Construction of explicit extension operators on general finite element grids. (English) Zbl 1015.65069
This paper deals with an approach for constructing energy-preserving explicit operators on quasi-uniform finite element grids defined in a general polygonal domain. A theoretical analysis of the energy-preserving property of this operator is presented. The authors describe one application of an explicit extension operator in the construction of non-overlapping domain decomposition methods with inexact subdomain solvers.
Reviewer: Pavol Chocholatý (Bratislava)
MSC:
| 65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |
| 35J25 | Boundary value problems for second-order elliptic equations |
References:
| [1] | Babuska, I.; Aziz, A., Survey lectures on the mathematical foundations of the finite element method, (Aziz, A., The Mathematical Foundations of Finite Element Method with Applications to Partial Differential Equations (1972), Academic Press: Academic Press New York) · Zbl 0268.65052 |
| [2] | Bergh, J.; Lofstrom, J., Interpolation Space: An Introduction (1976), Springer: Springer Berlin · Zbl 0344.46071 |
| [3] | Bramble, J. H.; Pasciak, J. E.; Vassilev, A. T., Analysis of non-overlapping domain decomposition algorithms with inexact solves, Math. Comp., 67, 1-19 (1998) · Zbl 0902.65067 |
| [4] | Brenner, S. C.; Scott, L. R., The Mathematical Theory of Finite Element Methods (1994), Springer: Springer Berlin · Zbl 0804.65101 |
| [5] | Brenner, S. C.; Sung, L. Y., Balancing domain decomposition for nonconforming plate elements, Numer. Math., 83, 25-52 (1999) · Zbl 0937.74060 |
| [6] | Chan, T. F.; Go, S.; Zou, J., Boundary treatments for multilevel methods on unstructured meshes, SIAM J. Sci. Comput., 21, 46-66 (1999) · Zbl 0947.65131 |
| [7] | Chan, T. F.; Zou, J., A convergence theory of multilevel additive Schwarz methods on unstructured meshes, Numer. Algorithms, 13, 365-398 (1996) · Zbl 0872.65097 |
| [8] | Chan, T. F.; Zou, J., Additive Schwarz domain decomposition methods for elliptic problems on unstructured meshes, Numer. Algorithms, 8, 329-346 (1994) · Zbl 0815.65127 |
| [9] | Ciarlet, P. G., The Finite Element Method for Elliptic Problems (1978), North-Holland: North-Holland Amsterdam · Zbl 0445.73043 |
| [10] | Dryja, M.; Widlund, O. B., Domain decomposition algorithms with small overlap, SIAM J. Sci. Comput., 15, 604-620 (1994) · Zbl 0802.65119 |
| [11] | Grisvard, P., Elliptic Problems in Nonsmooth Domains (1985), Pitman Advanced Publishing Program: Pitman Advanced Publishing Program Boston · Zbl 0695.35060 |
| [12] | Necas, J., Les methodes directes en theorie des equations elliptiques (1967), Academia: Academia Prague · Zbl 1225.35003 |
| [13] | Hasse, G.; Langer, U.; Meyer, A.; Nepomnyaschikh, S. V., Hierarchical extension operators and local multigrid methods in domain decomposition preconditioners, East-West J. Numer. Math., 2, 173-193 (1994) · Zbl 0849.65089 |
| [14] | Hasse, G.; Nepomnyaschikh, S. V., Explicit extension operators on hierarchical grids, East-West J. Numer. Math., 5, 231-248 (1997) · Zbl 0894.65061 |
| [15] | Lions, J. L.; Magenes, E., Nonhomogeneous Boundary Value Problems and Applications, I (1972), Springer: Springer Berlin · Zbl 0223.35039 |
| [16] | Matsokin, A. M.; Nepomnyaschikh, S. V., Norms in the space of traces of mesh functions, Sov. J. Numer. Anal. Math. Modelling, 3, 199-216 (1988) · Zbl 0825.65024 |
| [17] | Matsokin, A. M.; Nepomnyaschikh, S. V., The fictitious domain method and explicit continuation operators, Comput. Math. Math. Phys., 33, 45-59 (1993) · Zbl 0803.65117 |
| [19] | Nepomnyaschikh, S. V., Method of splitting into subspaces for solving elliptic boundary value problems in complex-form domains, Sov. J. Numer. Anal. Math. Modelling, 6, 151-168 (1991) · Zbl 0816.65096 |
| [20] | Nepomnyaschikh, S. V., Domain decomposition and fictitious domains methods for elliptic boundary value problems, (Keyes, D. E.; Chan, T. F.; Meurant, G.; Scroggs, J. S.; Voigt, R. G., Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations (1992), SIAM: SIAM Philadelphia, PA), 67-72 · Zbl 0770.65089 |
| [22] | Xu, J.; Zou, J., Non-overlapping domain decomposition methods, SIAM Rev., 40, 857-914 (1998) · Zbl 0913.65115 |
| [23] | Zlamal, M., On the finite element method, Numer. Math., 12, 394-409 (1968) · Zbl 0176.16001 |
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.
