OPTIMAL LOCAL APPROXIMATION SPACES FOR GENERALIZED FINITE ELEMENT METHODS WITH APPLICATION TO MULTISCALE PROBLEMS

被引:128
作者
Babuska, Ivo [1 ,2 ]
Lipton, Robert [3 ]
机构
[1] Univ Texas Austin, Inst Computat Engn & Sci, Austin, TX 78712 USA
[2] Univ Texas Austin, Dept Aerosp Engn, Austin, TX 78712 USA
[3] Louisiana State Univ, Dept Math, Baton Rouge, LA 70803 USA
基金
美国国家科学基金会;
关键词
generalized finite elements; partition of unity method; multiscale finite element method; fiber reinforced composites; Kolmogorov n-width; ELLIPTIC PROBLEMS; POROUS-MEDIA; FEM;
D O I
10.1137/100791051
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
The paper addresses a numerical method for solving second order elliptic partial differential equations that describe fields inside heterogeneous media. The scope is general and treats the case of rough coefficients, i.e., coefficients with values in L-infinity(Omega). This class of coefficients includes as examples media with microstructure as well as media with multiple nonseparated length scales. The approach taken here is based on the the generalized finite element method (GFEM) introduced in [I. Babuska, G. Caloz, and J. E. Osborn, SIAM J. Numer. Anal., 31 (1994), pp. 945-981] and elaborated in [I. Babuska, U. Banerjee, and J. Osborn, Int. J. Comput. Methods, 1 (2004), pp. 67103], [I. Babuska and J. Melenk, Internat. J. Numer. Methods Engrg., 40 (1997), pp. 727-758], and [T. Strouboulis, I. Babuska, and K. Copps, Comput. Methods Appl. Mech. Engrg., 181 (2001), pp. 43-69]. The GFEM is constructed by partitioning the computational domain Omega into a collection of preselected subsets w(i), i = 1, 2,... m, and constructing finite-dimensional approximation spaces Psi(i) over each subset using local information. The notion of the Kolmogorov n-width is used to identify the optimal local approximation spaces. These spaces deliver local approximations with errors that decay almost exponentially with the degrees of freedom N-i in the energy norm over w(i). The local spaces Psi(i) are used within the GFEM scheme to produce a finite-dimensional subspace S-N of H-1(Omega), which is then employed in the Galerkin method. It is shown that the error in the Galerkin approximation decays in the energy norm almost exponentially (i.e., superalgebraically) with respect to the degrees of freedom N. When length scales "separate" and the microstructure is sufficiently fine with respect to the length scale of the domain w(i), it is shown that homogenization theory can be used to construct local approximation spaces with exponentially decreasing error in the preasymtotic regime.
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页码:373 / 406
页数:34
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