This dataset and code support the manuscript 'Adjoint-aided homogenization for flows through heterogeneous membranes' by Kevin Wittkowski, Edouard Boujo, François Gallaire and Giuseppe A. Zampogna, under revision in the Journal of Fluid Mechanics, 2024. The data are Stokes flow simulations around an elliptical solid inclusion with periodic boundary conditions. Direct computations of the average flow quantities for several geometry parameters are compared with shape-sensitivity-based predictions obtained by solving a set of direct and adjoint equations around a circular solid inclusion. Please refer to the manuscript for further details. Contents: comsol_code/Mnn.mph, Mtt.mph: COMSOL Multiphysics 6.0 codes used to generate the datasets; README.txt: a .txt file describing the code usage; Mnn.txt,Mtt.txt: dataset of the direct evaluations of these two average microscopic quantities, input of the .m files; S1_Mnn.txt, S2_Mnn.txt, S1_Mtt.txt, S2_Mtt.txt: dataset containing the first and second order shape sensitivities computed on the reference circular geometry, input of the .m files; Mnn.m,Mtt.m: MatLab code used to compare the direct evaluations and adjoint predictions of the microscopic quantities; ellipse.m: Matlab function to generate the elliptical deformation; .fig files: Matlab figures, output of the .m files. They are used to compare the direct evaluation with the prediction based on the adjoint computations.