A Dilation-based Seamless Multiscale Method For Elliptic Problems

TL;DR

This paper introduces a novel dilation-based multiscale method for elliptic problems that reduces the need for scale separation, enabling efficient and accurate numerical homogenization in multiple dimensions.

Contribution

The paper extends seamless multiscale methods to elliptic problems using local dilation, providing a new approach that balances accuracy and computational tractability without full resolution.

Findings

Error estimates are established for the method.

Numerical experiments demonstrate promising results.

The approach effectively approximates multiscale elliptic operators.

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 \sepia{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.