The Finite Element Heterogeneous Multiscale Method: a computational strategy for multiscale PDEs
نویسنده
چکیده
Heterogeneous multiscale methods (HMM) have been introduced by E and Engquist [Commun. Math. Sci. 1 (2003), pp. 87-132] as a general methodology for the numerical computation of problems with multiple scales. In this paper we discuss finite element methods based on the HMM for multiscale partial differential equations (PDEs). We give numerous examples of such multiscale problems, including elliptic, parabolic and advection diffusion problems and discuss several applications in areas such as porous media flow, biology and material sciences. A detailed analysis of the methods as well as recent developments are discussed.
منابع مشابه
A FEM Multiscale Homogenization Procedure using Nanoindentation for High Performance Concrete
This paper aims to develop a numerical multiscale homogenization method for prediction of elasto-viscoplastic properties of a high performance concrete (HPC). The homogenization procedure is separated into two-levels according to the microstructure of the HPC: the mortar or matrix level and the concrete level. The elasto-viscoplastic behavior of individual microstructural phases of the matrix a...
متن کاملA Finite Element Heterogeneous Multiscale Method with Improved Control over the Modeling Error
Multiscale partial differential equations (PDEs) are difficult to solve by traditional numerical methods due to the need to resolve the small wavelengths in the media over the entire computational domain. We develop and analyze a Finite Element Heterogeneous Multiscale Method (FE-HMM) for approximating the homogenized solutions of multiscale PDEs of elliptic, parabolic, and hyperbolic type. Typ...
متن کاملApplication of M3GM in a Petroleum Reservoir Simulation
Reservoir formations exhibit a wide range of heterogeneity from micro to macro scales. A simulation that involves all of these data is highly time consuming or almost impossible; hence, a new method is needed to meet the computational cost. Moreover, the deformations of the reservoir are important not only to protect the uppermost equipment but also to simulate fluid pattern and petroleum produ...
متن کاملA stochastic heterogeneous multiscale method for porous media flow
A new multiscale algorithm is introduced based on the framework of the heterogeneous multiscale method. The mixed finite element method used ensures continuity of the flux within the entire domain. This method is shown to be free of “resonance error” and uses less memory than the mixed multiscale finite element method. To account for the highstochastic dimensionality of the permeability field, ...
متن کاملMultilevel Monte Carlo Methods for Stochastic Elliptic Multiscale PDEs
In this paper Monte Carlo Finite Element (MC FE) approximations for elliptic homogenization problems with random coefficients which oscillate on n ∈ N a-priori known, separated length scales are considered. The convergence of multilevel MC FE (MLMC FE) discretizations is analyzed. In particular, it is considered that the multilevel FE discretization resolves the finest physical length scale, bu...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2008