995)
Melenk, J.M., Sauter, S.: Wavenumber explicit convergence analysis for Galerkin discretizations of the Helmholtz equation. SIAM J. Numer. Anal. 49, 1210–1243 (2011)
Melenk, J.M., Parsania, A., Sauter, S.: General DG-methods for highly indefinite Helmholtz problems. J. Sci. Comput. 57(3), 536–581 (2013)
Ma, C., Alber, C., Scheichl, R.: Wavenumber explicit convergence of a multiscale GFEM for heterogeneous Helmholtz problems. ar**v:2112.10544 (2021)
Melenk, J.M., Sauter, S.A.: Wavenumber-explicit hp-FEM analysis for Maxwell’s equations with transparent boundary conditions. Found. Comp. Math. 21(1), 125–241 (2021)
Melenk, J.M., Sauter, S.A.: Wavenumber-explicit hp-FEM analysis for Maxwell’s equations with impedance boundary conditions. ar**v:2201.02602(2022)
McLean, W.C.H.: Strongly Elliptic Systems and Boundary Integral Equations. CUP (2000)
Morawetz, C.S., Ludwig, D.: An inequality for the reduced wave operator and the justification of geometrical optics. Comm. Pure Appl. Math. 21, 187–203 (1968)
Morawetz, C.S.: Decay for solutions of the exterior problem for the wave equation. Commun. Pure Appl. Math. 28(2), 229–264 (1975)
Nicaise, S., Tomezyk, J.: Convergence analysis of a hp-finite element approximation of the time-harmonic Maxwell equations with impedance boundary conditions in domains with an analytic boundary. Numer. Methods Partial Differ. Equ. 36(6), 1868–1903 (2020)
Ohlberger, M., Verfurth, B.: A new heterogeneous multiscale method for the Helmholtz equation with high contrast. Multiscale Model. Simul. 16(1), 385–411 (2018)
Peterseim, D.: Eliminating the pollution effect in Helmholtz problems by local subscale correction. Math. Comput. 86(305), 1005–1036 (2017)
Sayas, F. -J., Brown, T.S., Hassell, M.E.: Variational Techniques for Elliptic Partial Differential Equations: Theoretical Tools and Advanced Applications. CRC Press, Boca Raton (2019)
Sauter, S.A.: A refined finite element convergence theory for highly indefinite Helmholtz problems. Computing 78(2), 101–115 (2006)
Schatz, A.H.: An observation concerning Ritz-Galerkin methods with indefinite bilinear forms. Math. Comput. 28(128), 959–962 (1974)
Shannon, C.E.: Communication in the presence of noise. Proc. IRE 37(1), 10–21 (1949)
Spence, E.A., Kamotski, I.V., Smyshlyaev, V.P.: Coercivity of combined boundary integral equations in high-frequency scattering. Comm. Pure Appl. Math 68(9), 1587–1639 (2015)
Wu, H.: Pre-asymptotic error analysis of CIP-FEM and FEM for the Helmholtz equation with high wave number. Part I: linear version. IMA J. Numer. Anal. 34(3), 1266–1288 (2014)
Whittaker, E.T.: On the functions which are represented by the expansions of the interpolation-theory. Proc. R. Soc. Edinb. 35, 181–194 (1915)
Zhu, B., Wu, H.: Preasymptotic error analysis of the HDG method for Helmholtz equation with large wave number. J. Sci. Comput. 87(2), 1–34 (2021)
Zhu, L., Wu, H.: Preasymptotic error analysis of CIP-FEM and FEM for Helmholtz equation with high wave number. Part II: hp version. SIAM J. Numer. Anal. 51(3), 1828–1852 (2013)
Acknowledgements
The idea for this paper came out of the “Nachdiplom” lecture course I taught at ETH Zürich in Fall 2021; it is a pleasure to thank the Institute for Mathematical Research (FIM) and the Seminar for Applied Mathematics at ETH for their hospitality during that time. In particular, the proof of Theorem 5.1 presented in this paper was the result of Ralf Hiptmair asking me about the minimum technicalities needed for the proof in [32] of the analogous splitting in the variable-coefficient case. I also thank Martin Averseng for many useful discussions in his role as the teaching assistant for this course. Finally, I thank Alastair Spence (University of Bath) and Jared Wunsch (Northwestern University) for their insightful comments on a draft of this paper.
Funding
I acknowledge support from EPSRC grant EP/R005591/1.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author declares no competing interests.
Additional information
Communicated by: Jon Wilkening
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix: A. Proof of Theorem 2.2
Appendix: A. Proof of Theorem 2.2
Lemma A.1
(Morawetz identity for the Helmholtz operator [43, Section I.2]) If
with β and α real-valued C1 functions, then
Proof 1
This follows in a straightforward (but slightly involved) way by expanding the divergence on the right-hand side; for this done step-by-step, see, e.g. [51, Proof of Lemma 2.1]. □
The idea of the proof of Theorem 2.2 is to integrate the identity (A.2) over \(\mathbb {R}^{d}\) with v = u, α = (d − 1)/2, and β defined piecewise as β = R for r ≤ R and β = r for r ≥ R. The choice β = constant and α = (d − 1)/2 means that the non-divergence terms on the right-hand side of (A.2) become −|u|2 − k− 2|∇u|2; this is where we get \(\|u\|_{{H^{1}_{k}}(B_{R})}^{2}\) from. The choice β = r deals with the contribution from infinity (although this is not immediately clear from (A.2)). We therefore first look at the special case of (A.2) with β = r.
Lemma A.2
(Special case of (A.2) [42, Equation 1.2]) With \({\mathscr{L}} v\) and \({\mathscr{M}}_{\beta ,\alpha } v\) as in Lemma A.1, show that if \(\alpha \in \mathbb {R}\), then, with vr = x ⋅∇v/r,
Proof 2
This follows from (A.2) by choosing β = r and writing the term involving ∇β as
using the identity \(-2\Re {(z_{1}\overline {z_{2}})} = \vert z_{1}\vert ^{2} + \vert z_{2}\vert ^{2}-\vert z_{1} + z_{2}\vert ^{2} \). □
To integrate (A.3) over \(\mathbb {R}^{d}\setminus B_{R}\), we integrate it over \(B_{R_{1}}\setminus B_{R}\) and then send \(R_{1}\to \infty \). In preparation for this, we look at the boundary term on \(\partial B_{R_{1}}\).
Lemma A.3
Let
If u is an outgoing solution of \({\mathscr{L}} u=0\) in \(\mathbb {R}^{d}\setminus \overline {B_{R_{0}}}\), then, for all \(\alpha \in \mathbb {R}\),
The proof of Lemma A.3 requires the following classic result.
Theorem A.4
(Atkinson–Wilcox expansion (see, e.g. [13, Theorem 3.7])) If \(u\in H^{1}_{\text {loc}}(\mathbb {R}^{d}\setminus \overline {B_{R_{0}}})\) is an outgoing solution of k− 2Δu + u = 0 in \(\mathbb {R}^{d}\setminus \overline {B_{R_{0}}}\) for some R0 > 0, then there exist smooth functions Fn such that, for any R1 > R0,
where the sum in (A.6) (and all its derivatives) converges absolutely and uniformly.
Proof 3 (Proof of Lemma A.3)
By the definitions of Qr,α(v) (A.4) and \({\mathscr{M}}_{r, \alpha }v\) (A.1),
where we have again used the identity \(2\Re {(z_{1}\overline {z_{2}})} = \vert z_{1} + z_{2}\vert ^{2} - \vert z_{1}\vert ^{2} - \vert z_{2}\vert ^{2}\). We now claim that the term in large brackets in (A.7) is O(r−d− 1); if this is true, then
and thus (A.5) follows. By the Atkinson–Wilcox expansion (A.6), |u|2 = O(r1−d) and \(r^{-2}\vert {\mathscr{M}}_{r,\alpha }u\vert ^{2}= O(r^{-d-1})\). To prove the result, therefore, we only need to show that |∇u|2 −|ur|2 = O(r−d− 1). The quantity |∇u|2 −|ur|2 equals |∇Su|2 where ∇S is the surface gradient on |x| = r, which satisfies \(\nabla _{S} u = \nabla u - \widehat {x}u_{r}\). This differential operator is equal to 1/r multiplied by an operator acting only on \(\widehat {x}\), i.e. the angular variables; thus, |∇Su|2 is O(r−d− 1) and the proof is complete. □
We now integrate (A.3) over \(B_{R_{1}}\setminus B_{R}\), send \(R_{1}\to \infty \), and obtain an inequality involving the boundary term on ∂BR.
Lemma A.5
If u is an outgoing solution of \({\mathscr{L}} u=0\) in \(\mathbb {R}^{d}\setminus \overline {B_{R_{0}}}\), for some R0 > 0, then, for R > R0,
Proof 4
We integrate the identity (A.3) over \(B_{R_{1}}\setminus B_{R}\), where R1 > R, with v = u and then use the divergence theorem \({\int \limits }_{D} \nabla \cdot F = {\int \limits }_{\partial D} F\). The divergence theorem is valid for \(F\in C^{\infty }(\overline {D})\) and D Lipschitz (see, e.g. [41, Theorem 3.34]); we can use it here since, by elliptic regularity, \(u\in C^{\infty }(\overline {B_{R_{1}}\setminus B_{R}})\) (see, e.g. [41, Theorem 4.16]). This results in
Setting α = (d − 1)/2 eliminates the first term on the right-hand side of (A.9). Since |vr|≤|∇v|, the remaining terms on the right-hand side of (A.9) are non-negative, and thus
Sending \(R_{1} \to \infty \) and using (A.5), we obtain the result (A.8). □
Proof 5 (Proof of Theorem 2.2)
The plan is to integrate the identity (A.2) over BR with v = u, β = R, and α = (d − 1)/2, and then use the divergence theorem. We justify using the divergence theorem just as we did at the beginning of the proof of Lemma A.3 to find that if \(v\in C^{\infty }(\overline {D})\) then
We now claim that (A.10) holds for v ∈ H2(BR); this follows since \(C^{\infty }(\overline {D})\) is dense in H2(D) [41, Page 77], and, by the trace theorem (see, e.g. [41, Theorem 3.37]), (A.10) is continuous in v with respect to the topology of H2(BR). By Corollary 2.3, u ∈ H2(BR), and thus (A.10) holds with v = u. Using in (A.10) the definition of Q (A.4), the fact that u is an outgoing solution of the Helmholtz equation (2.1), and Lemma A.5, we find that
Thus
By the inequality |a + b|2 ≤ 2|a|2 + 2|b|2, the fact that |ikr + α|2 = k2r2 + α2, and the bound r ≤ R on BR,
Using this in (A.11), and recalling that α = (d − 1)/2, we find that \(\left \|{u}\right \|_{{H^{1}_{k}}(B_{R})} \leq C \left \|{f}\right \|_{L^{2}(B_{R})}\) with C the right-hand side of (2.7). □
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Spence, E.A. A simple proof that the hp-FEM does not suffer from the pollution effect for the constant-coefficient full-space Helmholtz equation. Adv Comput Math 49, 27 (2023). https://doi.org/10.1007/s10444-023-10025-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10444-023-10025-3