×

zbMATH — the first resource for mathematics

\(hp\) submeshing via non-conforming finite element methods. (English) Zbl 0971.65101
The mixed boundary value problem in a bounded polygonal domain \(\Omega\) in \({\mathbb R}^2\) is considered for the Poisson equation. The authors give a review of grid methods with discontinuous approximations (mortar element methods) and pay special attention to the analysis of two variants of non-conforming finite element methods with piecewise polynomial approximations on triangular cells and different ways to approximate the continuity of the function on the cutting line (interface). Estimates of convergence are suggested which depend on the grid parameter and degree of used polynomials.
The authors present numerical results in details and conclude the paper by a desription of possible generalization for three-dimensional domains. There are remarks about optimality of the methods “what is meant that the method should perform as well as the conforming method”, but it is clear that it can not take the place since approximations do not be belong to the original energy space. Besides, the correctness of approximating problems is rather weak since the constant in the inf-sup condition (3.2) might tend to 0.

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
PDF BibTeX XML Cite
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
[2] Babuska, I.; Kellog, R.B.; Pitkaranta, J., Direct and inverse error estimates for finite elements with mesh refinement, Numer. math., 33, 447-471, (1979) · Zbl 0423.65057
[3] Babuška, I.; Suri, M., The p and h – p versions of the finite element method: basic principles and properties, SIAM rev., 36, 578-632, (1994) · Zbl 0813.65118
[4] F.B. Belgacem, The mortar finite element method with Lagrange Multipliers, Numer. Math., 2000 (to appear) · Zbl 0944.65114
[5] Belgacem, F.B.; Maday, Y., Non-conforming spectral element methodology tuned to parallel implementation, Comput. meth. appl. mech. engrg., 116, 59-67, (1994) · Zbl 0841.65096
[6] F.B. Belgacem, P. Seshaiyer, M. Suri, Optimal convergence rates of hp mortar finite element methods for second-order elliptic problems, RAIRO M2AN, 2000 (to appear) · Zbl 0956.65106
[7] C. Bernardi, Y. Maday, A.T. Patera, Domain decomposition by the mortar element method, in: H.G. Kaper, M. Garbey (Eds.), Asymptotic and Numerical Methods for PDEs with Critical Parameters, 1993, pp. 269-286 · Zbl 0799.65124
[8] F. Brezzi, L.D. Marini, Macro hybrid elements and domain decomposition methods, Proc. Colloque en l’honneur du 60eme anniversaire de Jean Cea, Sophia-Antipolis, 1992 · Zbl 0845.65060
[9] Cassarin, M.; Widlund, O.B., A heirarchical preconditioner for the mortar finite element method, ETNA, electron. trans. numer. anal., 4, 75-88, (1996)
[10] Ciarlet, P.G., The finite element method for elliptic problems, (1978), North-Holland Amsterdam · Zbl 0445.73043
[11] M. Dauge, Elliptic boundary value problems on corner domains, Lecture Notes in Mathematics, vol. 1341, Springer, New York, 1988 · Zbl 0668.35001
[12] Demkowicz, L.; Oden, J.T.; Rachowicz, W.; Hardy, O., Toward a universal h-p adaptive element strategy, part I: constrained approximation and data structure, Comput. methods appl. mech. engrg., 77, 79-113, (1989) · Zbl 0723.73074
[13] M. Dorr, On the discretization of inter-domain coupling in elliptic boundary-value problems via the p version of the finite element method, in: T.F. Chan, R. Glowinski, J. Periaux, O.B. Widlund (Eds.), Domain Decomposition methods, SIAM, 1989
[14] Gui, W.; Babuška, I., The hp version of the finite element method in one dimension, Numer. math., 3, 577-657, (1986) · Zbl 0614.65088
[15] B. Guo, I. Babuška, The hp version of the finite element method, Parts 1 and 2 Comput. Mech. 1 (1986) 21-41; 203-220
[16] J.T. Oden, A. Patra, Y. Fend, Domain decomposition for adaptive hp finite element methods, Contemporary Mathematics, vol. 180, AMS, Providence, RI, 1994, pp. 295-301 · Zbl 0819.41008
[17] 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
[18] P. Seshaiyer, Non-Conforming hp finite element methods, Ph.D. Dissertation, University of Maryland Baltimore County, 1998 · Zbl 0909.65075
[19] P. Seshaiyer, M. Suri, Uniform hp convergence results for the mortar finite element method, Math. Comp., 2000 (to appear) · Zbl 0944.65113
[20] 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
[21] 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
[22] 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.