Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter January 30, 2018

L2-error analysis of an isoparametric unfitted finite element method for elliptic interface problems

  • Christoph Lehrenfeld EMAIL logo and Arnold Reusken


In the context of unfitted finite element discretizations the realization of high order methods is challenging due to the fact that the geometry approximation has to be sufficiently accurate. Recently a new unfitted finite element method was introduced which achieves a high order approximation of the geometry for domains which are implicitly described by smooth level set functions. This method is based on a parametric mapping which transforms a piecewise planar interface (or surface) reconstruction to a high order approximation. In the paper [C. Lehrenfeld and A. Reusken, IMA J. Numer. Anal. 38 (2018), No. 3, 1351–1387] an a priori error analysis of the method applied to an interface problem is presented. The analysis reveals optimal order discretization error bounds in the H1-norm. In this paper we extend this analysis and derive optimal L2-error bounds.

Classification: 65N30; 65N15; 65D05
  1. Funding: C. Lehrenfeld gratefully acknowledges funding by the German Science Foundation (DFG) within the project LE 3726/1-1.


[1] P. Bastian and C. Engwer, An unfitted finite element method using discontinuous Galerkin, Int. J. Numer. Methods Engrg. 79 (2009), NO. 12,1557–1576.10.1002/nme.2631Search in Google Scholar

[2] C. Bernardi, Optimal finite-element interpolation on curved domains, SIAM J. Numer. Anal. 26 (1989), No. 5,1212–1240.10.1137/0726068Search in Google Scholar

[3] T. Boiveau, E. Burman, S. Claus, and M. G. Larson, Fictitious domain method with boundary value correction using penaltyfree Nitsche method, Preprint arXiv:1610.04482 (2016).10.1515/jnma-2016-1103Search in Google Scholar

[4] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer, New York, 1994.10.1007/978-1-4757-4338-8Search in Google Scholar

[5] E. Burman, P. Hansbo, and M. G. Larson, A cut finite element method with boundary value correction, Preprint arXiv:1507.03096 (2015).10.1090/mcom/3240Search in Google Scholar

[6] K. W. Cheng and T.-P. Fries, Higher-order XFEM for curved strong and weak discontinuities, Int. J. Num. Meth. Engrg. 82 (2010), No. 5, 564–590.10.1002/nme.2768Search in Google Scholar

[7] C.-C. Chu, I. G. Graham, and T.-Y. Hou, A new multiscale finite element method for high-contrast elliptic interface problems, Math. Comput. 79 (2010), 1915–1955.10.1090/S0025-5718-2010-02372-5Search in Google Scholar

[8] K. Deckelnick, C. Elliott, and T. Ranner, Unfitted finite element methods using bulk meshes for surface partial differential equations, SIAM J. Numer. Anal. 52 (2014), 2137–2162.10.1137/130948641Search in Google Scholar

[9] K. Dréau, N. Chevaugeon, and N. Moës, Studied X-FEM enrichment to handle material interfaces with higher order finite element, Comput. Meth. Appl. Mech. Engrg. 199 (2010), No. 29, 1922–1936.10.1016/j.cma.2010.01.021Search in Google Scholar

[10] C. M. Elliott and T. Ranner, Finite element analysis for a coupled bulk-surface partial differential equation, IMA J. Numer. Anal. 33 (2013), 377–402.10.1093/imanum/drs022Search in Google Scholar

[11] A. Ern and J.-L. Guermond, Finite element quasi-interpolation and best approximation, ESAIM: M2AN51 (2017), 1367–1385.10.1051/m2an/2016066Search in Google Scholar

[12] T.-P. Fries and S. Omerović, Higher-order accurate integration of implicit geometries, Int. J. Num. Meth. Engrg. (2015).10.1002/nme.5121Search in Google Scholar

[13] J. Grande, C. Lehrenfeld, and A. Reusken, Analysis of a high order trace finite element method for PDEs on level set surfaces, SIAMJ. Numer. Anal. 56 (2017), No. 1, 228–255.10.1137/16M1102203Search in Google Scholar

[14] J. Grande and A. Reusken, A higher order finite element method for partial differential equations on surfaces, SIAM J. Numer. Anal. 54 (2016), No. 1, 388–414.10.1137/14097820XSearch in Google Scholar

[15] S. Groß, V. Reichelt, and A. Reusken, A finite element based level set method for two-phase incompressible flows, Comput. Visual. Sci. 9 (2006), 239–257.10.1007/s00791-006-0024-ySearch in Google Scholar

[16] S. Groß and A. Reusken, An extended pressure finite element space for two-phase incompressible flows, J. Comput. Phys. 224 (2007), 40–58.10.1016/ in Google Scholar

[17] A. Hansbo and P. Hansbo, An unfitted finite element method, based on Nitsche’s method for elliptic interface problems, Comp. Methods Appl. Mech. Engrg. 191 (2002), No. 47, 5537–5552.10.1016/S0045-7825(02)00524-8Search in Google Scholar

[18] J. Huang and J. Zou, Some new a priori estimates for second-order elliptic and parabolic interface problems, J. Diff. Equ. 184 (2002), 570–586.10.1006/jdeq.2001.4154Search in Google Scholar

[19] F. Kummer and M. Oberlack, An extension of the discontinuous Galerkin method for the singular Poisson equation, SIAM J. Sci. Comput. 35 (2013), A603–A622.10.1137/120878586Search in Google Scholar

[20] O. Ladyzhenskaya and N. Ural’tseva, Linear and Quasilinear Equations of Elliptic Type, Nauka, Moscow, 1973 (in Russian).Search in Google Scholar

[21] P. L. Lederer, C.-M. Pfeiler, C. Wintersteiger, and C. Lehrenfeld, Higher order unfitted FEM for Stokes interface problems, Proc. Appl. Math. Mech. 16 (2016), No. 1, 7–10.10.1002/pamm.201610003Search in Google Scholar

[22] C. Lehrenfeld, High order unfitted finite element methods on level set domains using isoparametric mappings, Comp. Methods Appl. Mech. Engrg. 300 (2016), 716 – 733.10.1016/j.cma.2015.12.005Search in Google Scholar

[23] C. Lehrenfeld, A higher order isoparametric fictitious domain method for level set domains, Preprint arXiv:1612.02561 (2016).10.1007/978-3-319-71431-8_3Search in Google Scholar

[24] C. Lehrenfeld and A. Reusken, Analysis of a high order unfitted finite element method for elliptic interface problems, IMA J. Numer. Anal. 38 (2018), No. 3, 1351–1387.10.1093/imanum/drx041Search in Google Scholar

[25] M. Lenoir, Optimal isoparametric finite elements and error estimates for domains involving curved boundaries, SIAM J. Numer. Anal. 23 (1986), No. 3, 562–580.10.1137/0723036Search in Google Scholar

[26] B. Müller, F. Kummer, and M. Oberlack, Highly accurate surface and volume integration on implicit domains by means of moment-fitting, Int.J. Num. Meth. Engrg. 96 (2013), No. 8, 512–528.10.1002/nme.4569Search in Google Scholar

[27] M. A. Olshanskii, A. Reusken, and J. Grande, A finite element method for elliptic equations on surfaces, SIAM J. Numeri. Anal. 47 (2009), No. 5, 3339–3358.10.1137/080717602Search in Google Scholar

[28] P. Oswald, On a BPX-preconditioner for ℙ1 elements, Computing51 (1993), 125–133.10.1007/BF02243847Search in Google Scholar

[29] J. Parvizian, A. Düster, and E. Rank, Finite cell method, Comput. Mech. 41 (2007), No. 1, 121–133.10.1007/s00466-007-0173-ySearch in Google Scholar

[30] R. I. Saye, High-order quadrature method for implicitly defined surfaces and volumes in hyperrectangles, SIAM J. Sci. Comput. 37 (2015), No. 2, A993–A1019.10.1137/140966290Search in Google Scholar

[31] J. Schöberl, C++11 Implementation of Finite Elements in NGSolve, Inst. for Analysis and Scientific Computing, Report No. ASC-2014-30, 2014.Search in Google Scholar

[32] Y Sudhakar and W. A. Wall, Quadrature schemes for arbitrary convex/concave volumes and integration of weak form in enriched partition of unity methods, Comp. Meth. Appl. Mech. Engrg. 258 (2013), 39–54.10.1016/j.cma.2013.01.007Search in Google Scholar

Received: 2017-08-31
Revised: 2017-12-15
Accepted: 2017-12-17
Published Online: 2018-01-30
Published in Print: 2019-06-26

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 29.2.2024 from
Scroll to top button