×

Trefftz method: A general theory. (English) Zbl 0978.65114

The article is concerned with an overview on (extensions of) the Trefftz method for the numerical solution of boundary value problems with linear ordinary or partial differential equations, where in addition, jumps can be prescribed at interior interfaces in the domain. The methods are based on a domain decomposition idea and on the choice of suitable basis and test functions. A distinction is made between “direct” and “indirect”, and between overlapping and non-overlapping methods.
Reviewer: M.Plum (Karlsruhe)

MSC:

65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
35J40 Boundary value problems for higher-order elliptic equations
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Ein Gegenstück zum Ritzschen Verfahren, Proc 2nd Int Cong Appl Mech Zurich, 1926, pp. 131-137.
[2] Jirousek, Comp Meth Appl Mech Eng 12 pp 77– (1977)
[3] Jirousek, Comp Meth Appl Mech Eng 14 pp 65– (1978)
[4] Celia, Numer Methods Partial Differential Eq 3 pp 117– (1987)
[5] Celia, Numer Methods Partial Differential Eq 5 pp 203– (1989)
[6] Celia, Adv Water Res 13 pp 187– (1990)
[7] Herrera, Numer Methods Partial Differential Eq 10 pp 205– (1994)
[8] Herrera, KINAM 3 pp 161– (1981)
[9] Boundary methods. An algebraic theory, Pitman Boston, 1984.
[10] Boundary methods for fluids, Vol. IV, and editors, Finite elements in fluids, Wiley, London, 1982, pp. 403-432.
[11] Trefftz method, editor, Topics in boundary element research, Vol. 1: Basic principles and applications, Springer?Verlag, New York, 1984, pp. 225-253.
[12] Herrera, Proc Nat Acad Sci USA 77 pp 4395– (1980)
[13] Herrera, Numer Methods Partial Differential Eq 1 pp 12– (1985)
[14] Herrera, Numer Methods Partial Differential Eq 1 pp 159– (1985)
[15] Herrera, Numer Methods Partial Differential Eq 1 pp 241– (1985)
[16] Some unifying concepts in applied mathematics, and editors, The merging of disciplines: New directions in pure, applied, and computational mathematics, Springer?Verlag, New York, 1986, pp. 79-88.
[17] Herrera, Numer Methods Partial Differential Eq 3 pp 199– (1987)
[18] Herrera, Numer Methods Partial Differential Eq 9 pp 431– (1993)
[19] Localized adjoint methods: a new discretization methodology, and editors, Computational methods in geosciences, SIAM, Philadelphia, 1992, pp. 66-77.
[20] Herrera, Adv Eng Soft 24 pp 43– (1995)
[21] Herrera, Polish Acad Sci 4 pp 369– (1997)
[22] Herrera, Numer Methods Partial Differential Eq 15 pp 709– (1999)
[23] Sánchez Sesma, Bull Seismol Soc Am 72 pp 473– (1982)
[24] and (editors), Domain decomposition methods, Proc Second Int Symp Domain Decomposition Methods, Los Angeles, California, SIAM, Philadelphia, 1989. · Zbl 0805.65116
[25] and (editors), Domain decomposition methods for partial differential equations, Proc Third Int Symp Domain Decomposition Methods Partial Differential Eq, Houston, Texas, SIAM, Philadelphia, 1990.
[26] and (editors), Domain decomposition methods for partial differential equations, Proc First Int Symp Domain Decomposition Methods Partial Differential Eq, Paris, France, SIAM, Philadelphia, 1988.
[27] and (editors), Domain decomposition methods for partial differential equations, Proc Fourth Int Symp Domain Decomposition Methods Partial Differential Eq, Moscow, SIAM, Philadelphia, 1991.
[28] and (editors), Computing methods in applied sciences and engineering, Proc Ninth Int Conf Comp Methods Appl Sci Eng, Paris, France, SIAM, Philadelphia, 1990.
[29] and (editors), Domain decomposition methods for partial differential equations, Proc Fifth Int Symp Domain Decomposition Methods Partial Differential Eq, Norfolk, Virginia, SIAM, Philadelphia, 1992.
[30] (editors), Domain decomposition methods in scientific and engineering computing, Proc Seventh Int Conf Domain Decomp, Penn State Univ, AMS, Providence, RI, 1994. · Zbl 0809.00026
[31] and (editors), Domain decomposition methods in science and engineering, Proc Sixth Int Conf Domain Decomposition, Como, Italy, AMS, Providence, RI, 1994. · Zbl 0785.00036
[32] Collocation from a broad perspective, Comp Meths Water Res, Vol. 2, et al., editors, Balkema, Rotterdam, pp. 661-667, 2000.
[33] Jirousek, Int J Numer Meth Eng 23 pp 651– (1986)
[34] Jirousek, Comp Struct 34 pp 51– (1990)
[35] Jirousek, Int J Numer Meth Eng 38 pp 2619– (1995)
[36] Jirousek, Comp Meth Appl Mech Eng · Zbl 0985.65148
[37] Jirousek, Int J Numer Meth Eng 24 pp 1367– (1987)
[38] Jirousek, Int J Numer Meth Eng 28 pp 431– (1989)
[39] and Application of the hybrid-Trefftz finite element model to thin shell analysis, Proc Euro Conf New Adv Comp Struc Mech, and (editors), Elsevier, Giens, France, 1991, pp. 547-554.
[40] Jirousek, Comp Struct 15 pp 575– (1982)
[41] Jirousek, Int J Numer Meth Eng 28 pp 211– (1989)
[42] and Hybrid-Trefftz p-element for 3-D axisymmetric problems of elasticity, Numer Methods Eng ’92, Proc First Euro Conf Numer Meth Eng, and (editors), Brussels, Elsevier, 1992, pp. 803-810.
[43] and Application of hybrid-Trefftz p-elements to stress analysis in shafts, Proc XI Polish Conf Comp Meth Mech Kielce?Cedzyna, Poland, 2 (1993), 983-990.
[44] and Application of hybrid-Trefftz element approach to transient heat conduction analysis, Int Rep LSC 94/10, Swiss Fed Inst Tech, Lausanne, 1994.
[45] Jirousek, Comp Struct 37 pp 217– (1990)
[46] Jirousek, Int J Numer Meth Eng 29 pp 391– (1990)
[47] and Mesh design and reliability assurance in hybrid-Trefftz p-element approach, Int Rep LSC 94/15, Swiss Fed Inst Tech, Laussanne, 1994.
[48] Jirousek, Arch Comp Meth Eng State Art Rev 3 pp 323– (1996)
[49] Jirousek, Comp Struct 63 pp 225– (1997)
[50] Zielinski, Int J Numer Methods Eng 24 pp 871– (1987)
[51] Petrolito, Comp Meth Appl Mech Eng 78 pp 331– (1990)
[52] Piltner, Int J Numer Methods Eng 33 pp 387– (1992)
[53] Piltner, J Elast 22 pp 45– (1989)
[54] Zielinski, Int J Numerical Meth Eng 21 pp 509– (1985)
[55] Berlanga, Applic Anal Int
[56] and Advanced calculus, Addison?Wesley, Reading, MA, 1968.
[57] On operator extensions: the algebraic theory approach, Advances in optimization and numerical analysis, Proc VI Workshop Optim Numer Anal, Oaxaca, Oax, México, January, 1992), Mathematics and its applications, Kluwer Academic, New York, 1992, pp. 155-163.
[58] and Operator ext. & Green?Herrera formulas, to appear.
[59] and Collocation: direct methods, to appear.
[60] and (1988), Numerical modeling in science and engineering, Wiley, New York, 1988. · Zbl 0636.65128
[61] On the Schwarz alternating method, First Int Symp Domain Decomposition Methods Partial Differential Eq, et al. (editors), 1987, pp. 1-42.
[62] Innovative discretization methodologies based on LAM, et al. (editors), Finite elements in fluids: New trends and applications, Vol. 2, Pineridge, 1993, pp. 1437-1447.
[63] and Transformations, transmutations, and kernel functions, Vol. 1, Longman Scientific & Technical, 1992. · Zbl 0827.35001
[64] Integral operators in the theory of linear partial differential equations, Ergeb Math Grenzgeb 23, Springer, Berlin, 1961; 2, rev. print, 1969. · Zbl 0093.28701
[65] New methods for solving elliptic equations, North Holland, Amsterdam, and Wiley, New York, 1967.
[66] Partial differential equations in the complex domain, Pitman, London. 1976a.
[67] Solution of boundary value problems by the method of integral operators, Pitman, London, 1976b.
[68] Analytic theory of partial differential equations, Pitman, Boston, 1980.
[69] Function theoretic methods in partial differential equations, Academic, New York. 1969. · Zbl 0187.35303
[70] Constructive methods for elliptic partial differential equations, Lecture Notes in Math, 365, 1974.
[71] and Methods of complex analysis in partial differential equations with applications, Wiley, New York, 1988.
[72] Complex integral operators in mathematical physics, Akademie?Verlag, Berlin, to appear.
[73] Herrera, Comp Meth Appl Mech Eng 30 pp 225– (1982)
[74] and Boundary methods. C-complete systems for biharmonic equations, editor, Boundary element methods, Springer?Verlag, Berlin, 1981, pp. 431-441.
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.