Sensitivity analysis and Taylor expansions in numerical homogenization problems

Summary.

Two-scale numerical homogenization problems are addressed, with particular application to the modified compressible Reynolds equation with periodic roughness. It is shown how to calculate sensitivities of the homogenized coefficients that come out from local problems. This allows for significant reduction 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-point-like ones. Numerical tests are reported showing the quantitative accuracy of low-order Taylor expansions in practical cases, independently of the shape and smoothness of the roughness function.