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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:[email protected]
Corresponding author.Email address:[email protected]
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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

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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