Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz hp-dGFEM
Ralf Hiptmair; Andrea Moiola; Ilaria Perugia; Christoph Schwab
- Volume: 48, Issue: 3, page 727-752
- ISSN: 0764-583X
Access Full Article
topAbstract
topHow to cite
topHiptmair, Ralf, et al. "Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz hp-dGFEM." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.3 (2014): 727-752. <http://eudml.org/doc/273280>.
@article{Hiptmair2014,
abstract = {We study the approximation of harmonic functions by means of harmonic polynomials in two-dimensional, bounded, star-shaped domains. Assuming that the functions possess analytic extensions to a δ-neighbourhood of the domain, we prove exponential convergence of the approximation error with respect to the degree of the approximating harmonic polynomial. All the constants appearing in the bounds are explicit and depend only on the shape-regularity of the domain and on δ. We apply the obtained estimates to show exponential convergence with rate O(exp(-b√N)), N being the number of degrees of freedom and b > 0, of a hp-dGFEM discretisation of the Laplace equation based on piecewise harmonic polynomials. This result is an improvement over the classical rate O(exp(-b3√N)), and is due to the use of harmonic polynomial spaces, as opposed to complete polynomial spaces.},
author = {Hiptmair, Ralf, Moiola, Andrea, Perugia, Ilaria, Schwab, Christoph},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {approximation by harmonic polynomials; exponential orders of convergence; hp-finite elements; exponential order of convergence; -finite elements},
language = {eng},
number = {3},
pages = {727-752},
publisher = {EDP-Sciences},
title = {Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz hp-dGFEM},
url = {http://eudml.org/doc/273280},
volume = {48},
year = {2014},
}
TY - JOUR
AU - Hiptmair, Ralf
AU - Moiola, Andrea
AU - Perugia, Ilaria
AU - Schwab, Christoph
TI - Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz hp-dGFEM
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 3
SP - 727
EP - 752
AB - We study the approximation of harmonic functions by means of harmonic polynomials in two-dimensional, bounded, star-shaped domains. Assuming that the functions possess analytic extensions to a δ-neighbourhood of the domain, we prove exponential convergence of the approximation error with respect to the degree of the approximating harmonic polynomial. All the constants appearing in the bounds are explicit and depend only on the shape-regularity of the domain and on δ. We apply the obtained estimates to show exponential convergence with rate O(exp(-b√N)), N being the number of degrees of freedom and b > 0, of a hp-dGFEM discretisation of the Laplace equation based on piecewise harmonic polynomials. This result is an improvement over the classical rate O(exp(-b3√N)), and is due to the use of harmonic polynomial spaces, as opposed to complete polynomial spaces.
LA - eng
KW - approximation by harmonic polynomials; exponential orders of convergence; hp-finite elements; exponential order of convergence; -finite elements
UR - http://eudml.org/doc/273280
ER -
References
top- [1] D.N. Arnold, F. Brezzi, B. Cockburn and L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal.39 (2002) 1749–1779. Zbl1008.65080MR1885715
- [2] I. Babuška and B.Q. Guo, The h-p version of the finite element method for domains with curved boundaries. SIAM J. Numer. Anal.25 (1988) 837–861. Zbl0655.65124MR954788
- [3] I. Babuška and B.Q. Guo, Regularity of the solution of elliptic problems with piecewise analytic data. I. Boundary value problems for linear elliptic equation of second order. SIAM J. Math. Anal. 19 (1988) 172–203. Zbl0647.35021MR924554
- [4] I. Babuška and B.Q. Guo, The h-p version of the finite element method for problems with nonhomogeneous essential boundary condition. Comput. Methods Appl. Mech. Engrg.74 (1989) 1–28. Zbl0723.73077MR1017747
- [5] I. Babuška and B.Q. Guo, Regularity of the solution of elliptic problems with piecewise analytic data. II. The trace spaces and application to the boundary value problems with nonhomogeneous boundary conditions. SIAM J. Math. Anal. 20 (1989) 763–781. Zbl0706.35028MR1000721
- [6] G.A. Baker, Finite element methods for elliptic equations using nonconforming elements. Math. Comput.31 (1977) 45–59. Zbl0364.65085MR431742
- [7] C.E. Baumann and J.T. Oden, A discontinuous hp finite element method for convection-diffusion problems. Comput. Methods Appl. Mech. Engrg.175 (1999) 311–341. Zbl0924.76051MR1702201
- [8] O. Cessenat and B. Després, Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz equation. SIAM J. Numer. Anal.35 (1998) 255–299. Zbl0955.65081MR1618464
- [9] P.J. Davis, Interpolation and approximation, Republication, with minor corrections, of the 1963 original, with a new preface and bibliography. Dover Publications Inc., New York (1975). Zbl0329.41010MR380189
- [10] J. Douglas, Jr. and T. Dupont, Interior penalty procedures for elliptic and parabolic Galerkin methods, in Computing methods in applied sciences (Second Internat. Sympos., Versailles, 1975), Vol. 58. Lect. Notes in Phys. Springer, Berlin (1976) 207–216. MR440955
- [11] T.A. Driscoll and L.N. Trefethen, Schwarz-Christoffel mapping, in vol. 8 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2002). Zbl1003.30005MR1908657
- [12] L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL (1992). Zbl0804.28001MR1158660
- [13] C. Farhat, I. Harari and U. Hetmaniuk, A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime. Comput. Methods Appl. Mech. Eng.192 (2003) 1389–1419. Zbl1027.76028MR1963058
- [14] G. Gabard, P. Gamallo and T. Huttunen, A comparison of wave-based discontinuous Galerkin, ultra-weak and least-square methods for wave problems. Int. J. Numer. Methods Engrg.85 (2011) 380–402. Zbl1217.76047MR2779291
- [15] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, 2nd edition. Springer-Verlag (1983). Zbl1042.35002MR737190
- [16] P. Grisvard, Elliptic problems in nonsmooth domains, in vol. 24 of Monogr. Stud. Math. Pitman, Boston (1985). Zbl0695.35060MR775683
- [17] P. Henrici, Applied and computational complex analysis, Power series-integration-conformal mapping-location of zeros, in vol. 1 of Pure and Applied Mathematics. John Wiley & Sons, New York (1974). Zbl0313.30001MR372162
- [18] P. Henrici, Applied and computational complex analysis, Discrete Fourier analysis-Cauchy integrals-construction of conformal maps-univalent functions, in vol. 3 of Pure and Applied Mathematics. John Wiley & Son, New York (1986). Zbl0578.30001MR822470
- [19] R. Hiptmair, A. Moiola and I. Perugia, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version. SIAM J. Numer. Anal.49 (2011) 264–284. Zbl1229.65215MR2783225
- [20] R. Hiptmair, A. Moiola and I. Perugia, Trefftz discontinuous Galerkin methods for acoustic scattering on locally refined meshes. Appl. Numer. Math. (2013). Available at http://dx.doi.org/10.1016/j.apnum.2012.12.004 Zbl1288.65166MR3191479
- [21] R. Hiptmair, A. Moiola and I. Perugia, Plane wave discontinuous Galerkin methods: exponential convergence of the hp-version, Technical report 2013-31 (http://www.sam.math.ethz.ch/reports/2013/31), SAM-ETH Zürich, Switzerland (2013). Submitted to Found. Comput. Math. Zbl1165.65076
- [22] R. Hiptmair, A. Moiola, I. Perugia and C. Schwab, Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz hp-dGFEM, Technical report 2012-38 (http://www.sam.math.ethz.ch/reports/2012/38), SAM-ETH, Zürich, Switzerland (2012). Zbl1295.31004
- [23] T. Huttunen, P. Monk and J. P. Kaipio, Computational aspects of the ultra-weak variational formulation. J. Comput. Phys.182 (2002) 27–46. Zbl1015.65064MR1936802
- [24] F. Li, On the negative-order norm accuracy of a local-structure-preserving LDG method. J. Sci. Comput.51 (2012) 213–223. Zbl06046085MR2891952
- [25] F. Li and C.-W. Shu, A local-structure-preserving local discontinuous Galerkin method for the Laplace equation. Methods Appl. Anal.13 (2006) 215–233. Zbl1134.65081MR2381547
- [26] A.I. Markushevich, Theory of functions of a complex variable. Vol. I, II, III, english edition. Translated and edited by Richard A. Silverman. Chelsea Publishing Co., New York (1977). Zbl0357.30002MR444912
- [27] J.M. Melenk, On Generalized Finite Element Methods. Ph.D. thesis. University of Maryland (1995). MR2692949
- [28] A.I. Markushevich, Operator adapted spectral element methods I: harmonic and generalized harmonic polynomials. Numer. Math.84 (1999) 35–69. Zbl0941.65112MR1724356
- [29] J. M. Melenk and I. Babuška, The partition of unity finite element method: basic theory and applications. Comput. Methods Appl. Mech. Engrg.139 (1996) 289–314. Zbl0881.65099MR1426012
- [30] A. Moiola, Trefftz-discontinuous Galerkin methods for time-harmonic wave problems, Ph.D. thesis, Seminar for applied mathematics. ETH Zürich (2011). Available at: http://e-collection.library.ethz.ch/view/eth:4515.
- [31] A. Moiola, R. Hiptmair and I. Perugia, Vekua theory for the Helmholtz operator. Z. Angew. Math. Phys.62 (2011) 779–807. Zbl1266.35016MR2843917
- [32] P. Monk and D.Q. Wang, A least squares method for the Helmholtz equation. Comput. Methods Appl. Mech. Engrg.175 (1999) 121–136. Zbl0943.65127MR1692914
- [33] R. Nevanlinna and V. Paatero, Introduction to complex analysis. Translated from the German by T. Kövari and G.S. Goodman. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. (1969). Zbl0502.30001MR239056
- [34] B. Rivière, M. F. Wheeler and V. Girault, Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems I. Comput. Geosci. 3 (1999) 337–360 (2000). Zbl0951.65108MR1750076
- [35] C. Schwab, p- and hp-Finite Element Methods. Theory and Applications in Solid and Fluid Mechanics. Numerical Mathematics and Scientific Computation. Clarendon Press, Oxford (1998). Zbl0910.73003MR1695813
- [36] I.N. Vekua, New methods for solving elliptic equations. North Holland (1967). Zbl0146.34301MR212370
- [37] J.L. Walsh, Interpolation and approximation by rational functions in the complex domain, 5th edition, vol. XX of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, R.I. (1969). Zbl0146.29902MR218588JFM61.0315.01
- [38] R. Webster, Convexity, Oxford Science Publications. Oxford University Press, New York (1994). Zbl0835.52001MR1443208
- [39] T.P. Wihler, Discontinuous Galerkin FEM for Elliptic Problems in Polygonal Domains. Ph.D. thesis, Swiss Federal Institute of Technology Zurich (2002). Available at: http://e-collection.library.ethz.ch/view/eth:26201.
- [40] T.P. Wihler, P. Frauenfelder and C. Schwab, Exponential convergence of the hp-DGFEM for diffusion problems. p-FEM2000: p and hp finite element methods–mathematics and engineering practice (St. Louis, MO). Comput. Math. Appl. 46 (2003) 183–205. Zbl1059.65095MR2015278
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.