A unified framework for mesh refinement in random and physical space

TL;DR

This paper introduces a versatile mesh refinement framework applicable to both random and physical spaces, eliminating the need for explicit reduced models and demonstrating effectiveness across various complex PDE examples.

Contribution

The work presents a novel, model-agnostic mesh refinement framework applicable to random and physical spaces, expanding previous methods that relied on explicit reduced models.

Findings

Effective in refining meshes for complex PDEs

Versatile application to both random and physical spaces

Demonstrates efficiency and adaptability in numerical examples

Abstract

In recent work we have shown how an accurate reduced model can be utilized to perform mesh refinement in random space. That work relied on the explicit knowledge of an accurate reduced model which is used to monitor the transfer of activity from the large to the small scales of the solution. Since this is not always available, we present in the current work a framework which shares the merits and basic idea of the previous approach but does not require an explicit knowledge of a reduced model. Moreover, the current framework can be applied for refinement in both random and physical space. In this manuscript we focus on the application to random space mesh refinement. We study examples of increasing difficulty (from ordinary to partial differential equations) which demonstrate the efficiency and versatility of our approach. We also provide some results from the application of the new framework to physical space mesh refinement.