#### Title

Optimal local approximation spaces for generalized finite element methods with application to multiscale problems

#### Document Type

Article

#### Publication Date

5-17-2011

#### Abstract

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∞(Ω). 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. 67- 103], [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 O into a collection of preselected subsets ωi, i = 1, 2, . . .m, and constructing finite-dimensional approximation spaces ωi 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 Ni in the energy norm over ωi. The local spaces Ψi are used within the GFEM scheme to produce a finite-dimensional subspace SN of H1(Ω), 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 ωi, it is shown that homogenization theory can be used to construct local approximation spaces with exponentially decreasing error in the preasymtotic regime. © 2011 Society for Industrial and Applied Mathematics.

#### Publication Source (Journal or Book title)

Multiscale Modeling and Simulation

#### First Page

373

#### Last Page

406

#### Recommended Citation

Babuska, I., & Lipton, R.
(2011). Optimal local approximation spaces for generalized finite element methods with application to multiscale problems.* Multiscale Modeling and Simulation**, 9* (1), 373-406.
https://doi.org/10.1137/100791051