Year: 2025
Type: article
Abstract: . Many numerical methods for multiscale differential equations require a scale separation between the larger and the smaller scales to achieve accuracy and computational efficiency. In the area of multiscale dynamical systems, so-called seamless methods have been introduced to reduce the requirement of scale separation. We will translate these methods to numerical homogenization problems and extend the technique to multiple dimensions. The initial step is to prove that a one-dimensional second-order elliptic operator with oscillatory coefficients can be rewritten as a multiscale dynamical system. Inspired by this, multiscale elliptic operators in higher dimensions are approximated by a novel approach based on local dilation, which provides a middle ground for balancing intractability and accuracy without the need for full resolution. The dilation operator can be further generalized to preserve important structures by properly decomposing the coefficient field. Error estimates are developed, and promising numerical results of different examples are included.
DOI: 10.1137/24m1668755
Authors: Ziheng Chen, Björn Engquist
Institution: The University of Texas at Austin
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W4411435085