Multiscale-Spectral GFEM and Optimal Oversampling

TL;DR

This paper discusses the implementation and efficiency improvements of the Multiscale Spectral GFEM, demonstrating contrast-independent exponential convergence and proposing strategies to reduce computational costs in generating local bases.

Contribution

It introduces cost-effective strategies for constructing local bases in MS-GFEM while maintaining high accuracy and convergence rates.

Findings

Demonstrates contrast-independent exponential convergence of MS-GFEM solutions.

Proposes strategies that reduce computational costs without sacrificing accuracy.

Develops a nearly optimal local basis based on boundary partition of unity and A-harmonic extensions.

Abstract

In this work we address the Multiscale Spectral Generalized Finite Element Method (MS-GFEM) developed in [I. Babu\v{s}ka and R. Lipton, Multiscale Modeling and Simulation 9 (2011), pp. 373--406]. We outline the numerical implementation of this method and present simulations that demonstrate contrast independent exponential convergence of MS-GFEM solutions. We introduce strategies to reduce the computational cost of generating the optimal oversampled local approximating spaces used here. These strategies retain accuracy while reducing the computational work necessary to generate local bases. Motivated by oversampling we develop a nearly optimal local basis based on a partition of unity on the boundary and the associated A-harmonic extensions.