Non-nested multi-grid solvers for mixed divergence-free Scott-Vogelius discretizations

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Date
2007
Volume
1261
Issue
Journal
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Publisher
Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik
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Abstract

Studying high-dimensional Hamiltonian systems with microstructure, it is an important and challenging problem to identify reduced macroscopic models that describe some effective dynamics on large spatial and temporal scales. This paper concerns the question how reasonable macroscopic Lagrangian and Hamiltonian structures can by derived from the microscopic system. In the first part we develop a general approach to this problem by considering non-canonical Hamiltonian structures on the tangent bundle. This approach can be applied to all Hamiltonian lattices (or Hamiltonian PDEs) and involves three building blocks: (i) the embedding of the microscopic system, (ii) an invertible two-scale transformation that encodes the underlying scaling of space and time, (iii) an elementary model reduction that is based on a Principle of Consistent Expansions. In the second part we exemplify the reduction approach and derive various reduced PDE models for the atomic chain. The reduced equations are either related to long wave-length motion or describe the macroscopic modulation of an oscillatory microstructure.

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Keywords
Non-Nested Multi-Grid, Stabilized Finite Elements, Navier-Stokes Equations, LBB-Stability
Citation
Linke, A., Matthies, G., & Tobiska, L. (2007). Non-nested multi-grid solvers for mixed divergence-free Scott-Vogelius discretizations (Vol. 1261). Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik.
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