Least-squares Mixed Generalized Multiscale Finite Element Methods For Partial Differential Equations

In this doctoral dissertation,we present least-square mixed generalized multi-scale finite element method to approximate elliptic problems with multiscale and high-contrast diffusion coefficients.We attempt to give accurate approximation for both pressure and velocity.A mixed formulation is considered such that both pressure and velocity are approximated simultaneously.This formulation arises naturally in many applications such as flows in porous media.Due to the mul-tiscale nature of the solutions,using model reduction is required to efficiently obtain approximate solutions.There are many multiscale approaches for elliptic problems in mixed formulation.These approaches include numerical homogeniza-tion and mixed multiscale finite element method,which aim to obtain a coarse accurate representation of the velocity without using an accurate representation for pressure.It has been a challenging task to construct a method giving accurate representation for both pressure and velocity.The goal in this paper is to construct multiscale basis functions for both pressure and velocity.First,we will apply the framework of Generalized Multiscale Finite Element Method(GMsFEM),and design systematic strategies for the construction of basis.The construction involves snapshot spaces and dimension reduction via local spec-tral problems.The mixed formulation is minimized in the sense of least-squares.The compatibility condition for multiscale finite element spaces of the pressure and velocity is not required in the least-squares mixed form.This gives more flexibility for the construction of multiscale basis functions for velocity and pres-sure.Convergence analysis is carried out for the least-squares mixed GMsFEM.Several numerical examples for various permeability fields are presented to show the performance of the presented method.The numerical results show that the least-squares mixed GMsFEM can give accurate approximation for both pressure and velocity using only a few basis functions per coarse neighborhood.Next,we present two kinds of adaptive least squares mixed generalized multi-scale finite element methods(GMsFEMs)for solving an elliptic problem in highly heterogeneous porous media.An offline adaptive method is developed through it-eratively enriching the local velocity and pressure multiscale basis functions based on residual-based error indicators.In addition,an online adaptive method is also proposed to create the new multiscale basis functions in the online stage for both velocity and pressure.The enriched basis functions are computed by the residual and maximizing the reduction in error.The offline adaptive method attempts to provide a good approximation space based on the heterogeneity of coefficient,and the online adaptive method aims at providing a good approximation space by tak-ing the given source into account.Both of the adaptive methods can achieve a bet-ter approximation than the uniform enrichment method using the same number of basis functions.Convergence analysis is carried out for the adaptive least-squares GMsFEMs.The analysis suggests that,by choosing a suitable number of initial basis functions for velocity and pressure,the online adaptive method can render a faster convergence rate compared with the offline adaptive method and the uni-form enrichment.A few numerical results are presented to confirm the analysis and the performance of the presented adaptive multiscale methods.Finally,we apply least-square mixed GMsFEM to solve parabolic equations.The offline and online stages are shown in detail.By choosing a proper parame-ter,we can give an uncoupling method.In the online computation,we solve the problem in coarse scale which significantly reduce the computational time.The numerical results show our method can give a good approximation on the solu-tion when using less basis functions.Beside,it is stable by using backward Euler method when discretizing in time.