zbMATH — the first resource for mathematics

Interior boundary conditions in the Schwarz alternating method for the Trefftz method. (English) Zbl 1182.74255
Summary: New types of the Schwarz alternating methods (SAMs) with overlapping and non-overlapping are proposed, by using different interior boundary conditions, such as the Dirichlet, the Neumann and the Robin conditions. Those SAMs using different interior boundary conditions are called the mixed SAMs. This paper consists of two parts. In the first part, for a simple continuous model: the Laplacian solutions on a sectorial domain, the convergence rates of the mixed SAMs are derived in detail for different interior boundary conditions. Compared with the classic overlapping SAM with the Dirichlet conditions, some of mixed SAMs with the interior Neumann or Robin conditions will converge faster. Moreover, a number of new and better SAMs than those in the existing literature are explored in this paper. In the second part, for solving Motz’s problem, the Trefftz method using particular solutions and the finite difference method (FDM) are combined, and the mixed SAMs are applied to implement the algorithms into parallel. Numerical results by the mixed SAMs are given to support the analysis made from the simple models. Hence, we may employ some mixed SAMs proposed in this paper to speed the SAM convergence rates.

74S30 Other numerical methods in solid mechanics (MSC2010)
74S20 Finite difference methods applied to problems in solid mechanics
Full Text: DOI
[1] Badea, L., A generalization of the Schwarz alternating method to an arbitrary number of subdomain, Numer math, 55, 61-81, (1989) · Zbl 0633.65029
[2] Chan, T.F.; Hou, T.Y.; Lions, P.L., Geometry related convergence results for domain decomposition algorithm problems, SIAM J numer anal, 28, 378-391, (1991) · Zbl 0724.65109
[3] Douglas, J.; Leme, P.J.P.; Roberts, J.E.; Wang, J., A parallel iterative procedure applicable to the approximate solution of second order partial differential equations by mixed element methods, Numer math, 65, 95-108, (1993) · Zbl 0813.65122
[4] Douglas, J.; Huang, C.S., An accelerated domain decomposition procedure based on Robin transmission conditions, Bit, 37, 678-686, (1997) · Zbl 0886.65114
[5] Dryja, M.; Widlund, O.B., Domain decomposition algorithms with small overlap, SIAM J sci comput, 15, 604-620, (1994) · Zbl 0802.65119
[6] Dryja, M.; Smith, B.F.; Widlund, O.B., Schwarz analysis of iterating substructing alternating algorithms for elliptic problems is three dimensions, SIAM J numer anal, 31, 1662-1694, (1994) · Zbl 0818.65114
[7] Guo, B.; Cao, W., Additive Schwarz methods for the h – p version of the finite element method in two dimensions, SIAM J sci comput, 18, 1267-1288, (1997) · Zbl 0892.65072
[8] Hageman, L.A.; Young, D.M., Applied iterative methods, (1981), Academic Press New York · Zbl 0459.65014
[9] Hall, C.A.; Porsching, T.A., Numerical analysis of partial differential equations, (1990), Prentice-Hall Englewood Cliffs, NJ · Zbl 0807.65093
[10] Holst, M.; Vandewalle, S., Schwarz methods to symmetrize or not to symmetrize, SIAM J numer anal, 34, 699-722, (1997) · Zbl 0880.65090
[11] Huang, C.S.; Wang, N.C., An accelerated domain decomposition procedure for mixed Schwarz alternating method, Technical report, (2003), National Sun Yat-sen University Kaohsiung, Taiwan
[12] Li, Z.C., The Schwarz alternating method for the singularity problems, SIAM J sci comput, 15, 5, 1064-1082, (1994) · Zbl 0813.65129
[13] Li, Z.C., A nonconforming combined method for solving Laplace’s boundary value problems with singularities, Numer math, 49, 475-497, (1986) · Zbl 0586.65075
[14] Li, Z.C., Penalty combinations of the Ritz-Galerkin and finite difference methods for singularity problems, J comput appl math, 81, 1-13, (1997) · Zbl 0886.65106
[15] Li, Z.C., Combined methods for elliptic equations with singularities, infinities, (1998), Kluwer Boston, London · Zbl 0909.65079
[16] Lions, P.L., On the Schwarz alternating method I, (), 1-61
[17] Lions, P.L., On the Schwarz alternating method II, stochastic interpretation and order properties, (), 47-70
[18] Lions, P.L., On the Schwarz alternating method III, a variant for non-overlapping subdomains, (), 202-223
[19] Lu TT, Hu HY, Li ZC. Highly accurate solutions of Motz’s and cracked beam problems, accepted by Engineering Analysis with Boundary Elements; 2004;8:1387-1403. · Zbl 1074.74655
[20] Lü, T.; Shin, T.M.; Liem, C., Domain decomposition methods: techniques of numerical solution for PDE, (1992), Scientific Publishing Beijing, (in Chinese)
[21] Mathew, T.P.; Mathew, T.P., Schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems, part II, Numer math, Theory numer math, 65, 469-492, (1993) · Zbl 0801.65107
[22] Matsokin, A.M.; Nepomnyaschikh, S.V., A Schwarz alternating method in a subspace, Soviet math, 29, 78-84, (1985) · Zbl 0611.35017
[23] Miller, K., Numerical analogy to the Schwarz alternating procedure, Numer math, 7, 91-103, (1965) · Zbl 0154.41201
[24] Pararino, L.F., Additive Schwarz methods for the p-version finite element method, Numer math, 66, 493-515, (1994) · Zbl 0791.65083
[25] Schwarz, H.A., Uber einige abbildungsaufgaben, Ges math abh, 1, 65-83, (1869)
[26] Smith, B.; Bjorstad, P.; Gropp, W., Domain decomposition, parallel multilevel methods for elliptic partial differential equations, (1996), Cambridge University Press Cambridge, MA · Zbl 0857.65126
[27] Varga, R.S., Matrix iterative analysis, (1962), Prentice-Hall Englewood Cliffs, NY · Zbl 0133.08602
[28] Yanik, E.G., A Schwarz alternating procedure using spline collection methods, Int J numer meth eng, 28, 621-627, (1989) · Zbl 0676.65113
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.