×
Seshaiyer, Padmanabhan; Smith, Philip

A non-conforming finite element method for sub-meshing. (English) Zbl 1033.65106

Appl. Math. Comput. 139, No. 1, 85-100 (2003).
The authors study non-conforming finite elements in the discretization of the Laplacian, where the solution domain is subdivided by regions with different degrees of refinement. The use of non-conforming elements is of special interest when the different levels of refinement do not match across the boundaries of the subdivision.
The authors analyse a model problem consisting of the Laplacian in a domain formed by the union of two rectangles with mixed Dirichlet and Neumann boundary conditions. For the transition between the two subdomains interface elements are used. The main result of the paper is a bound for the error in the energy norm by the sum of the approximation and the consistency error. Some numerical results are presented for both the \(h\)- and the \(p\)-version of finite elements.
Reviewer: Rolf Dieter Grigorieff (Berlin)

Cited in 4 Documents

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N15 Error bounds for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs

Keywords:

submeshing; non-conforming finite element; \(h\)-version; \(p\)-version; error estimate; Laplace equation; mesh refinement; mixed Dirichlet and Neumann boundary conditions; numerical results
Cite Review PDF
Full Text: DOI

References:

[1] M.A. Aminpour, S.L. McClearly, J.B. Ransom, A global/local analysis method for treating details in structural design. in: Proceedings of the Third NASA advanced composites technology conference, NASA CP-3178, vol. 1, Part 2, 1992, pp. 967-986; M.A. Aminpour, S.L. McClearly, J.B. Ransom, A global/local analysis method for treating details in structural design. in: Proceedings of the Third NASA advanced composites technology conference, NASA CP-3178, vol. 1, Part 2, 1992, pp. 967-986
[2] Babuška, I.; Suri, M., The \(p\) and hp versions of the finite element method: basic principles and properties, SIAM Rev., 36, 578-632 (1994) · Zbl 0813.65118
[3] Ben Belgacem, F., The mortar finite element method with Lagrange multipliers, Numer. Math., 84, 173-197 (1999) · Zbl 0944.65114
[4] Ben Belgacem, F.; Maday, Y., Non-conforming spectral element methodology tuned to parallel implementation, Comp. Meth. Appl. Mech. Engrg., 116, 59-67 (1994) · Zbl 0841.65096
[5] Ben Belgacem, F.; Seshaiyer, P.; Suri, M., Optimal convergence of hp mortar finite element methods for second order elliptic problems, RAIRO Math. Mod. Numer. Anal., 34, 591-608 (2000) · Zbl 0956.65106
[6] Bernardi, C.; Maday, Y.; Patera, A. T., Domain decomposition by the mortar element method, (Kaper, H. G.; Garbey, M., Asymptotic and Numerical Methods for PDEs with Critical Parameters (1993)), 269-286 · Zbl 0799.65124
[7] F. Brezzi, L.D. Marini, Macro hybrid elements and domain decomposition methods, in: Proc. Colloque en l’honneur du 60eme anniversaire de Jean Cea, Sophia-Antipolis, 1992; F. Brezzi, L.D. Marini, Macro hybrid elements and domain decomposition methods, in: Proc. Colloque en l’honneur du 60eme anniversaire de Jean Cea, Sophia-Antipolis, 1992 · Zbl 0845.65060
[8] M. Dorr. On the discretization of inter-domain coupling in elliptic boundary-value problems via the \(p\); M. Dorr. On the discretization of inter-domain coupling in elliptic boundary-value problems via the \(p\)
[9] Farhat, C.; Roux, F. X., A method of finite element tearing and interconnecting and its parallel solution algorithm, Int. J. Numer. Meth. Engng., 32, 6, 1205-1228 (1991) · Zbl 0758.65075
[10] Oullette, D., Schur complements and statistics, LAA, 36, 187-295 (1981) · Zbl 0455.15012
[11] Raviart, P. A.; Thomas, J. M., Primal hybrid finite element methods for 2nd order elliptic equations, Math. Comp., 31, 391-396 (1977) · Zbl 0364.65082
[12] J.E. Schiermeier, J.M. Housner, M.A. Aminpour, W.J. Stroud, The application of interface elements to dissimilar meshes in global/local analysis, in: Proceedings of the 1996 MSC World Users’ Conference, 1996; J.E. Schiermeier, J.M. Housner, M.A. Aminpour, W.J. Stroud, The application of interface elements to dissimilar meshes in global/local analysis, in: Proceedings of the 1996 MSC World Users’ Conference, 1996
[13] J.E. Schiermeier, J.M. Housner, M.A. Aminpour, W.J. Stroud, Interface elements in global/local analysis-Part 2: Surface interface elements, in: Proceedings of the 1997 MSC World Users’ Conference, 1997; J.E. Schiermeier, J.M. Housner, M.A. Aminpour, W.J. Stroud, Interface elements in global/local analysis-Part 2: Surface interface elements, in: Proceedings of the 1997 MSC World Users’ Conference, 1997
[14] Schur, I., Uber Potenzreihen, die im Innern des Enheitskreises beschrankt sind, J. Reine Angew. Math., 147, 205 (1917) · JFM 46.0475.01
[15] Seshaiyer, P.; Suri, M., Convergence results for non-conforming hp methods: the mortar finite element method, Cont. Math., 218, 467-473 (1998)
[16] Seshaiyer, P.; Suri, M., Uniform hp convergence results for the mortar finite element method, Math. Comp., 69, 521-546 (2000) · Zbl 0944.65113
[17] Seshaiyer, P.; Suri, M., hp submeshing via non-conforming finite element methods, Comp. Meth. Appl. Mech. Engrg., 189, 1011-1030 (2000) · Zbl 0971.65101
[18] Swann, H., On the use of lagrange multipliers in domain decomposition for solving elliptic problems, Math. Comp., 60, 49-78 (1993) · Zbl 0795.65073
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.