The $hp$-version of the boundary element method with quasi-uniform meshes in three dimensions

Bespalov, Alexei ; Heuer, Norbert

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 42 (2008), p. 821-849 / Harvested from Numdam

Résumé

We prove an a priori error estimate for the $hp$-version of the boundary element method with hypersingular operators on piecewise plane open or closed surfaces. The underlying meshes are supposed to be quasi-uniform. The solutions of problems on polyhedral or piecewise plane open surfaces exhibit typical singularities which limit the convergence rate of the boundary element method. On closed surfaces, and for sufficiently smooth given data, the solution is $H^1$-regular whereas, on open surfaces, edge singularities are strong enough to prevent the solution from being in $H^1$. In this paper we cover both cases and, in particular, prove an a priori error estimate for the $h$-version with quasi-uniform meshes. For open surfaces we prove a convergence like $O(h^{1/2}p^{-1})$, $h$ being the mesh size and $p$ denoting the polynomial degree. This result had been conjectured previously.

Détail

Publié le : 2008-01-01

MR 2454624 | Zbl 1154.41300

DOI : https://doi.org/10.1051/m2an:2008025
Classification:  41A10,  65N15,  65N38

Bibliographie

[1] M. Ainsworth and L. Demkowicz, Explicit polynomial preserving trace liftings on a triangle. Math. Nachr. (to appear). | Zbl pre05553712

[2] M. Ainsworth and D. Kay, The approximation theory for the $p$-version finite element method and application to non-linear elliptic PDEs. Numer. Math. 82 (1999) 351-388. | MR 1692127 | Zbl 0938.65122

[3] M. Ainsworth and K. Pinchedez, The $hp$-MITC finite element method for the Reissner-Mindlin plate problem. J. Comput. Appl. Math. 148 (2002) 429-462. | MR 1936149 | Zbl 1058.74076

[4] M. Ainsworth, W. Mclean and T. Tran, The conditioning of boundary element equations on locally refined meshes and preconditioning by diagonal scaling. SIAM J. Numer. Anal. 36 (1999) 1901-1932. | MR 1712149 | Zbl 0947.65125

[5] I. Babuška and B.Q. Guo, Optimal estimates for lower and upper bounds of approximation errors in the $p$-version of the finite element method in two dimensions. Numer. Math. 85 (2000) 219-255. | MR 1754720 | Zbl 0970.65117

[6] I. Babuška and M. Suri, The $h$-$p$ version of the finite element method with quasiuniform meshes. RAIRO Modél. Math. Anal. Numér. 21 (1987) 199-238. | Numdam | MR 896241 | Zbl 0623.65113

[7] I. Babuška and M. Suri, The optimal convergence rate of the $p$-version of the finite element method. SIAM J. Numer. Anal. 24 (1987) 750-776. | MR 899702 | Zbl 0637.65103

[8] I. Babuška and M. Suri, The treatment of nonhomogeneous Dirichlet boundary conditions by the $p$-version of the finite element method. Numer. Math. 55 (1989) 97-121. | MR 987158 | Zbl 0673.65066

[9] I. Babuška, R.B. Kellogg and J. Pitkäranta, Direct and inverse error estimates for finite elements with mesh refinement. Numer. Math. 33 (1979) 447-471. | MR 553353 | Zbl 0423.65057

[10] J. Bergh and J. Löfström, Interpolation Spaces, Grundlehren der mathematischen Wissenschaften 223. Springer-Verlag, Berlin (1976). | MR 482275 | Zbl 0344.46071

[11] A. Bespalov and N. Heuer, The $p$-version of the boundary element method for hypersingular operators on piecewise plane open surfaces. Numer. Math. 100 (2005) 185-209. | MR 2135781 | Zbl 1082.65129

[12] A. Bespalov and N. Heuer, The $p$-version of the boundary element method for weakly singular operators on piecewise plane open surfaces. Numer. Math. 106 (2007) 69-97. | MR 2286007 | Zbl 1117.65160

[13] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). | MR 520174 | Zbl 0383.65058

[14] M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results. SIAM J. Math. Anal. 19 (1988) 613-626. | MR 937473 | Zbl 0644.35037

[15] L. Demkowicz, Polynomial exact sequences and projection-based interpolation with applications to Maxwell equations, in Mixed Finite Elements, Compatibility Conditions and Applications, D. Boffi and L. Gastaldi Eds., Lecture Notes in Mathematics 1939, Springer-Verlag (2008). | MR 2459075 | Zbl 1143.78366

[16] L. Demkowicz and I. Babuška, $p$ interpolation error estimates for edge finite elements of variable order in two dimensions. SIAM J. Numer. Anal. 41 (2003) 1195-1208. | MR 2034876 | Zbl 1067.78016

[17] V.J. Ervin and N. Heuer, An adaptive boundary element method for the exterior Stokes problem in three dimensions. IMA J. Numer. Anal. 26 (2006) 297-325. | MR 2218635 | Zbl 1096.65120

[18] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman Publishing Inc., Boston (1985). | MR 775683 | Zbl 0695.35060

[19] B.Q. Guo, Approximation theory for the $p$-version of the finite element method in three dimensions. Part 1: Approximabilities of singular functions in the framework of the Jacobi-weighted Besov and Sobolev spaces. SIAM J. Numer. Anal. 44 (2006) 246-269. | MR 2217381 | Zbl 1115.65117

[20] B.Q. Guo and N. Heuer, The optimal rate of convergence of the $p$-version of the boundary element method in two dimensions. Numer. Math. 98 (2004) 499-538. | MR 2088925 | Zbl 1060.65117

[21] B.Q. Guo and N. Heuer, The optimal convergence of the $h$-$p$ version of the boundary element method with quasiuniform meshes for elliptic problems on polygonal domains. Adv. Comp. Math. 24 (2006) 353-374. | MR 2222275 | Zbl 1103.65109

[22] N. Heuer and F. Leydecker, An extension theorem for polynomials on triangles. Calcolo 45 (2008) 69-85. | MR 2424648

[23] N. Heuer, M. Maischak and E.P. Stephan, Exponential convergence of the $hp$-version for the boundary element method on open surfaces. Numer. Math. 83 (1999) 641-666. | MR 1728223 | Zbl 0946.65120

[24] J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications I. Springer-Verlag, New York (1972). | Zbl 0223.35039

[25] P. Monk, On the $p$- and $hp$-extension of Nédélec’s curl-conforming elements. J. Comput. Appl. Math. 53 (1994) 117-137. | MR 1305972 | Zbl 0820.65066

[26] J. Nečas, Les Méthodes Directes en Théorie des Équations Elliptiques. Academia, Prague (1967). | MR 227584

[27] C. Schwab, $p$- and $hp$-Finite Element Methods. Clarendon Press, Oxford (1998). | MR 1695813 | Zbl 0910.73003

[28] C. Schwab and M. Suri, The optimal $p$-version approximation of singularities on polyhedra in the boundary element method. SIAM J. Numer. Anal. 33 (1996) 729-759. | MR 1388496 | Zbl 0854.65108

[29] E.P. Stephan, Boundary integral equations for screen problems in $\mathbb {R}^3$. Integr. Equ. Oper. Theory 10 (1987) 257-263. | MR 878247 | Zbl 0653.35016

[30] E.P. Stephan, The $h$-$p$ boundary element method for solving $2$- and $3$-dimensional problems. Comput. Methods Appl. Mech. Engrg. 133 (1996) 183-208. | MR 1399637 | Zbl 0918.73290

[31] E.P. Stephan and M. Suri, The $h$-$p$ version of the boundary element method on polygonal domains with quasiuniform meshes. RAIRO Modél. Math. Anal. Numér. 25 (1991) 783-807. | Numdam | MR 1135993 | Zbl 0744.65073

[32] T. Von Petersdorff, Randwertprobleme der Elastizitätstheorie für Polyeder - Singularitäten und Approximation mit Randelementmethoden. Ph.D. thesis, Technische Hochschule Darmstadt, Germany (1989). | Zbl 0709.73009

[33] T. Von Petersdorff and E.P. Stephan, Regularity of mixed boundary value problems in $\mathbb {R}^3$ and boundary element methods on graded meshes. Math. Methods Appl. Sci. 12 (1990) 229-249. | MR 1043756 | Zbl 0722.35017