Field conserving adaptive mesh refinement (AMR) scheme on massively parallel adaptive octree meshes

TL;DR

This paper introduces a scalable, field-conserving coarsening scheme for parallel octree-based adaptive mesh refinement, improving conservation and accuracy in long-term PDE simulations.

Contribution

It proposes a new coarsening operator that enforces global conservation during mesh coarsening in parallel AMR, enhancing solution accuracy and stability.

Findings

The method reduces conservation error compared to standard injection.

It maintains solution quality in phase-field models.

The approach is computationally efficient and scalable.

Abstract

Adaptive mesh refinement (AMR) is widely used to efficiently resolve localized features in time-dependent partial differential equations (PDEs) by selectively refining and coarsening the mesh. However, in long-horizon simulations, repeated intergrid interpolations can introduce systematic drift in conserved quantities, especially for variational discretizations with continuous basis functions. While interpolation from parent-to-child during refinement in continuous Galerkin (CG) discretizations is naturally conservative, the standard injection-based child-to-parent coarsening interpolation is generally not. We propose a simple, scalable field-conserving coarsening operator for parallel, octree-based AMR. The method enforces discrete global conservation during coarsening by first computing field conserving coarse-element values at quadrature points and then recovering coarse nodal degrees of freedom via an $L^2$ projection (mass-matrix solve), which simultaneously controls the $L_2$ error. We evaluate the approach on mass-conserving phase-field models, including the Cahn--Hilliard and Cahn--Hilliard--Navier--Stokes systems, and compare against injection in terms of conservation error, solution quality, and computational cost.