Colloquium

Modern structures such as airplane wings exhibit complicated sub structures and make use of composite materials in their construction. The high cost of experimental tests for these hierarchical structures is driving a trend toward virtual testing. This requires the development of multi-scale numerical methods capable of handling large degrees of freedom spread across different length scales. In this talk we review multi-scale numerical methods and introduce the theory of the Kolmogorov n-width as a means to identify optimal local basis functions for use in multi-scale finite element methods. We are able to identify a spectral basis with nearly exponential convergence with respect to the dimension of the approximation space. The convergence result is shown to hold in a very general setting. This is joint work with Ivo Babuska.