Spectral methods; Pseudospectral methods; Stability; Boundary conditions

A new class of p version FEM for elliptic problems with microstructure is developed. Based on arguments from the theory of n-widths, the existence of subspaces with favorable approximation properties for solution sets of PDEs is deduced. The construction of such subspaces is addressed for problems with (patch-wise) periodic microstructure. Families of adapted spectral shape functions are exhibited which give exponential convergence for smooth data, independently of the coefficient regularity. Some theoretical results on the spectral approach in homogenization are presented. Numerical results show robust exponential convergence in all cases.