Hybrid Localized Spectral Decomposition for multiscale problems

TL;DR

This paper introduces a hybrid spectral decomposition method for multiscale elliptic problems with heterogeneous coefficients, providing efficient, localized basis functions and optimal error estimates for both low- and high-contrast cases.

Contribution

It develops a novel hybrid localized spectral decomposition approach that handles high-contrast multiscale problems with minimal regularity assumptions and provides dimension-independent, extendable solutions.

Findings

Achieves quasi-optimal error estimates for low-contrast problems.

Enriches solution space with local eigenvalue problems for high-contrast coefficients.

Method is dimensionally independent and applicable to elasticity problems.

Abstract

We consider a finite element method for elliptic equation with heterogeneous and possibly high-contrast coefficients based on primal hybrid formulation. A space decomposition as in FETI and BDCC allows a sequential computations of the unknowns through elliptic problems and satisfies equilibrium constraints. One of the resulting problems is non-local but with exponentially decaying solutions, enabling a practical scheme where the basis functions have an extended, but still local, support. We obtain quasi-optimal a priori error estimates for low-contrast problems assuming minimal regularity of the solutions. To also consider the high-contrast case, we propose a variant of our method, enriching the space solution via local eigenvalue problems and obtaining optimal a priori error estimate that mitigates the effect of having coefficients with different magnitudes and again assuming no regularity of the solution. The technique developed is dimensional independent and easy to extend to other problems such as elasticity.