Composite finite elements for elliptic interface problems

Peterseim, Daniel

doi:10.1090/S0025-5718-2014-02815-9

Composite finite elements for elliptic interface problems
HTML articles powered by AMS MathViewer

by Daniel Peterseim;

Math. Comp. 83 (2014), 2657-2674

DOI: https://doi.org/10.1090/S0025-5718-2014-02815-9

Published electronically: February 26, 2014

PDF | Request permission

Abstract:

A Composite Finite Element method approximates linear elliptic boundary value problems with discontinuous diffusion coefficient at possibly high contrast. The discontinuity appears at some interface that is not necessarily resolved by the underlying finite element mesh. The method is non-conforming in the sense that shape functions preserve continuity across the interface in only an approximate way. However, the method allows balancing this non-conformity error and the error of the best approximation in such a way that the total discretization error (in energy norm) decreases linear with regard to the mesh size and independent of contrast.

References

Roland Becker, Erik Burman, and Peter Hansbo, A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity, Comput. Methods Appl. Mech. Engrg. 198 (2009), no. 41-44, 3352–3360. MR 2571349, DOI 10.1016/j.cma.2009.06.017
Matthias Blumenfeld, The regularity of interface-problems on corner-regions, Singularities and constructive methods for their treatment (Oberwolfach, 1983) Lecture Notes in Math., vol. 1121, Springer, Berlin, 1985, pp. 38–54. MR 806385, DOI 10.1007/BFb0076261
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 3rd ed., Texts in Applied Mathematics, vol. 15, Springer, New York, 2008. MR 2373954, DOI 10.1007/978-0-387-75934-0
C.-C. Chu, I. G. Graham, and T.-Y. Hou, A new multiscale finite element method for high-contrast elliptic interface problems, Math. Comp. 79 (2010), no. 272, 1915–1955. MR 2684351, DOI 10.1090/S0025-5718-2010-02372-5
Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 520174
Monique Dauge and Serge Nicaise, Oblique derivative and interface problems on polygonal domains and networks, Comm. Partial Differential Equations 14 (1989), no. 8-9, 1147–1192. MR 1017069, DOI 10.1080/03605308908820649
W. Hackbusch and S. A. Sauter, Composite finite elements for problems containing small geometric details. Part II: Implementation and numerical results, Computing and Visualization in Science 1 (1997), 15–25.
W. Hackbusch and S. A. Sauter, Composite finite elements for the approximation of PDEs on domains with complicated micro-structures, Numer. Math. 75 (1997), no. 4, 447–472. MR 1431211, DOI 10.1007/s002110050248
Anita Hansbo and Peter Hansbo, An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems, Comput. Methods Appl. Mech. Engrg. 191 (2002), no. 47-48, 5537–5552. MR 1941489, DOI 10.1016/S0045-7825(02)00524-8
R. B. Kellogg, Singularities in interface problems, Numerical Solution of Partial Differential Equations, II (SYNSPADE 1970) (Proc. Sympos., Univ. of Maryland, College Park, Md., 1970) Academic Press, New York-London, 1971, pp. 351–400. MR 289923
D. Leguillon and E. Sánchez-Palencia, Computation of singular solutions in elliptic problems and elasticity, John Wiley & Sons, Ltd., Chichester; Masson, Paris, 1987. MR 995254
K. Lemrabet, An interface problem in a domain of $\textbf {R}^{3}$, J. Math. Anal. Appl. 63 (1978), no. 3, 549–562. MR 493687, DOI 10.1016/0022-247X(78)90059-8
Keddour Lemrabet, Régularité de la solution d’un problème de transmission, J. Math. Pures Appl. (9) 56 (1977), no. 1, 1–38 (French). MR 509312
Jingzhi Li, Jens Markus Melenk, Barbara Wohlmuth, and Jun Zou, Optimal a priori estimates for higher order finite elements for elliptic interface problems, Appl. Numer. Math. 60 (2010), no. 1-2, 19–37. MR 2566075, DOI 10.1016/j.apnum.2009.08.005
M. Ohlberger and M. Rumpf, Hierarchical and adaptive visualization on nested grids, Computing 59 (1997), no. 4, 365–385. MR 1486386, DOI 10.1007/BF02684418
D. Peterseim, The composite mini element: A mixed FEM for the Stokes equations on complicated domains, Ph.D. thesis, Universität Zürich, http://www.dissertationen.uzh.ch/, 2007.
Daniel Peterseim, Robustness of finite element simulations in densely packed random particle composites, Netw. Heterog. Media 7 (2012), no. 1, 113–126. MR 2908612, DOI 10.3934/nhm.2012.7.113
Daniel Peterseim and Stefan A. Sauter, The composite mini element-coarse mesh computation of Stokes flows on complicated domains, SIAM J. Numer. Anal. 46 (2008), no. 6, 3181–3206. MR 2439507, DOI 10.1137/070704356
Daniel Peterseim and Stefan A. Sauter, Finite element methods for the Stokes problem on complicated domains, Comput. Methods Appl. Mech. Engrg. 200 (2011), no. 33-36, 2611–2623. MR 2812028, DOI 10.1016/j.cma.2011.04.017
D. Peterseim and S. Sauter, Finite elements for elliptic problems with highly varying, nonperiodic diffusion matrix, Multiscale Model. Simul. 10 (2012), no. 3, 665–695. MR 3022017, DOI 10.1137/10081839X
M. Rech, Composite finite elements: An adaptive two-scale approach to the non-conforming approximation of Dirichlet problems on complicated domains, Ph.D. thesis, Universität Zürich, 2006.
M. Rech, S. Sauter, and A. Smolianski, Two-scale composite finite element method for Dirichlet problems on complicated domains, Numer. Math. 102 (2006), no. 4, 681–708. MR 2207285, DOI 10.1007/s00211-005-0654-x
S. A. Sauter and R. Warnke, Extension operators and approximation on domains containing small geometric details, East-West J. Numer. Math. 7 (1999), no. 1, 61–77. MR 1683936
S. A. Sauter and R. Warnke, Composite finite elements for elliptic boundary value problems with discontinuous coefficients, Computing 77 (2006), no. 1, 29–55. MR 2207954, DOI 10.1007/s00607-005-0150-2
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, NJ, 1970. MR 290095