Ivo Babuska

Ivo Babuska, Robert Lipton

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 $L^\infty(Ω)$. This class of coefficients includes as examples media with micro-structure as well as media with multiple non-separated length scales. The approach taken here is based on the the generalized finite element method (GFEM) introduced in \cite{107}, and elaborated in \cite{102}, \cite{103} and \cite{104}. The GFEM is constructed by partitioning the computational domain $Ω$ into to a collection of preselected subsets $ω_{i},i=1,2,..m$ and constructing finite dimensional approximation spaces $Ψ_{i}$ 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 $ω_i$. The local spaces $% Ψ_{i}$ are used within the GFEM scheme to produce a finite dimensional subspace $S^N$ of $H^{1}(Ω)$ 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., super-algebraicly) 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 $ω_i$ it is shown that homogenization theory can be used to construct local approximation spaces with exponentially decreasing error in the pre-asymtotic regime.

Ivo Babuska, Mohammad Motamed

We study mathematical and computational models for computing the deformation of fiber-reinforced cross-plied laminates due to external forces. This requires an understanding of both micro-structural effects and different sources of uncertainty in the problem. We first show that the uncertainties in the problem are of both statistical (aleatoric) and systematic (epistemic) types and that current multiscale stochastic models, such as stationary random fields, which are based on precise probability theory, are not capable of correctly characterizing uncertainty in fiber composites. Next, we motivate the applicability of models based on imprecise uncertainty theory and present a novel fuzzy-stochastic model, which can more accurately describe uncertainties in fiber composites. The new model is constructed by combining stochastic fields and fuzzy variables through a simple calibration-validation approach. Finally, we construct a global-local multiscale algorithm for efficiently computing output quantities of interest. The method aims at approximating required quantities, such as displacements and stresses, in regions of relatively small size, e.g. hot spots or zones. The algorithm uses the concept of representative volume elements and computes a global solution to construct a local approximation that captures the microscale features of the solution. The results are based on and backed by real experimental data.

Ivo Babuska, Victor Nistor

We study the approximation properties of a harmonic function $u \in H\sp{1-k}(Ω)$, $k > 0$, on relatively compact sub-domain $A$ of $Ω$, using the Generalized Finite Element Method. For smooth, bounded domains $Ω$, we obtain that the GFEM--approximation $u_S$ satisfies $\|u - u_S\|_{H\sp{1}(A)} \le C h^γ\|u\|_{H\sp{1-k}(Ω)}$, where $h$ is the typical size of the ``elements'' defining the GFEM--space $S$ and $γ\ge 0 $ is such that the local approximation spaces contain all polynomials of degree $k + γ+ 1$. The main technical result is an extension of the classical super-approximation results of Nitsche and Schatz \cite{NitscheSchatz72} and, especially, \cite{NitscheSchatz74}. It turns out that, in addition to the usual ``energy'' Sobolev spaces $H^1$, one must use also the negative order Sobolev spaces $H\sp{-l}$, $l \ge 0$, which are defined by duality and contain the distributional boundary data.