zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
The partition of unity finite element method: Basic theory and applications. (English) Zbl 0881.65099
Summary: The paper presents the basic ideas and the mathematical foundation of the partition of unity finite element method (PUFEM). We will show how the PUFEM can be used to employ the structure of the differential equation under consideration to construct effective and robust methods. Although the method and its theory are valid in $n$ dimensions, a detailed and illustrative analysis will be given for a one-dimensional model problem. We identify some classes of non-standard problems which can profit highly from the advantages of the PUFEM and conclude this paper with some open questions concerning implementational aspects of the PUFEM.

MSC:
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
35J25Second order elliptic equations, boundary value problems
WorldCat.org
Full Text: DOI
References:
[1] Aziz, A. K.; Babuška, I. M.: Mathematical foundations of the finite element method with applications to partial differential equations. (1972)
[2] I. Babuška and J.E. Osborn, Private communication.
[3] Bergman, S.: Integral operators in the theory of linear partial differential equations. (1961) · Zbl 0093.28701
[4] Freund, R. W.: A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems. SIAM J. Sci. comput. 14, No. 2, 470-482 (1993) · Zbl 0781.65022
[5] Gradshtein, I. S.: Table of integrals, series and products. (1980)
[6] Herrera, I.: Boundary methods: an algebraic theory. (1984) · Zbl 0549.35004
[7] Kress, R.: Linear integral equations. (1989) · Zbl 0671.45001
[8] Melenk, J. M.: On generalized finite element methods. Ph.d. thesis (1995)
[9] Mergelyan, S. N.: Uniform approximation to functions of a complex variable. 3 of 1, 294-391 (1962)
[10] Muskhelishvili, N. I.: Some basic problems of the mathematical theory of elasticity. (1963) · Zbl 0124.17404
[11] Oh, Hae-Soo; Babuška, I.: The p-version of the finite element method for the elliptic boundary value problems with interfaces. Comput. methods appl. Mech. engrg. 97, 211-231 (1992) · Zbl 0762.65059
[12] Oh, Hae-Soo; Babuška, I.: The method of auxiliary mapping for the finite element solutions of elasticity problems containing singularities. J. comput. Phys. 121, No. 2, 193-212 (1995) · Zbl 0833.73061
[13] Babuška, I.; Caloz, G.; Osborn, J.: Special finite element methods for a class of second order elliptic problems with rough coefficients. SIAM J. Numer. anal. 31, 945-981 (1994) · Zbl 0807.65114
[14] Babuška, I.; Ihlenburg, F.; Paik, E.; Sauter, S.: A generalized finite element method for solving the Helmholtz equation in two dimensions with minimal pollution. Comput. methods appl. Mech. engrg. 128, 325-360 (1995) · Zbl 0863.73055
[15] I. Babuška and J.M. Melenk, The partition of unity method, Int. J. Numer. Methods Engrg., in press. · Zbl 0949.65117
[16] I. Babuška and Z. Zhang, The partition of unity finite element method for the elastically supported beam, to appear.
[17] Szegö, G.: Über polynome, die zu einer gegebenen kurve der komplexen ebene gehören. Mathematische zeitschrift 9, 218-270 (1921) · Zbl 48.0374.04
[18] Thomson, L. L.; Pinsky, P. M.: A Galerkin least squares finite element method for the two-dimensional Helmholtz equation. Int. J. Numer. methods engrg. 38, 371-397 (1995) · Zbl 0844.76060
[19] Vekua, I. N.: New methods for solving elliptic equations. (1967) · Zbl 0146.34301
[20] Walsh, J. L.: Interpolation and approximation by rational functions in the complex domain. Colloquium publications 20 (1960) · Zbl 0106.28104