Consistent Splitting in p-Multigrid DG

• Consistent splitting schemes are rigorously designed multilevel algorithms that ensure error reduction and uniform convergence in p-multigrid DG methods.
• They employ norm-preserving inter-level transfers, Chebyshev smoothers, and coarse-grid corrections to maintain optimal solver performance across mesh sizes and polynomial degrees.
• Numerical experiments confirm that non-inherited splitting reduces the smoothing cost to O(p) while delivering robust preconditioning for elliptic and interface problems.

A consistent splitting scheme is a rigorously designed multilevel algorithm in which the inter-level transfer operators, smoothing procedures, and coarse-level corrections are constructed to ensure that error reduction and convergence estimates do not deteriorate with respect to discretization parameters such as mesh size or polynomial degree. In the context of high-order discontinuous Galerkin (DG) discretizations—most notably for elliptic problems—such schemes are crucial to achieve optimal solver performance, especially in the $p$-version and $hp$-version of DG methods. The archetypal example is the $p$-multigrid W-cycle algorithm for symmetric interior penalty DG (SIPDG) discretizations, where consistent splitting refers both to the selection of the discrete hierarchy (with inherited or non-inherited bilinear forms) and the design of transfer and smoothing operators that preserve uniform convergence rates and minimize inter-level pollution.

1. Mathematical Formulation of Consistent Splitting in $p$-Multigrid DG Methods

In high-order SIPDG discretizations, the consistent splitting scheme operates on a nested hierarchy of broken polynomial spaces. On a Cartesian mesh $\mathcal{T}_h$ of size $h$, each element $T$ employs tensor-product Legendre bases $\Psi_T^\alpha$ generated from reference-cube polynomials $L_\alpha(\xi)$, giving local spaces $$Q_k(T) = \mathrm{span}\{\Psi_T^\alpha:\|\alpha\|_\infty\le k\}$$ up to degree $k\le p$. The global broken space is $$U_k(\mathcal{T}_h) = \{ v\in L^2(\Omega) : v|_T\in Q_k(T) \ \forall T\} \subset U_{k+1}(\mathcal{T}_h)$$.

The SIPDG bilinear form for $(u, v) \in U_k(\mathcal{T}_h)$ is

$$a_k(u,v) = \sum_{T\in\mathcal{T}_h} \int_T \nabla u\cdot\nabla v - \sum_{F\in\mathcal{S}_h} \int_F \{\nabla u\}\cdot[v] - \sum_{F\in\mathcal{S}_h} \int_F \{\nabla v\}\cdot[u] + \sum_{F\in\mathcal{S}_h} \int_F \alpha [u]\cdot[v],$$

with a penalty parameter $\alpha = \alpha_0 k^2 / h$ for optimal coercivity. The mesh-dependent norm is

$$\|v\|_{1,k,h}^2 = \sum_T \|\nabla v\|_{L^2(T)}^2 + \frac{k^2}{h} \sum_F \|[v]\|_{L^2(F)}^2.$$

The multigrid architecture is based on the nested spaces $U_1 \subset U_2 \subset \cdots \subset U_p$ and associated operators:

• Prolongation $P_{k-1}^k$: natural injection $U_{k-1} \to U_k$,
• Restriction $R_k^{k-1}$: $L^2$-adjoint of prolongation, i.e., $(R_k^{k-1} v, w)_{L^2} = (v, P_{k-1}^k w)_{L^2}$.

This structure underpins the consistent splitting: coarse-grid corrections and error components are exactly decomposed with respect to the nested bases, and transfer operators are norm-preserving.

2. $p$-Multigrid W-Cycle Algorithm and Smoother Consistency

The core consistent splitting scheme in this context is the $p$-multigrid W-cycle. At each level $k$, the algorithm proceeds as follows:

1. Pre-smoothing: Apply $m$ steps of a Chebyshev polynomial smoother $G_{k,m}$, constructed as a degree-$m$ stability-optimized polynomial $P_m(\omega_k D_k^{-1}A_k)$, where $A_k$ is the stiffness matrix and $D_k$ its diagonal.
2. Coarse correction: Compute the coarse-level residual $r_{k-1}=R_k^{k-1}(f_k - A_k v_k^m)$, recursively apply two W-cycles at level $k-1$, and prolongate the correction.
3. Post-smoothing: Apply $m$ additional steps with $G_{k,m}$.

The Chebyshev smoother is critical for consistency; the polynomial $P_m$ satisfies: $$\sup_{x\in[0,1]} |\sqrt{x} P_m(x)| = \frac{1}{2m+1}, \quad \sup |P_m(x)| = 1,$$ with scaling parameter $\omega_k\lesssim 1/(k+1)$. Its cost is one sparse matrix-vector product per smoothing step plus $O(1)$ operations.

If the level-$k$ operators $A_k$ are assembled anew at each level (non-inherited), then the splitting is fully consistent in both algebraic and functional terms.

3. Convergence Theory and Effect of Splitting Consistency

Rigorous convergence analysis leverages the precise action of the splitting. For the two-level error propagation operator $E_{k,m}^{\mathrm{TL}}$, the approximation and smoothing estimates yield

$$\|E_{k,m}^{\mathrm{TL}}\|_{A_k,1} \lesssim \frac{k}{m},$$

implying that choosing $m \gtrsim k$ ensures contraction uniformly in $h$ and $p$ (Lei et al., 17 Sep 2025). The $W$-cycle generalizes this result recursively: $$\|E_{k,m}^W\|_{A_k,1} \lesssim \|E_{k,m}^{\mathrm{TL}}\|_{A_k,1} + O\left(\frac{k^2}{m^2}\right),$$ thus $m>C_0 k$ is sufficient for convergence independent of the discretization.

The distinction between non-inherited and inherited forms is essential:

• Non-inherited splitting: Each $A_k$ assembled at its own level; the smoothing cost required for uniform convergence is $m = O(p)$—a major reduction over previous $O(p^2)$ results. This usage ensures full consistency in the splitting, and theoretical error reduction reflects the fine-level spectral properties uniformly at all levels.
• Inherited splitting: Operators $\tilde{A}_{k-1}$ are defined via restriction, leading to an approximation error $O(hp/k^2)$ and smoothing constants $O(kp/(hm))$, requiring $m \gtrsim p^2/k$. Thus, inconsistencies in the splitting (in the mapping of stiffness forms between levels) degrade convergence efficiency.

4. Numerical Verification and Practical Implications

Extensive numerical experiments confirm the theoretical predictions:

• In 1D, with mesh-size and polynomial degree $p$ varied, using $m=p$ smoothing steps yields a convergence rate $\rho \sim 0.75$ independent of $h,p$.
• In 1D–3D Poisson and interface problems, W-cycle contraction numbers $\rho\approx 0.7$–0.8 hold across uniform and non-inherited hierarchies. Operator complexity remains $O(1)$ in $h$ and grows only mildly in $p$.
• Applying the $p$-multigrid W-cycle as a preconditioner to GMRES reduces iteration counts from $O(100\text{–}1000)$ (no preconditioning) to $O(10\text{–}20)$ independent of $h$ and $p$.

In cases of unfitted finite element meshes (geometries generated by a “big-element-condition” merging for smooth interfaces), the p-multigrid-preconditioned solvers are robust—iteration counts for 3D elliptic interface problems remain stable regardless of $p$ or mesh refinement.

Hierarchy Type	Required Smoothing Steps $m$	Uniform Convergence in $h,p$?
Non-inherited	$O(p)$	Yes
Inherited	$O(p^2)$	Not always

5. Extensions, Limitations, and Recommended Practice

The consistent splitting paradigm generalizes to other high-order DG discretizations and application domains. Key recommendations include:

• When possible, use non-inherited operators at each $p$-level to guarantee optimality and minimal smoothing cost.
• Choose penalty parameter $\alpha = \alpha_0 k^2/h$ with $\alpha_0$ in the range 5–10 for stability and coercivity.
• For inherited forms, be prepared for increased smoothing expenditure (as $p$ grows), unless problem-specific deflation or coarse correction mechanisms are included to compensate for the loss in splitting consistency.
• For unfitted or cut-cell meshes, combine p-multigrid preconditioning with mesh merging/aggregation to prevent small-cell-induced ill-conditioning.

In summary, a consistent splitting scheme—precisely constructed inter-level transfer, smoothing, and correction operators—is the foundation of scalable, uniformly convergent multigrid solvers for $p$-version SIPDG discretizations. Theoretical analysis and numerical evidence in (Lei et al., 17 Sep 2025) show that such schemes enable $O(p)$ cost scaling and independence from mesh-size, thereby making high-order DG methods competitive and robust for elliptic and interface problems.