An analysis of Feng’s and other symmetric local absorbing boundary conditions
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 1, pp. 257-273.

With symmetric local absorbing boundary conditions for the Helmholtz equation scattering problems can be solved on a truncated domain, where the outgoing radiation condition is approximated by a Dirichlet-to-Neumann map with higher tangential derivatives on its outer boundary. Feng’s conditions are symmetric local absorbing boundary conditions, which are based on an asymptotic expansion of the coefficients of the exact Dirichlet-to-Neumann map for large radia of the circular outer boundary. In this article we analyse the well-posedness of variational formulations with symmetric local absorbing boundary conditions in general and show how the modelling error introduced by Feng’s conditions depends on the radius of the truncated domain.

Reçu le :
DOI : 10.1051/m2an/2014029
Classification : 65N30, 35J25, 78M30
Mots-clés : Absorbing boundary conditions, Feng’s conditions, error analysis
Schmidt, Kersten 1 ; Heier, Christian 1

1 Research Center MATHEON, TU Berlin, 10623 Berlin, Germany.
@article{M2AN_2015__49_1_257_0,
     author = {Schmidt, Kersten and Heier, Christian},
     title = {An analysis of {Feng{\textquoteright}s} and other symmetric local absorbing boundary conditions},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {257--273},
     publisher = {EDP-Sciences},
     volume = {49},
     number = {1},
     year = {2015},
     doi = {10.1051/m2an/2014029},
     zbl = {1457.65214},
     mrnumber = {3342200},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2014029/}
}
TY  - JOUR
AU  - Schmidt, Kersten
AU  - Heier, Christian
TI  - An analysis of Feng’s and other symmetric local absorbing boundary conditions
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2015
SP  - 257
EP  - 273
VL  - 49
IS  - 1
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2014029/
DO  - 10.1051/m2an/2014029
LA  - en
ID  - M2AN_2015__49_1_257_0
ER  - 
%0 Journal Article
%A Schmidt, Kersten
%A Heier, Christian
%T An analysis of Feng’s and other symmetric local absorbing boundary conditions
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2015
%P 257-273
%V 49
%N 1
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an/2014029/
%R 10.1051/m2an/2014029
%G en
%F M2AN_2015__49_1_257_0
Schmidt, Kersten; Heier, Christian. An analysis of Feng’s and other symmetric local absorbing boundary conditions. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 1, pp. 257-273. doi : 10.1051/m2an/2014029. http://archive.numdam.org/articles/10.1051/m2an/2014029/

M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. ninth dover printing, tenth gpo printing ed. Dover, New York (1964). | MR | Zbl

R.A. Adams,Sobolev spaces. Academic Press, New York, London (1975). | MR

W. Arendt, G. Metafune, D. Pallara and S. Romanell, The Laplacian with Wentzell-Robin boundary conditions on spaces of continuous functions. Semigroup Forum 67 (2003) 247–261. | DOI | MR | Zbl

W. Bao and H. Han, High-order local artificial boundary conditions for problems in unbounded domains. Comput. Methods Appl. Mech. Engrg. 188 (2000) 455–471. | DOI | MR | Zbl

A. Bayliss, M. Gunzburger and E. Turkel, Boundary conditions for the numerical solution of elliptic equations in exterior regions. SIAM J. Appl. Math. 42 (1982) 430–451. | DOI | MR | Zbl

V. Bonnaillie-Nol, M. Dambrine, F. Hrau and G. Vial, On generalized Ventcel’s type boundary conditions for Laplace operator in a bounded domain. SIAM J. Math. Anal. 42 (2010) 931–945. | DOI | MR | Zbl

D. Braess, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, 3th eds. Cambridge University Press (2007). | Zbl

Concepts Development Team. Webpage of Numerical C++ Library Concepts 2. http://www.concepts.math.ethz.ch (2014).

K.-J. Engel, The Laplacian on C(ω ¯) with generalized Wentzell boundary conditions. Arch. Math. (Basel) 81 (2003) 548–558. | DOI | MR | Zbl

B. Engquist and A. Majda, Absorbing boundary conditions for numerical simulation of waves. Proc. Nat. Acad. Sci. USA 74 (1977) 1765. | DOI | MR | Zbl

W. Feller, The parabolic differential equations and the associated semi-groups of transformations. Ann. Math. 55 (1952) 468–519. | DOI | MR | Zbl

W. Feller, Generalized second order differential operators and their lateral conditions. Illinois J. Math. 1 (1957) 459–504. | DOI | MR | Zbl

K. Feng, Finite element method and natural boundary reduction. In Proc. of the International Congress of Mathematicians (1983) 1439–1453. | MR | Zbl

P. Frauenfelder and C. Lage, Concepts – An Object-Oriented Software Package for Partial Differential Equations. Math. Model. Numer. Anal. 36 (2002) 937–951. | DOI | Numdam | MR | Zbl

C. Geuzaine and J.-F. Remacle, Gmsh: A 3-d finite element mesh generator with built-in pre-and post-processing facilities. Int. J. Numer. Methods Engrg. 79 (2009) 1309–1331. | DOI | MR | Zbl

D. Givoli, Non-reflecting boundary conditions. J. Comput. Phys. 94 (1991) 1–29. | DOI | MR | Zbl

D. Givoli, Numerical methods for problems in infinite domains. Elsevier, Amsterdam New York (1992). | MR | Zbl

D. Givoli and J.B. Keller, Special finite elements for use with high-order boundary conditions. Comput. Methods Appl. Mech. Engrg. 119 (1994) 199–213. | DOI | MR | Zbl

D. Givoli, I. Patlashenko and J.B. Keller, High-order boundary conditions and finite elements for infinite domains. Comput. Methods Appl. Mech. Engrg. 143 (1997) 13–39. | DOI | MR | Zbl

H. Han and X. Wu, A survey on artificial boundary method. Sci. China Math. 56 (2013) 2439–2488. | DOI | MR | Zbl

F. Ihlenburg, Finite Element Analysis of Acoustic Scattering. Springer-Verlag (1998). | MR | Zbl

A.L. Koh, A.I. Fernández-Domínguez, D.W. Mccomb, S.A. Maier and J.K. Yang, High-resolution mapping of electron-beam-excited plasmon modes in lithographically defined gold nanostructures. Nano letters 11 (2011) 1323–1330. | DOI

R. Leis, Initial Boundary Value Problems in Mathematical Physics. B. G. Teubner Gmbh (1986). | MR | Zbl

W. McLean, Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press (2000). | MR | Zbl

J.-F. Remacle, J. Lambrechts, B. Seny, E. Marchandise, A. Johnen and C. Geuzainet, Blossom-quad: A non-uniform quadrilateral mesh generator using a minimum-cost perfect-matching algorithm. Int. J. Numer. Methods Engrg. 89 (2012) 1102–1119. | DOI | MR | Zbl

S. Sauter and C. Schwab, Boundary element methods. Springer-Verlag, Heidelberg (2011). | MR | Zbl

K. Schmidt, J. Diaz and C. Heier, Non-conforming Galerkin finite element methods for symmetric local absorbing boundary conditions. In preparation.

K. Schmidt and P. Kauf, Computation of the band structure of two-dimensional photonic crystals with hp finite elements. Comput. Methods Appl. Mech. Engr. 198 (2009) 1249–1259. | DOI | MR | Zbl

A. Venttsel’, On boundary conditions for multidimensional diffusion processes. Theory Probab. Appl. 4 (1959) 164–177. | DOI | MR | Zbl

A.D. Venttsel’, Semigroups of operators that correspond to a generalized differential operator of second order. Dokl. Akad. Nauk SSSR (N.S.) 111 (1956) 269–272. | MR | Zbl

M. Wang, C. Engström, K. Schmidt and C. Hafner, On high-order FEM applied to canonical scattering problems in plasmonics. J. Comput. Theor. Nanosci. 8 (2011) 1–9. | DOI

Cité par Sources :