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

TL;DR

This work identifies mass-conservation drift as a key issue in long-duration simulations using octree AMR with continuous Galerkin discretizations, where standard injection-based coarsening breaks global invariants. It introduces a field-conserving coarsening operator that first enforces conservation at coarse-element quadrature points via a local $L^2$ projection and then recovers coarse nodal DOFs through a global $L^2$ projection (mass-matrix solve), applicable to CG with Lagrange bases of arbitrary order. The method is validated on mass-conserving phase-field models, including the Cahn–Hilliard and CHNS systems, across 2D and 3D, showing exact mass conservation under AMR and substantially reduced coarsening-induced energy mismatch, with modest overhead and good scalability. This approach enhances the robustness of long-time multiphysics simulations with AMR and can be extended to conserve additional integral quantities in future work. Overall, the paper provides a practical, scalable solution to enforce discrete global conservation in CG-based octree AMR, improving reliability and physical fidelity in complex simulations.

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.

Figures (16)

• 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 $[\int_{\Omega} g(x) \; d\Omega ]_{\mathcal{M}_O}$ = $[\int_{\Omega} g(x) \; d\Omega ]_{\mathcal{M}_R}$ = 10.6284 , where $\mathcal{M}_O$ is the original mesh and $\mathcal{M}_R$ is the refined mesh.
• 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. $[\int_{\Omega} g(x) \; d\Omega ]_{\mathcal{M}_O}$ = 10.6284, whereas $[\int_{\Omega} g(x) \; d\Omega ]_{\mathcal{M}_C}$ = 10.6036 , where $\mathcal{M}_O$ is the original mesh and $\mathcal{M}_C$ is the coarsened mesh.
• Figure 3: Figure showing Gauss quadrature points and nodal points for (a) Q1 and (b) Q2 elements at two levels of refinement in 1D.
• Figure 4: Mass conserving interpolation: Figure showing the redistribution of $\phi$ on the coarse mesh when the solution is projected from the fine mesh, such that $\int_{\Omega} g(x) d\Omega$ remains constant. $[\int_{\Omega} g(x) \; d\Omega ]_{\mathcal{M}_O}$ = $[\int_{\Omega} g(x) \; d\Omega ]_{\mathcal{M}_C}$ = 10.6284 , where $\mathcal{M}_O$ is the original mesh and $\mathcal{M}_C$ is the coarsened mesh.
• Figure 5: Interpolation using quadratic basis function: Figure comparing the coarsening of a function $g(x) = |\cos(2\pi x)| + 10$ defined on a fine mesh with 8 quadratic elements to a coarse mesh with 4 quadratic elements using (a) injection and (b) mass-conserving interpolation. The integral of the function over the domain $\Omega=[0,1]$ on the original refined mesh (=10.6367) is preserved up to numerical precision using the proposed mass-conserving interpolation scheme, whereas injection results in a noticeable deviation.