Exponentially Convergent Multiscale Finite Element Method

TL;DR

This paper reviews ExpMsFEM, a multiscale finite element method that achieves nearly exponential convergence for PDEs in complex media without relying on partition of unity functions, enabling efficient high-frequency wave simulations.

Contribution

It introduces a systematic enrichment of the approximation space in ExpMsFEM that avoids partition of unity functions and achieves exponential convergence rates.

Findings

Achieves nearly exponential convergence with respect to basis functions

Separates online and offline computations for efficiency

Applicable to high-frequency wave propagation in heterogeneous media

Abstract

We provide a concise review of the exponentially convergent multiscale finite element method (ExpMsFEM) for efficient model reduction of PDEs in heterogeneous media without scale separation and in high-frequency wave propagation. ExpMsFEM is built on the non-overlapped domain decomposition in the classical MsFEM while enriching the approximation space systematically to achieve a nearly exponential convergence rate regarding the number of basis functions. Unlike most generalizations of MsFEM in the literature, ExpMsFEM does not rely on any partition of unity functions. In general, it is necessary to use function representations dependent on the right-hand side to break the algebraic Kolmogorov $n$-width barrier to achieve exponential convergence. Indeed, there are online and offline parts in the function representation provided by ExpMsFEM. The online part depends on the right-hand side locally and can be computed in parallel efficiently. The offline part contains basis functions that are used in the Galerkin method to assemble the stiffness matrix; they are all independent of the right-hand side, so the stiffness matrix can be used repeatedly in multi-query scenarios.