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

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.