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(-b 3√N)), and is due to the use of harmonic polynomial spaces, as opposed to complete polynomial spaces.
Mots-clés : approximation by harmonic polynomials, exponential orders of convergence, hp-finite elements
@article{M2AN_2014__48_3_727_0,
author = {Hiptmair, Ralf and Moiola, Andrea and Perugia, Ilaria and Schwab, Christoph},
title = {Approximation by harmonic polynomials in star-shaped domains and exponential convergence of {Trefftz} $hp${-dGFEM}},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {727--752},
publisher = {EDP-Sciences},
volume = {48},
number = {3},
year = {2014},
doi = {10.1051/m2an/2013137},
zbl = {1295.31004},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2013137/}
}
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 PY - 2014 SP - 727 EP - 752 VL - 48 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2013137/ DO - 10.1051/m2an/2013137 LA - en ID - M2AN_2014__48_3_727_0 ER -
%0 Journal Article %A Hiptmair, Ralf %A Moiola, Andrea %A Perugia, Ilaria %A Schwab, Christoph %T Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz $hp$-dGFEM %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2014 %P 727-752 %V 48 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2013137/ %R 10.1051/m2an/2013137 %G en %F M2AN_2014__48_3_727_0
Hiptmair, Ralf; Moiola, Andrea; Perugia, Ilaria; Schwab, Christoph. Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz $hp$-dGFEM. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 3, pp. 727-752. doi : 10.1051/m2an/2013137. http://archive.numdam.org/articles/10.1051/m2an/2013137/
[1] , , and , Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2002) 1749-1779. | MR | Zbl
[2] and , The h-p version of the finite element method for domains with curved boundaries. SIAM J. Numer. Anal. 25 (1988) 837-861. | MR | Zbl
[3] and , 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. | MR | Zbl
[4] and , 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. | MR | Zbl
[5] and , 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. | MR | Zbl
[6] , Finite element methods for elliptic equations using nonconforming elements. Math. Comput. 31 (1977) 45-59. | MR | Zbl
[7] and , A discontinuous hp finite element method for convection-diffusion problems. Comput. Methods Appl. Mech. Engrg. 175 (1999) 311-341. | MR | Zbl
[8] and , Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz equation. SIAM J. Numer. Anal. 35 (1998) 255-299. | MR | Zbl
[9] , Interpolation and approximation, Republication, with minor corrections, of the 1963 original, with a new preface and bibliography. Dover Publications Inc., New York (1975). | MR | Zbl
[10] and , 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. | MR
[11] and , Schwarz-Christoffel mapping, in vol. 8 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2002). | MR | Zbl
[12] and , Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL (1992). | MR | Zbl
[13] , and , 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. | MR | Zbl
[14] , and , 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. | MR | Zbl
[15] and , Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, 2nd edition. Springer-Verlag (1983). | MR | Zbl
[16] , Elliptic problems in nonsmooth domains, in vol. 24 of Monogr. Stud. Math. Pitman, Boston (1985). | MR | Zbl
[17] , 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). | MR | Zbl
[18] , 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). | MR | Zbl
[19] , and , Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version. SIAM J. Numer. Anal. 49 (2011) 264-284. | MR | Zbl
[20] , and , 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 | MR | Zbl
[21] , and , 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. | Numdam | Zbl
[22] , , and , 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).
[23] , and , Computational aspects of the ultra-weak variational formulation. J. Comput. Phys. 182 (2002) 27-46. | MR | Zbl
[24] , On the negative-order norm accuracy of a local-structure-preserving LDG method. J. Sci. Comput. 51 (2012) 213-223. | MR
[25] and , A local-structure-preserving local discontinuous Galerkin method for the Laplace equation. Methods Appl. Anal. 13 (2006) 215-233. | MR | Zbl
[26] , 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). | MR | Zbl
[27] , On Generalized Finite Element Methods. Ph.D. thesis. University of Maryland (1995). | MR
[28] , Operator adapted spectral element methods I: harmonic and generalized harmonic polynomials. Numer. Math. 84 (1999) 35-69. | MR | Zbl
[29] and , The partition of unity finite element method: basic theory and applications. Comput. Methods Appl. Mech. Engrg. 139 (1996) 289-314. | MR | Zbl
[30] , 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] , and , Vekua theory for the Helmholtz operator. Z. Angew. Math. Phys. 62 (2011) 779-807. | MR | Zbl
[32] and , A least squares method for the Helmholtz equation. Comput. Methods Appl. Mech. Engrg. 175 (1999) 121-136. | MR | Zbl
[33] and , 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). | MR | Zbl
[34] , and , Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems I. Comput. Geosci. 3 (1999) 337-360 (2000). | MR | Zbl
[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). | MR | Zbl
[36] , New methods for solving elliptic equations. North Holland (1967). | MR | Zbl
[37] , 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). | JFM | MR | Zbl
[38] , Convexity, Oxford Science Publications. Oxford University Press, New York (1994). | MR | Zbl
[39] , 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] , and , 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. | MR | Zbl
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