Sensitivity analysis and Taylor expansions in numerical homogenization problems

Two-scale numerical homogenization problems are addressed, with particular application to the modified compressible Reynolds equation with periodic ro ughness. It is shown how to calculate sensitivities of the homogenized coef ficients that come out from local problems. This allows for significant red uction of the computational cost by two means: The construction of accurate Taylor expansions, and the implementation of rapidly convergent nonlinear algorithms (such as Newton's) instead of fixed-pointlike ones. Numerical te sts are reported showing the quantitative accuracy of low-order Taylor expa nsions in practical cases, independently of the shape and smoothness of the roughness function.