Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-10T14:46:38.811Z Has data issue: false hasContentIssue false
  • English
  • Français

Evaluating Local Approximations of the L2-Orthogonal Projection Between Non-Nested Finite Element Spaces

Published online by Cambridge University Press:  28 May 2015

Thomas Dickopf  and
Rolf Krause
Thomas Dickopf*
Affiliation:
Università della Svizzera italiana (USI, University of Lugano), Institute of Computational Science, Via G. Buffi 13, 6904 Lugano, Switzerland
Rolf Krause*
Affiliation:
Università della Svizzera italiana (USI, University of Lugano), Institute of Computational Science, Via G. Buffi 13, 6904 Lugano, Switzerland
*
Corresponding author.Email address:thomas.dickopf@usi.ch
Corresponding author.Email address:thomas.dickopf@usi.ch
Get access

Abstract

We present quantitative studies of transfer operators between finite element spaces associated with unrelated meshes. Several local approximations of the global L2-orthogonal projection are reviewed and evaluated computationally. The numerical studies in 3D provide the first estimates of the quantitative differences between a range of transfer operators between non-nested finite element spaces. We consider the standard finite element interpolation, Clément’s quasi-interpolation with different local polynomial degrees, the global L2-orthogonal projection, a local L2-quasi-projection via a discrete inner product, and a pseudo-L2-projection defined by a Petrov-Galerkin variational equation with a discontinuous test space. Understanding their qualitative and quantitative behaviors in this computational way is interesting per se; it could also be relevant in the context of discretization and solution techniques which make use of different non-nested meshes. It turns out that the pseudo-L2-projection approximates the actual L2-orthogonal projection best. The obtained results seem to be largely independent of the underlying computational domain; this is demonstrated by four examples (ball, cylinder, half torus and Stanford Bunny).

Keywords

65D0565F1065N3065N5065N55Finite elementsunstructured meshesnon-nested spacestransfer operatorsinterpolationprojection
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 7 , Issue 3 , August 2014 , pp. 288 - 316
Copyright
Copyright © Global Science Press Limited 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)Google Scholar
[2]Ainsworth, H.: Octree C++ General Component (2005)Google Scholar
[3]Apel, T.: Anisotropic Finite Elements: Local Estimates and Applications. Advances in Numerical Mathematics. Teubner, Stuttgart (1999)Google Scholar
[4]Apel, T.: Interpolation in h-version finite element spaces. In: E., Stein, R., de Borst, T.J.R., Hughes (eds.) Encyclopedia of Computational Mechanics. Vol. 1. Fundamentals, pp. 55–72. Wiley, Chichester (2004)Google Scholar
[5]Bank, R.E., Dupont, T.F., Yserentant, H.: The hierarchical basis multigrid method. Numer. Math. 52(4), 427–458 (1988)CrossRefGoogle Scholar
[6]Barber, C.M., Dobkin, D.P., Huhdanpaa, H.: The quickhull algorithm for convex hulls. ACM Trans. Math. Softw. 22(4), 469–483 (1996)CrossRefGoogle Scholar
[7]Bastian, P., Birken, K., Johannsen, K., Lang, S., Neuβ, N., Rentz-Reichert, H., Wieners, C.: UG –a flexible software toolbox for solving partial differential equations. Comput. Vis. Sci. 1(1), 27–40 (1997)CrossRefGoogle Scholar
[8]Ben Belgacem, F.: The mortar finite element method with Lagrange multipliers. Numer. Math. 84(2), 173–197 (1999)Google Scholar
[9]Bernardi, C., Maday, Y., Patera, A.T.: A new nonconforming approach to domain decomposition: the mortar element method. In: H., Brezis, J.L., Lions (eds.) Nonlinear Partial Differential Equations and Their Applications, Pitman Res. Notes Math. Ser., vol. 299, pp. 13–51. Harlow: Longman Scientific & Technical, New York (1994)Google Scholar
[10]Braess, D.: Finite Elemente. Springer, Berlin (2007)CrossRefGoogle Scholar
[11]Braess, D., Verfürth, R.: Multigrid methods for nonconforming finite element methods. SIAM J. Numer. Anal. 27(4), 979–986 (1990)CrossRefGoogle Scholar
[12]Bramble, J.H., Pasciak, J.E., Steinbach, O.: On the stability of the L2-projection in H1(Ω). Math. Comp. 71(237), 147–156 (2002)Google Scholar
[13]Bramble, J.H., Pasciak, J.E., Vassilevski, P.S.: Computational scales of Sobolev norms with applications to preconditioning. Math. Comp. 69(230), 463–480 (2000)Google Scholar
[14]Bramble, J.H., Pasciak, J.E., Wang, J., Xu, J.: Convergence estimates for multigrid algorithms without regularity assumptions. Math. Comp. 57(195), 23–45 (1991)CrossRefGoogle Scholar
[15]Bramble, J.H., Pasciak, J.E., Xu, J.: The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms. Math. Comp. 56(193), 1–34 (1991)CrossRefGoogle Scholar
[16]Bramble, J.H., Xu, J.: Some estimates for a weighted L2 projection. Math. Comp. 56(194), 463–476 (1991)Google Scholar
[17]Brandt, A.: Multi-level adaptive solutions to boundary-value problems. Math. Comp. 31(138), 333–390 (1977)CrossRefGoogle Scholar
[18]Brenner, S.C.: Convergence of nonconforming V-cycle and F-cycle multigrid algorithms for second order elliptic boundary value problems. Math. Comp. 73(247), 1041–1066 (2004)Google Scholar
[19]Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, vol. 15. Springer, Berlin (2002)Google Scholar
[20]X.-C., Cai: The use of pointwise interpolation in domain decomposition methods with non-nested meshes. SIAM J. Sci. Comput. 16(1), 250–256 (1995)Google Scholar
[21]Carstensen, C: Quasi-interpolation and a posteriori error analysis in finite element methods. Math. Model. Numer. Anal. 33(6), 1187–1202 (1999)CrossRefGoogle Scholar
[22]Carstensen, C: Merging the Bramble-Pasciak-Steinbach and the Crouzeix-Thomée criterion for H-stability of the L2-projection onto finite element spaces. Math. Comp. 71(237), 157–163 (2002)Google Scholar
[23]Carstensen, C: Clément interpolation and its role in adaptive finite element error control. In: E., Koelink, J., van Neerven, B., de Pagter, G., Sweers (eds.) Partial Differential Equations and Functional Analysis - The Philippe Cléement Festschrift, Oper. Theory Adv. Appl., vol. 168, pp. 27–43. Birkhäuser, Basel (2006)CrossRefGoogle Scholar
[24]Chan, T.F., Smith, B.F., Zou, J.: Overlapping Schwarz methods on unstructured meshes using non-matching coarse grids. Numer. Math. 73(2), 149–167 (1996)CrossRefGoogle Scholar
[25]Ciarlet, P.G.: The Finite Element Method for Elliptic Problems, Studies in Mathematics and its Applications,> vol. 4. North-Holland, Amsterdam (1978)Google Scholar
[26]Clement, P.: Approximation by finite element functions using local regularization. RAIRO Anal. Numéer. 9(R-2), 77–84 (1975)Google Scholar
[27]Cohen, A., Daubechies, I., Feauveau, J.C.: Biorthogonal bases of compactly supported wavelets. Comm. Pure Appl. Math. 45(5), 485–560 (1992)CrossRefGoogle Scholar
[28]Crouzeix, M., Thoméee,V., : The stability in L p and Wl of the L 2-projection onto finite element function spaces. Math. Comp. 48(178), 521–532 (1987)Google Scholar
[29]Dahmen, W., Kunoth, A., Urban, K.: Biorthogonal spline wavelets on the interval - stability and moment conditions. Appl. Comput. Harmon. Anal. 6(2), 132–196 (1999)CrossRefGoogle Scholar
[30]Dickopf, T.: Multilevel methods based on non-nested meshes. Ph.D. thesis, University of Bonn (2010). http://hss.ulb.uni-bonn.de/2010/2365Google Scholar
[31]Dickopf, T.: Nodal interpolation between first-order finite element spaces in 1d is uniformly H1-stable. In: A., Cangiani, R.L., Davidchack, E., Georgoulis, A., Gorban, J., LevesleyM., Tretyakov (eds.) Numerical Mathematics and Advanced Applications, Proceedings of ENUMATH 2011, pp. 419–427. Springer, Berlin (2012)Google Scholar
[32]Dickopf, T., Krause, R.: Efficient simulation of multi-body contact problems on complex geometries: a flexible decomposition approach using constrained minimization. Int. J. Numer. Methods Engrg. 77(13), 1834–1862 (2009)CrossRefGoogle Scholar
[33]Dickopf, T., Krause, R.: Weak information transfer between non-matching warped interfaces. In: M., Bercovier, M.J., Gander, R., Kornhuber, O.B., Widlund (eds.) Domain Decomposition Methods in Science and Engineering XVIII, Lect. Notes Comput. Sci. Eng., vol. 70, pp. 283–290. Springer, Berlin (2009)Google Scholar
[34]J., Douglas jun., Dupont, T.F., Wahlbin, L.: The stability in Lq of the L2-projection into finite element function spaces. Numer. Math. 23(3), 193–197 (1975)Google Scholar
[35]Evans, L.C.: Partial Differential Equations, Graduate Studies in Mathematics, vol. 19. AMS, Providence, RI, USA (1998)Google Scholar
[36]Falletta, S.: The approximate integration in the mortar method constraint. In: O.B., Widlund, D.E., Keyes (eds.) Domain Decomposition Methods in Science and Engineering XVI, Lect. Notes Comput. Sci. Eng., vol. 55, pp. 555–563. Springer, Berlin (2007)Google Scholar
[37]Flemisch, B., Wohlmuth, B.: Stable Lagrange multipliers for quadrilateral meshes of curved interfaces in 3D. Comput. Methods Appl. Mech. Eng. 196(8), 1589–1602 (2007)CrossRefGoogle Scholar
[38]Gander, M.J., Japhet, C.: An algorithm for non-matching grid projections with linear complexity. In: M., Bercovier, M.J., Gander, R., Kornhuber, O.B., Widlund (eds.) Domain Decomposition Methods in Science and Engineering XVIII, Lect. Notes Comput. Sci. Eng., vol. 70, pp. 185–192. Springer, Berlin (2009)Google Scholar
[39]Griebel, M.: Multilevelmethoden als Iterationsverfahren über Erzeugendensystemen. Teubner Skripten zur Numerik. Teubner, Stuttgart (1994)Google Scholar
[40]Hackbusch, W.: Multi-Grid Methods and Applications, Springer Series in Computational Mathematics, vol. 4. Springer, Berlin (1985)Google Scholar
[41]Kim, C., Lazarov, R.D., Pasciak, J.E., Vassilevski, P.S.: Multiplier spaces for the mortar finite element method in three dimensions. SIAM J. Numer. Anal. 39(2), 519–538 (2001)CrossRefGoogle Scholar
[42]Knyazev, A.V.: Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method. SIAM J. Sci. Comput. 23(2), 517–541 (2001)CrossRefGoogle Scholar
[43]Lamichhane, B.P.: Higher order mortar finite elements with dual lagrange multiplier spaces and applications. Ph.D. thesis, University of Stuttgart (2006)Google Scholar
[44]Lamichhane, B.P., Wohlmuth, B.: Biorthogonal bases with local support and approximation properties. Math. Comp. 76(257), 233–249 (2007)CrossRefGoogle Scholar
[45]Maday, Y., Rapetti, F., Wohlmuth, B.: The influence of quadrature formulas in 2d and 3d mortar element methods. In: L.F., Pavarino, A., Toselli (eds.) Recent Developments in Domain Decomposition Methods, Lect. Notes Comput. Sci. Eng., vol. 23, pp. 203–221. Springer, Berlin (2002)Google Scholar
[46]Oswald, P.: Multilevel Finite Element Approximation. Theory and Applications. Teubner Skripten zur Numerik. Teubner, Stuttgart (1994)Google Scholar
[47]Oswald, P.: Intergrid transfer operators and multilevel preconditioners for nonconforming discretizations. Appl. Numer. Math. 23(1), 139–158 (1997)CrossRefGoogle Scholar
[48]Oswald, P.: Optimality of multilevel preconditioning for nonconforming P1 finite elements. Numer. Math. 111(2), 267–291 (2008)CrossRefGoogle Scholar
[49]Oswald, P., Wohlmuth, B.: On polynomial reproduction of dual FE bases. In: N., Debit, M., Garbey, R., Hoppe, J., Périaux, D.E., Keyes, Y., Kuznetsov (eds.) Thirteenth International Conference on Domain Decomposition Methods, pp. 85–96. CIMNE, Barcelona (2002)Google Scholar
[50]Ruge, J.W., Stüben, K.: Algebraic multigrid. In: S.F., Mccormick (ed.) Multigrid Methods, Frontiers in Applied Mathematics, vol. 3, pp. 73–130. SIAM, Philadelphia, PA, USA (1987)Google Scholar
[51]Sandia National Laboratories: CUBIT (2012). http://cubit.sandia.govGoogle Scholar
[52]Schenk, O., Gärtner, K.: Solving unsymmetric sparse systems of linear equations with PARDISO. Future Generation Computer Systems 20(3), 475–487 (2004)CrossRefGoogle Scholar
[53]Schenk, O., Gärtner, K.: On fast factorization pivoting methods for sparse symmetric indefinite systems. Electron. Trans. Numer. Anal. 23, 158–179 (2006)Google Scholar
[54]Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54(190), 483–493 (1990)CrossRefGoogle Scholar
[55]Stanford 3D Scanning Repository: Stanford Bunny (1994). http://graphics.stanford.edu/data/3DscanrepGoogle Scholar
[56]Steinbach, O.: On a generalized L2-projection and some related stability estimates in Sobolev spaces. Numer. Math. 90(4), 775–786 (2002)CrossRefGoogle Scholar
[57]Steinbach, O.: Stability Estimates for Hybrid Coupled Domain Decomposition Methods, Lecture Notes in Mathematics, vol. 1809. Springer, Berlin (2003)Google Scholar
[58]Toselli, A., Widlund, O.B.: Domain Decomposition Methods –Algorithms and Theory, Springer Ser. Comput. Math., vol. 34. Springer, Berlin (2005)Google Scholar
[59]Trottenberg, U., Oosterlee, C.W., Schüller, A.: Multigrid. Academic Press, Orlando (2001)Google Scholar
[60]Verfürth, R.: Error estimates for some quasi-interpolation operators. Math. Model. Numer. Anal. 33(4), 695–713 (1999)CrossRefGoogle Scholar
[61]Vujičić, M.: Linear Algebra Thoroughly Explained. Springer, Berlin (2008)CrossRefGoogle Scholar
[62]Wohlmuth, B.: A mortar finite element method using dual spaces for the lagrange multiplier. SIAM J. Numer. Anal. 38(3), 989–1012 (2000)CrossRefGoogle Scholar
[63]Wohlmuth, B.: Discretization Methods and Iterative Solvers Based on Domain Decomposition, Lect. Notes Comput. Sci. Eng., vol. 17. Springer, Berlin (2001)Google Scholar
[64]Xu, J.: Theory of multilevel methods. Ph.D. thesis, Cornell University (1989)Google Scholar
[65]Xu, J.: The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids. Computing 56(3), 215–235 (1996)CrossRefGoogle Scholar
[66]Yserentant, H.: Old and new convergence proofs for multigrid methods. Acta Numerica 2, 285–326 (1993)CrossRefGoogle Scholar