Direct and Iterative Methods for Numerical Homogenization
نویسندگان
چکیده
Elliptic problems with oscillating coefficients can be approximated up to arbitrary accuracy by using sufficiently fine meshes, i.e., by resolving the fine scale. Well-known multiscale finite elements [5, 9] can be regarded as direct numerical homogenization methods in the sense that they provide approximations of the corresponding (unfeasibly) large linear systems by much smaller systems while preserving the fine-grid discretization accuracy (model reduction). As an alternative, we present iterative numerical homogenization methods that provide approximations up to fine-grid discretization accuracy and discuss differences and commonalities. Acknowledgements This research has been funded by Deutsche Forschungsgemeinschaft (DFG) through grant CRC 1114.
منابع مشابه
Numerical Homogenization of Elliptic Multiscale Problems by Subspace Decomposition
Numerical homogenization tries to approximate solutions of elliptic partial differential equations with strongly oscillating coefficients by the solution of localized problems over small subregions. We develop and analyze a rapidly convergent iterative method for numerical homogenization that shares this feature with existing approaches and is modeled after the Schwarz method. The method is hig...
متن کاملNumerical homogenization methods
Numerical homogenization methods Synonyms multiscale methods for homogenization problems, upscaling methods, representative volume element methods Definition Numerical homogenization methods are techniques for finding numerical solutions of partial differential equations (PDEs) with rapidly oscillating coefficients (multiple scales). In mathematical analysis, homogenization can be defined as a ...
متن کاملPreconditioned Generalized Minimal Residual Method for Solving Fractional Advection-Diffusion Equation
Introduction Fractional differential equations (FDEs) have attracted much attention and have been widely used in the fields of finance, physics, image processing, and biology, etc. It is not always possible to find an analytical solution for such equations. The approximate solution or numerical scheme may be a good approach, particularly, the schemes in numerical linear algebra for solving ...
متن کاملA stable iteration to the matrix inversion
The matrix inversion plays a signifcant role in engineering and sciences. Any nonsingular square matrix has a unique inverse which can readily be evaluated via numerical techniques such as direct methods, decomposition scheme, iterative methods, etc. In this research article, first of all an algorithm which has fourth order rate of convergency with conditional stability will be proposed. ...
متن کامل