Conservative and accurate solution transfer between high-order and low-order refined finite element spaces

TL;DR

This paper introduces general transfer operators for high-order and low-order finite element spaces that are conservative, accurate, and applicable to complex meshes, enabling improved coupling and adaptive refinement.

Contribution

The paper develops and proves properties of new transfer operators between high-order and low-order finite element spaces, applicable to complex geometries and mesh refinement.

Findings

Operators are conservative and high-order accurate

Numerical experiments confirm theoretical properties

Applications in adaptive mesh refinement and multi-discretization coupling

Abstract

In this paper we introduce general transfer operators between high-order and low-order refined finite element spaces that can be used to couple high-order and low-order simulations. Under natural restrictions on the low-order refined space we prove that both the high-to-low-order and low-to-high-order linear mappings are conservative, constant preserving and high-order accurate. While the proofs apply to affine geometries, numerical experiments indicate that the results hold for more general curved and mixed meshes. These operators also have applications in the context of coarsening solution fields defined on meshes with nonconforming refinement. The transfer operators for $H^1$ finite element spaces require a globally coupled solve, for which robust and efficient preconditioners are developed. We present several numerical results confirming our analysis and demonstrate the utility of the new mappings in the context of adaptive mesh refinement and conservative multi-discretization coupling.