Grégoire Allaire introduces the concept of two-scale convergence to better describe sequences of oscillating functions. He proves that bounded sequences in $ L^2(\Omega) $ are relatively compact with respect to this new type of convergence. A corrector-type theorem is established, allowing sequences to be replaced by their two-scale limit up to a strongly convergent remainder in $ L^2(\Omega) $. These results are particularly useful for homogenization of partial differential equations with periodically oscillating coefficients. A new method for proving convergence is proposed, an alternative to the energy method of Tartar. The two-scale convergence method is demonstrated on examples, including linear and nonlinear second-order elliptic equations. The paper discusses homogenization of partial differential equations with periodically oscillating coefficients. It introduces the two-scale convergence method, which is more efficient than the energy method. The method is self-contained, allowing the homogenized equation and convergence to be found in a single process. It also introduces the two-scale homogenized problem, which is a combination of the usual homogenized and cell equations. This method is particularly effective for problems with periodic oscillations and provides a rigorous justification for the first term in the ansatz. The two-scale convergence method is applied to linear second-order elliptic equations, leading to the two-scale homogenized problem, which is a combination of the usual homogenized equation and the cell problem. The method is shown to be effective in recovering previous results and providing a new formulation of the limit problem. The paper also discusses the generalization of two-scale convergence to other spaces and the application of the method to various types of equations.Grégoire Allaire introduces the concept of two-scale convergence to better describe sequences of oscillating functions. He proves that bounded sequences in $ L^2(\Omega) $ are relatively compact with respect to this new type of convergence. A corrector-type theorem is established, allowing sequences to be replaced by their two-scale limit up to a strongly convergent remainder in $ L^2(\Omega) $. These results are particularly useful for homogenization of partial differential equations with periodically oscillating coefficients. A new method for proving convergence is proposed, an alternative to the energy method of Tartar. The two-scale convergence method is demonstrated on examples, including linear and nonlinear second-order elliptic equations. The paper discusses homogenization of partial differential equations with periodically oscillating coefficients. It introduces the two-scale convergence method, which is more efficient than the energy method. The method is self-contained, allowing the homogenized equation and convergence to be found in a single process. It also introduces the two-scale homogenized problem, which is a combination of the usual homogenized and cell equations. This method is particularly effective for problems with periodic oscillations and provides a rigorous justification for the first term in the ansatz. The two-scale convergence method is applied to linear second-order elliptic equations, leading to the two-scale homogenized problem, which is a combination of the usual homogenized equation and the cell problem. The method is shown to be effective in recovering previous results and providing a new formulation of the limit problem. The paper also discusses the generalization of two-scale convergence to other spaces and the application of the method to various types of equations.