Many problems in engineering and science are represented by nonlinear partial differential equations (PDEs) with high contrast parameters and multiple scales. Solving these equations involves expensive computational cost because the fine-grid needs to resolve smallest scales and high contrast. In such cases, reduced-order methods are often needed. Reduced-order methods can be divided into local reduction methods and global reduction methods. Local reduced-order methods such as upscaling, Multiscale Finite Element Method (MsFEM) and Generalized Multiscale Finite Element Method (GMsFEM) divide the computational domain into coarse grids, where each grid contains small-scale heterogeneities and high contrast, and represents the computations for macroscopic simulations. In local model reduction, reduced-order models are constructed in each coarse region. Some known approaches, such as homogenization and numerical homogenization, are developed for problems with and without scale separation, respectively. Global reduced-order models, such as Proper Orthogonal Decomposition (POD), construct the reduced-order models via global finite element basis functions. These basis functions are constructed by solving many forward problems that can be expensive. In this dissertation, we propose global-local model reduction methods. The idea of global-local model reduction methods is to approximate the global basis functions locally and adaptively. However, in the case of nonlinear systems, additional interpolation techniques are required, such as Discrete Empirical Interpolation Method (DEIM). We propose a general global-local approach that uses the GMsFEM to construct adaptive approximation for the global basis functions. The developments of these methods require adaptive offline and adaptive online reduced-order model strategies, which we pursue in this work. We consider the applications to nonlinear ow problems, such as nonlinear Forch-heimer flow. In this case, we construct multiscale basis functions for the velocity field following mixed GMsFEM. In addition, we present a local online adaptive method for the basis enrichment of the function space based on an error indicator depending on the local residual norm. Finally, we propose a global online adaptive method to add new global basis functions to the POD subspace. We use local error indicators and solve the global residual problem using the GMsFEM, with local online adaptation.
