Evaluation of Local Multiscale Approximation Spaces for Partition of Unity Methods

The simulation of the behavior of heterogeneous and composite materials poses a number of challenges to numerical methods e.g. due to the presence of discontinuous material coefficients. Moreover, the material properties of fibers and inclusions are significantly different from those of the surrounding matrix. Thus, the gradients of the solution feature a substantial discontinuity at the material interface between inclusions and matrix. Hence, materials with many fine scale inclusions need a very high resolution mesh in the context of traditional finite element (FE) analysis. However, many approaches within the context of numerical homogenization have been proposed to tackle and overcome this need for a large number of degrees of freedom. To this end, either discontinuous coefficients are replaced by smooth effective coefficients or, standard FE shape functions are replaced by more complex, numerically computed shape functions while the overall quality of the approximation is retained. In this paper we study two-dimensional examples of heat transfer and (linear) elasticity in composite materials using a number of different homogenization approaches with the overall goal of evaluating and comparing their performance when used for the construction of multiscale enrichment functions for a partition of unity method (PUM).