Variational Multiscale Stabilization and the Exponential Decay of Fine-scale Correctors

2015/05/28 by Daniel Peterseim, Peterseim, Daniel
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Composite Material Mechanics

paper · pdf · doi:10.48550/arxiv.1505.07611

Abstract

This paper addresses the variational multiscale stabilization of standard finite element methods for linear partial differential equations that exhibit multiscale features. The stabilization is of Petrov-Galerkin type with a standard finite element trial space and a problem-dependent test space based on pre-computed fine-scale correctors. The exponential decay of these correctors and their localisation to local cell problems is rigorously justified. The stabilization eliminates scale-dependent pre-asymptotic effects as they appear for standard finite element discretizations of highly oscillatory problems, e.g., the poor L² approximation in homogenization problems or the pollution effect in high-frequency acoustic scattering.

Related

• Error analysis of a variational multiscale stabilization for convection-dominated diffusion equations in 2d
• A variational multiscale method derived from an adaptive stabilized conforming finite element method via residual minimization on dual norms
• Eliminating the pollution effect in Helmholtz problems by local subscale\n correction
• A Unified Framework for Multiscale Modeling using the Mori-Zwanzig\n Formalism and the Variational Multiscale Method
• Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods