[Paper Review] Field conserving adaptive mesh refinement (AMR) scheme on massively parallel adaptive octree meshes

Kumar Saurabh, Makrand A. Khanwale+3 more|arXiv (Cornell University)|Feb 8, 2026

Solidification and crystal growth phenomena0 citations

TL;DR

The paper introduces a field-conserving coarsening operator for parallel octree-based AMR in continuous Galerkin discretizations, ensuring discrete global conservation during coarsening and evaluating it on Cahn–Hilliard–type systems.

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.

Motivation & Objective

• Motivate mass conservation in long-horizon multiphysics simulations using AMR.
• Develop a conservative coarsening operator that guarantees discrete global conservation during AMR cycles.
• Provide a scalable algorithm compatible with CG discretizations on octree meshes.
• Demonstrate conservation and accuracy improvements on phase-field models such as CH and CHNS.

Proposed method

• Use field-conserving coarsening by first enforcing conservation at coarse-element quadrature points, then recover coarse nodal DOFs via an L2 projection (mass-matrix solve).
• Require that refinement (parent-to-child) is conservative, while coarsening (child-to-parent) uses a local L2 projection to the coarse grid that preserves the global integral over each coarse element.
• Construct a restriction operator that maps fine-element Gauss-point values to coarse-element Gauss-point values via a local L2 projection, yielding a diagonal (or easily invertible) mass matrix in standard Gauss-Legendre setups.
• Assemble the coarse nodal values by solving a global L2 projection (M G = b) of the coarse-point data onto the coarse finite element space, ensuring mass conservation.
• Extend the approach to arbitrary polynomial orders (Pp/Qp) and to multiple dimensions using tensor products of the 1D restriction operator, with a Kronecker-structured extension for 2D/3D.

Figure 1 : Refinement: Figure representing the refinement case. The red marker represents the extra points that are added on the refined mesh. The value of $\bigg[\int_{\Omega}g(x)\;d\Omega\bigg]_{\mathcal{M}_{O}}$ = $\bigg[\int_{\Omega}g(x)\;d\Omega\bigg]_{\mathcal{M}_{R}}$ = 10.6284 , where $\math

Experimental results

Research questions

• RQ1 How can coarsening in CG on octree AMR be made mass-conserving without sacrificing accuracy?
• RQ2 Does the proposed conservative coarsening scheme reduce mass drift and preserve invariants over long time integrations compared to injection-based coarsening?
• RQ3 What is the impact of the conservative coarsening on convergence rates and computational cost for phase-field PDEs?
• RQ4 Is the method scalable to massively parallel, high-resolution simulations with single-level AMR updates?

Key findings

• Conservative coarsening preserves the global integral (mass) to numerical precision across AMR cycles.
• For MMS diffusion tests, the conservative scheme maintains optimal convergence rates (≈2 for linear, ≈3 for quadratic) with slightly lower errors than injection in the linear case.
• In mass drift tests, the conservative scheme eliminates drift at each coarsening step, while injection shows nonzero drift that improves only with finer meshes.
• Quadratic and linear bases show comparable or improved accuracy under the conservative coarsening, with marginal differences in the error magnitudes.
• The method is demonstrated on CH and CHNS-type models, highlighting mass conservation benefits in long-horizon, phase-field simulations.

Figure 2 : Coarsening by Injection : Figure demonstrating the coarsening procedure by injection. The gray shaded region shows the area corresponding to the difference between two mesh. $\bigg[\int_{\Omega}g(x)\;d\Omega\bigg]_{\mathcal{M}_{O}}$ = 10.6284, whereas $\bigg[\int_{\Omega}g(x)\;d\Omega\big