Numerical solution of the stationary diffusion equation by means of homogenization

For elliptic partial differential equations with periodically oscillating coefficients quadratic L2-convergence of a corrected asymptotic expansion, which is motivated by the theory of homogenization, is proven in the one-dimensional case. In the two-dimensional case the rate of convergence and its dependency on the symmetry of the diffusion coefficient is numerically analysed. The correction of the asymptotic expansion on a locally refined grid is then embedded inside a two-grid method and numerically compared with a classical PCG-method.