Multiscale Partition of Unity

Abstract

We introduce a new Partition of Unity Method for the numerical homogenization of elliptic partial differential equations with arbitrarily rough coefficients. We do not restrict to a particular ansatz space or the existence of a finite element mesh. The method modifies a given partition of unity such that optimal convergence is achieved independent of oscillation or discontinuities of the diffusion coefficient. The modification is based on an orthogonal decomposition of the solution space while preserving the partition of unity property. This precomputation involves the solution of independent problems on local subdomains of selectable size. We deduce quantitative error estimates for the method that account for the chosen amount of localization. Numerical experiments illustrate the high approximation properties even for 'cheap' parameter choices.

Proceedings for Seventh International Workshop on Meshfree Methods for Partial Differential Equations, 18 pages, 3 figures

Cited by in corpus (10)

• A reduced basis localized orthogonal decomposition
• A Multiscale Method for Porous Microstructures
• A combined multiscale finite element method based on the LOD technique for the multiscale elliptic problems with singularities
• Multiscale mixed finite elements
• On Multiscale Methods in Petrov-Galerkin formulation
• Convergence analysis of an operator-compressed multiscale finite element method for Schrödinger equations with multiscale potentials
• Generalized Rough Polyharmonic Splines for Multiscale PDEs with Rough Coefficients
• Semi-Lagrangian Subgrid Reconstruction for Advection-Dominant Multiscale Problems
• Upscaling Singular Sources in Weighted Sobolev Spaces by Sub-Grid Corrections
• The Localized Reduced Basis Multiscale method for two-phase flows in porous media