A Concurrent Global-local Numerical Method for Multiscale PDEs

TL;DR

This paper introduces a hybrid numerical method for multiscale PDEs that efficiently captures both global macroscopic behavior and local microscopic details, improving accuracy without significantly increasing computational cost.

Contribution

A new concurrent global-local numerical method for multiscale PDEs that combines microscopic and homogenized coefficients, with proven convergence and demonstrated effectiveness.

Findings

Method's convergence proven for bounded coefficients

Achieves comparable cost to heterogeneous multiscale methods

Numerical results confirm accuracy and efficiency

Abstract

We present a new hybrid numerical method for multiscale partial differential equations, which simultaneously captures the global macroscopic information and resolves the local microscopic events over regions of relatively small size. The method couples concurrently the microscopic coefficients in the region of interest with the homogenized coefficients elsewhere. The cost of the method is comparable to the heterogeneous multiscale method, while being able to recover microscopic information of the solution. The convergence of the method is proved for problems with bounded and measurable coefficients, while the rate of convergence is established for problems with rapidly oscillating periodic or almost-periodic coefficients. Numerical results are reported to show the efficiency and accuracy of the proposed method.