Adaptive multigrid for high-order discontinuous Galerkin methods based on the full approximation scheme

Abstract: We propose an adaptive multigrid (MG) method for discontinuous Galerkin formulations of elliptic problems using Brandt's full approximation scheme (FAS). Unlike common approaches, this method achieves local $hp$-refinement of hexahedral meshes without the need for hanging nodes. The core component of the FAS-MG method is an overlapping Schwarz smoother, which is optionally accelerated by a Krylov method. This smoother is designed for unstructured curvilinear meshes but maintains a tensor-product structure for fast diagonalization. Numerical experiments demonstrate the exceptional efficiency of the FAS-MG method. Dedicated studies confirm its robustness against high aspect ratios, element deformation, and irregular mesh topology. We also verify its capability for dynamic parallel mesh adaptation using the wave-front benchmark of Červený, Dobrev, and Kolev (SIAM J. Sci. Comp. 41, 2019). Finally, we present preliminary results of extending the method to incompressible Navier-Stokes problems.

• The paper develops an FAS multigrid framework for adaptive high-order DG-SEM that avoids hanging nodes by coupling refinement levels through transfer operators over frozen elements.
• The solver combines weighted overlapping Schwarz smoothing, fast diagonalization, and Krylov acceleration to maintain near-constant performance for aspect ratios up to 48 and on irregular curved meshes.
• The method reduces residuals by ten orders of magnitude in one V-cycle on a 963-million-DOF compressor-cascade problem and cuts adaptive degrees of freedom by 93–98% compared with global refinement.

Overview and contribution

This paper develops an adaptive multigrid (MG) method for elliptic problems discretized with the discontinuous Galerkin spectral element method (DG-SEM) on hexahedral meshes. The central algorithmic device is Brandt's full approximation scheme (FAS), adapted so that each mesh level extends only over its refinement zone rather than covering the entire domain. This eliminates hanging nodes and transition elements entirely: consecutive levels are coupled through transfer operators acting on frozen elements that interpolate their values from parent leaves. To the author's knowledge, this is the first application of FAS multigrid to high-order adaptive methods beyond prior work on one-dimensional space-time DG-SEM for conservation laws.

The second key component is a weighted overlapping Schwarz smoother (WAS) that generalizes earlier Cartesian-grid work to unstructured curvilinear meshes while preserving the tensor-product structure required for fast diagonalization (FD). Robustness against anisotropy is obtained by accelerating the Schwarz method with Golub–Ye inexact preconditioned conjugate gradients (IPCG), yielding the SCG smoother. Combined with full multigrid (FMG) startup, this produces the FAS-FMG solver whose efficiency and robustness are demonstrated across more than 4000 parameter studies and several challenging test cases.

Multilevel FAS-DG formulation

The model problem is a scalar reaction-diffusion equation with interior penalty DG discretization on conforming hexahedral meshes with tensor-product nodal bases of degree $p$. The multilevel mesh is built by successive $h$- or $p$-refinement (not simultaneous), with all elements on a level sharing one polynomial degree — a technical restriction adopted to keep data structures simple. Elements are classified as active or frozen; active elements are further split into twigs (regularly refined) and leaves. Refinement rules enforce a layer of frozen elements around every refinement zone, enclosing the active children.

The FAS-DG system couples levels directly: frozen-element coefficients are interpolated from parent leaves, twig right-hand sides combine projected child solutions with restricted child residuals, and leaves replicate the original discrete equations. Two properties follow. First, upon convergence the formulation is fully consistent with the single-mesh DG system and conservative on every level. Second, twig coefficients depend only on projected child solutions and restricted residuals, not on leaves or frozen elements, which allows the twig and leaf subsystems to be solved sequentially at reduced cost. A caveat stated explicitly: inter-level coupling via transfer operators is generally not conservative — small scales leaving a refinement zone cannot re-enter unless representable on the coarser mesh; whether this helps or hurts remains an open question.

For fine-to-coarse solution projection, both embedded interpolation and $L^2$ projection were tested; they perform nearly identically, with $L^2$ projection slightly degrading $h$-multigrid convergence rates.

Weighted Schwarz smoother with fast diagonalization

The smoother partitions the domain into element-centered overlapping subdomains including face, edge, and vertex neighbors — unlike the face-only subdomains of Vu et al., which sacrifice information from edge/vertex couplings. On unstructured curvilinear meshes, each subdomain problem is approximated by a Cartesian surrogate: the central element map is linearized to a parallelepiped with volume-preserving edge rescaling, variable diffusivity is replaced by its element average, and irregular neighbor nodes are merged arithmetically or zero-filled. The resulting tensor-product operator is inverted exactly by FD at $O(p)$ cost per node, with eigenvectors drawn from a small precomputed set determined only by boundary-condition type per face. Subdomain corrections are combined using the quintic weighting function introduced in prior work by the author, which previously reduced iteration counts by up to a factor of 3 relative to uniform weighting.

The design deliberately trades exactness of the subproblem solve for applicability of FD. On severely distorted meshes this approximation can fail: on the "diamond" mesh with irregular topology, WAS-based methods do not converge at all, whereas SCG-accelerated variants remain robust. Krylov acceleration on every level (rather than only the top level, as in classical CG-SEM hybrids) is what provides robustness against high aspect ratios, deformation, and irregularity; Chebyshev acceleration was rejected because adaptation changes the extremal eigenvalue repeatedly.

Numerical performance

On uniform Cartesian grids ($p = 4$ to 32, over 4000 configurations varying penalty $\mu_\star \in [2,10]$, overlap $\delta \in [0.04,0.16]$, smoothing parameters), the fastest configurations reduce the residual by ten orders of magnitude in remarkably few cycles:

Method	$h$0	$h$1	equivalent V-cycles	$h$2 (µs/DOF/core)
$h$3-FAS-FMG-SCG(3,1,2)	32	1	1.7	0.55
$h$4-FAS-FMG-WAS(1,1,1)	8	1	1.0	0.29
$h$5-FAS-FMG-WAS(1,1,1)	16	1	1.0	0.34

$h$6-FAS-FMG-WAS is the most efficient method generally, with $h$7-FAS-FMG-SCG preferred at very high degree. Performance is insensitive to the penalty parameter, consistent with independent findings by Fehn et al.

Against aspect ratios up to AR = 48, unaccelerated WAS becomes unstable and CS-CG-WAS iteration counts escalate prohibitively. In contrast, FAS-FMG-SCG(12,4,8) maintains $h$8 and normalized runtimes below 5 µs/DOF across the whole range — practically constant cost where comparable Chebyshev-Jacobi approaches report factor-of-10 cost increases at AR = 50. On curved spherical-shell meshes, runtimes increase moderately (roughly a factor of two); on the irregular diamond mesh, $h$9-FAS-FMG-SCG suffers only minor loss while pure WAS fails outright.

The compressor-cascade blade case combines all difficulties: aspect ratios up to ~8, curvature, distortion, variable diffusivity, and strong size variation. With global refinement to 963 million leaf DOFs, FAS-FMG-SCG(6,2,4) reduces the residual ten orders of magnitude in a single V-cycle at 1.54 µs/DOF on 32 processes, with excellent strong scalability up to 1024 ranks. Local $p$0-refinement near the blade saves about 93% of memory and 90% of runtime at 2.12 µs/DOF, though parallel efficiency degrades modestly beyond 32 processes because frozen-element workload is not accounted for in partitioning — an acknowledged shortcoming of the current SFC strategy.

Adaptive refinement benchmark

Using the spherical wave-front benchmark of Červený, Dobrev, and Kolev with dynamic parallel adaptation driven by a discrete energy-norm error indicator, the adaptive FAS-FMG solver saves between 93% and 98% of DOFs relative to global refinement, with savings increasing as tolerance tightens. For $p$1, results agree closely with MFEM reference data; $p$2 becomes substantially more efficient below errors of $p$3. Mesh adaptation and operator setup together consume only about 10% of solver time, and SFC-based level-by-level repartitioning limits imbalance to at most 8 elements per level while costing less than a single V-cycle. Solver times drop into the 1–4 µs/DOF/core range once meshes grow past the initial sparse state.

Comparison to the state of the art

Two claims distinguish this work from recent alternatives. First, against Chebyshev-Jacobi smoothers (Fehn et al.), the reported speedup reaches a factor of 20 or more for $p$4 on comparable hardware, attributed to higher algorithmic efficiency; at low degrees the methods are comparable. Second, against Schwarz-Chebyshev preconditioners under anisotropy, the SCG-smoothed FAS method is described as practically immune to aspect ratio, whereas competitors incur roughly order-of-magnitude cost growth. Third, unlike global-coarsening or local-smoothing adaptive multigrid schemes operating on a single globally refined mesh, the FAS approach avoids hanging-node operator modifications and constructs coarse levels naturally during refinement. These comparisons rest on cross-study hardware normalization (the reference CPU has 1.35× peak per-core performance but lower bandwidth), which the paper acknowledges makes direct comparison difficult.

Limitations and open questions

Several restrictions are conceded explicitly. All elements on a level share one polynomial degree, excluding fully general $p$5-adaptivity within a level. The surrogate-domain construction assumes the linearized element geometry and averaged coefficients remain adequate; on the diamond mesh the WAS variant fails entirely, so Krylov acceleration is effectively mandatory on irregular meshes. The discrete energy norm used for adaptation underestimates error at low resolution, requiring heuristic forced refinement of front-cut elements. Inter-level transfer non-conservation is unresolved. Parallel efficiency with local refinement degrades because frozen-element load is excluded from balancing, and the optimal partitioning strategy for small lower-level partitions is left open. Finally, application to Navier–Stokes problems is limited to preliminary results: a cylinder flow at Reynolds number 100 on a five-level mesh with 6.34 million velocity-pressure DOFs, without systematic validation.

Conclusion

The paper delivers a complete adaptive multigrid framework for high-order DG-SEM that combines FAS-based local $p$6-refinement without hanging nodes, a tensor-product Schwarz smoother retaining fast diagonalization on unstructured curvilinear meshes, and IPCG acceleration for robustness. The headline quantitative results — single-V-cycle convergence to $p$7 residual reduction on a 963-million-DOF industrial mesh, near-invariance to aspect ratios up to 48, and 93–98% DOF savings under dynamic adaptation — support the claim that Krylov-accelerated weighted Schwarz smoothers outperform polynomial smoothers at high order. Remaining questions concern within-level $p$8-variation, conservation across refinement boundaries, load-aware partitioning including frozen elements, and rigorous validation on turbulent flow problems.