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Approximation by harmonic polynomials in star-shaped domainsand exponential convergence of Trefftz hp-dGFEM∗

Published online by Cambridge University Press:  11 February 2014

Ralf Hiptmair ,
Andrea Moiola ,
Ilaria Perugia  and
Christoph Schwab
Ralf Hiptmair
Affiliation:
Seminar of Applied Mathematics, ETH Zürich, 8092 Zürich, Switzerland.. [email protected]
Andrea Moiola
Affiliation:
Department of Mathematics and Statistics, University of Reading, Whiteknights, RG6 6AX, UK.; [email protected]
Ilaria Perugia
Affiliation:
Faculty of Mathematics, University of Vienna, 1090 Wien, Austria.; [email protected]
Christoph Schwab
Affiliation:
Seminar of Applied Mathematics, ETH Zürich, 8092 Zürich, Switzerland.; [email protected]
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Abstract

We study the approximation of harmonic functions by means of harmonic polynomials intwo-dimensional, bounded, star-shaped domains. Assuming that the functions possessanalytic extensions to a δ-neighbourhood of the domain, we proveexponential convergence of the approximation error with respect to the degree of theapproximating harmonic polynomial. All the constants appearing in the bounds are explicitand depend only on the shape-regularity of the domain and on δ. We applythe obtained estimates to show exponential convergence with rate O(exp(-b√N)), N being the number of degrees offreedom and b > 0, of a hp-dGFEM discretisation ofthe Laplace equation based on piecewise harmonic polynomials. This result is animprovement over the classical rate O(exp(-b 3√N)), and is due to the use of harmonic polynomialspaces, as opposed to complete polynomial spaces.

Keywords

Approximation by harmonic polynomialsexponential orders of convergencehp-finite elements
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 3 , May 2014 , pp. 727 - 752
Copyright
© EDP Sciences, SMAI, 2014

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References

Arnold, D.N., Brezzi, F., Cockburn, B. and Marini, L.D., Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2002) 1749–1779. Google Scholar
Babuška, I. and Guo, B.Q., The h-p version of the finite element method for domains with curved boundaries. SIAM J. Numer. Anal. 25 (1988) 837–861. Google Scholar
Babuška, I. and Guo, B.Q., 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. Google Scholar
Babuška, I. and Guo, B.Q., 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. Google Scholar
Babuška, I. and Guo, B.Q., 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. Google Scholar
Baker, G.A., Finite element methods for elliptic equations using nonconforming elements. Math. Comput. 31 (1977) 45–59. Google Scholar
Baumann, C.E. and Oden, J.T., A discontinuous hp finite element method for convection-diffusion problems. Comput. Methods Appl. Mech. Engrg. 175 (1999) 311–341. Google Scholar
Cessenat, O. and Després, B., Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz equation. SIAM J. Numer. Anal. 35 (1998) 255–299. Google Scholar
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).
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.
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).
L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL (1992).
Farhat, C., Harari, I. and Hetmaniuk, U., 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. Google Scholar
Gabard, G., Gamallo, P. and Huttunen, T., 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. Google Scholar
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, 2nd edition. Springer-Verlag (1983).
P. Grisvard, Elliptic problems in nonsmooth domains, in vol. 24 of Monogr. Stud. Math. Pitman, Boston (1985).
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).
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).
Hiptmair, R., Moiola, A. and Perugia, I., Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version. SIAM J. Numer. Anal. 49 (2011) 264–284. Google Scholar
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
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.
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).
Huttunen, T., Monk, P. and Kaipio, J. P., Computational aspects of the ultra-weak variational formulation. J. Comput. Phys. 182 (2002) 27–46. Google Scholar
Li, F., On the negative-order norm accuracy of a local-structure-preserving LDG method. J. Sci. Comput. 51 (2012) 213–223. Google Scholar
Li, F. and Shu, C.-W., A local-structure-preserving local discontinuous Galerkin method for the Laplace equation. Methods Appl. Anal. 13 (2006) 215–233. Google Scholar
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).
J.M. Melenk, On Generalized Finite Element Methods. Ph.D. thesis. University of Maryland (1995).
Markushevich, A.I., Operator adapted spectral element methods I: harmonic and generalized harmonic polynomials. Numer. Math. 84 (1999) 35–69. Google Scholar
Melenk, J. M. and Babuška, I., The partition of unity finite element method: basic theory and applications. Comput. Methods Appl. Mech. Engrg. 139 (1996) 289–314. Google Scholar
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.
Moiola, A., Hiptmair, R. and Perugia, I., Vekua theory for the Helmholtz operator. Z. Angew. Math. Phys. 62 (2011) 779–807. Google Scholar
Monk, P. and Wang, D.Q., A least squares method for the Helmholtz equation. Comput. Methods Appl. Mech. Engrg. 175 (1999) 121–136. Google Scholar
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).
Rivière, B., Wheeler, M. F. and Girault, V., Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems I. Comput. Geosci. 3 (1999) 337–360 (2000). Google Scholar
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).
I.N. Vekua, New methods for solving elliptic equations. North Holland (1967).
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).
R. Webster, Convexity, Oxford Science Publications. Oxford University Press, New York (1994).
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.
Wihler, T.P., Frauenfelder, P. and Schwab, C., 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. Google Scholar