# Rigorous Theory for Accurate Multi-scale Analysis with Non Well-seperated Length Scales

The Generalized Finite Element Method (GFEM) is a domain decomposition method based upon a partition of unity of a computational domain. This method is initiated and developed in (Babuska, Caloz, and Osborne, 1994), (Babuska and Melenk, 1997), (Babuska, Banerjee, and Osborne, 2002). This approach is well suited to modeling heterogeneous elastic media and is a Galerkin scheme based upon choosing a finite dimensional approximation space inside each subdomain that supports the partition of unity. For example if the subdomains are triangles and the partition of unity functions are the classic hat functions then the GFEM is the well known linear Finite Element (FE) approximation. The approximation property of the FE basis improves as we refine the triangular mesh. On the other hand the GFEM approximation can improve if we increase the dimension of the local approximation space associated with each subdomain. This feature allows for an approximation theory based on the dimension of the approximation space which is not possible in either the classic FE approximation or in homogenization theory. This provides a new opportunity for developing an approximation theory of heterogeneous media in the absence of scale separation. This approach is distinct from classical homogenization which requires a separation of length scales between the length scale of heterogeneity and that of the boundary loading and body forces.

It is demonstrated in (Babuska and Melenk, 1997) that the global error is controlled by the local approximation error. So the principle theoretical question surrounding GFEM is to find an optimal local approximation space for each subdomain participating in the partition of unity and to estimate its rate of convergence. These questions are answered in collaboration with I. Babuuska in (Babuska and Lipton, 2011) and with I. Babuuska and my Ph.D. student X. Huang in (Babuska, Xu, and Lipton, 2014). The first paper (Babuska and Lipton, 2011) solves this problem for scalar problems and the second (Babuska, Xu, and Lipton, 2014). applies the methodology to linear elasticity.

In these papers we consider Neumann boundary conditions and inside the domain $D$ the displacement $u$ is a solution of $$ {\rm div}(\mathbb{A}(x)e(u))=0.$$ Here $\mathbb{A}(x)=\mathbb{A}_{ijkl}(x)$ is an elastic tensor with $L^\infty$ elements and satisfies ellipticity and boundedness $$0\leq c_1|e|^2\leq \mathbb{A}e:e\leq c_2|e|^2. $$ A function satisfying the homogeneous equation is said to be $A$-harmonic. We consider a local domain $S$ associated with the partition of unity. We assume that $S\subset \subset D$ is compactly contained inside the the domain of interest $D$. (Local domains touching the boundary of $D$ can be handled in a modified but similar way.) We then take a slightly larger domain $S^*$ such that $S\subset \subset S^*\subset \subset D$ and consider all $N$ dimensional spaces of $A$-harmonic functions on $S^*$. We ask of all such $N$ dimensional subspaces can we find the one that approximates the solution the best in the energy norm restricted to $S$. Since we consider the energy norm we are locally approximating the solution up to a constant but the constant can be determined. It turns out that the restriction operator $Ru=u_{|_{\scriptscriptstyle{S}}}$ is a compact on operator on the space of $A$-harmonic functions defined on $S^*$. The best basis with optimal approximation properties is shown to be the span of the first $N$ eigenfunctions of $RR^*$ and the convergence rate is given by the square root of the $N^{th}$ singular value of $R$. This singular value can be estimated using an iterated application of the Caccioppoli inequality and it decays almost exponentially with respect to $N$. From this we deduce that the approximation for the optimal global approximation decreases nearly exponentially with the dimension of local approximation spaces.

We call the associated numerical method the Multiscale Spectral Generalized Finite Element Method (MS-GFEM).
This method serves two roles: 1) it provides a dimensionally reduced coarse scale version of the original physical problem and
2) it allows for an efficient and accurate post processing step to resolve local fields at lower length scales.
This feature is distinct from pure upscaling methods where only coarse scale field information is obtained.
The simplest multiscale method is the well known method of periodic homgenization with correctors, see for
example (Besounssan, Lions, and Papanicolau, 1978). ** However for non periodic problems with non well separated length scales the homogenization approach
is no longer applicable.** On the other hand MS-GFEM **is designed for such problems** and *provides computational recovery
of coarse scale information through a global solve while providing selective resolution of local fields as a post-processing step.*
The scheme is highly parallel and can be implemented on on large parallel machines.

*This work is supported by NSF Grant DMS-1211066*

The work discussed here has appeared in I. Babuska and R. Lipton, Optimal Local Approximation Spaces for Generalized Finite Element Methods with Application to Multiscale Problems. Multiscale Modeling and Simulation, SIAM 9 (2011) 373-406. and I. Babuska, Xu Huang, and R. Lipton, Machine Computation Using the Exponentially Convergent Multiscale Spectral Generalized Finite Element Method. (With Ivo Babuska and Xu Huang). Mathematical Modeling and Numerical Analysis (M2AN), 48 Number 2 (2014) 493-515.

*This work is supported by NSF Grant DMS-1211066*

## References

- I. Babuska, G. Caloz, and J. E. Osborn, Special finite element methods for a class of second order elliptic problems with rough coefficients, SIAM J. Numer. Anal. 31 (1994), pp. 945-981.
- I. Babuska and J. Melenk, The partition of unity finite element method, Internat. J. Numer. Methods Engrg., 40 (1997), pp. 727-758.
- I. Babuska, U. Banerjee, and J. Osborn, Generalized finite element methods-main ideas, results and perspective, Int. J. Comput. Methods, 1 (2004), pp. 67-103.
- I. Babuska and R. Lipton, Optimal Local Approximation Spaces for Generalized Finite Element Methods with Application to Multiscale Problems. Multiscale Modeling and Simulation, SIAM 9 (2011) 373-406.
- I. Babuska, Xu Huang, and R. Lipton, Machine Computation Using the Exponentially Convergent Multiscale Spectral Generalized Finite Element Method. (With Ivo Babuska and Xu Huang). Mathematical Modeling and Numerical Analysis (M2AN), 48 Number 2 (2014) 493-515.
- A. Besounssan, J. L. Lions, and G. C. Papanicolau, Asymptotic Analysis for Periodic Structures, North Holland, Amsterdam, 1978.