Capturing Small Scales in Elliptic Problems Using a Residual-Free Bubbles Finite Element Method
- 1 January 2003
- journal article
- Published by Society for Industrial & Applied Mathematics (SIAM) in Multiscale Modeling & Simulation
- Vol. 1 (3) , 485-503
- https://doi.org/10.1137/s1540345902411402
Abstract
In this work we study the residual-free bubbles (RFB) finite element method for solv- ing second order elliptic equations with rapidly varying coefficients. The RFB technique is closely related to both the multiscale finite element method (MsFEM) introduced by Hou, Wu, and Cai (Math. Comp., 68 (1999), pp. 913-943) and the upscaling procedures which are very common in the engineering literature for solving this kind of partial differential equation. We also introduce a vari- ation of the RFB method, based on macrobubbles and referred to as the residual-free macrobubbles (RFMB) method, which gives more accurate numerical solutions. In the case of periodic coefficients we are able to prove a priori error estimates for the methods. Eventually, we test the numerical methods on model problems.Keywords
This publication has 8 references indexed in Scilit:
- Augmented spaces, two‐level methods, and stabilizing subgridsInternational Journal for Numerical Methods in Fluids, 2002
- Residual-free bubbles for advection-diffusion problems: the general error analysisNumerische Mathematik, 2000
- Convergence of a Nonconforming Multiscale Finite Element MethodSIAM Journal on Numerical Analysis, 2000
- Global and Local Error Analysis for the Residual-Free Bubbles Method Applied to Advection-Dominated ProblemsSIAM Journal on Numerical Analysis, 2000
- First-order corrections to the homogenised eigenvalues of a periodic composite medium. A convergence proofProceedings of the Royal Society of Edinburgh: Section A Mathematics, 1997
- CHOOSING BUBBLES FOR ADVECTION-DIFFUSION PROBLEMSMathematical Models and Methods in Applied Sciences, 1994
- A relationship between stabilized finite element methods and the Galerkin method with bubble functionsComputer Methods in Applied Mechanics and Engineering, 1992
- Convergence estimates for product iterative methods with applications to domain decompositionMathematics of Computation, 1991