Conservativeness of multilevel coupling

Determine whether the generally non-conservative coupling between mesh levels in the full-approximation storage discontinuous Galerkin formulation is advantageous or disadvantageous, particularly because small scales may leave a refinement zone without entering the enclosing coarser region unless they are representable on the coarser mesh.

Background

The paper develops a full-approximation storage multigrid method for high-order discontinuous Galerkin spectral-element discretizations on locally refined hexahedral meshes. Unlike formulations that use a global finest-level mesh with hanging nodes or transition elements, the proposed method stores solutions on multiple refinement levels and couples them through transfer operators and frozen elements.

Although the DG formulation remains conservative on each individual mesh level, the authors note that interlevel coupling through transfer operators is generally not conservative. In particular, fine-scale structures can leave a refinement zone but are not transferred into the surrounding coarse region when the coarse discretization cannot represent those structures. The paper explicitly leaves unresolved whether this loss of interlevel conservativeness is beneficial or harmful.