Incomplete Interior Penalty Galerkin (IIPG) Method

• IIPG is a discontinuous Galerkin method that omits symmetric consistency terms to streamline computation while maintaining controlled stability.
• The formulation reduces face integrations, achieving practical efficiency without compromising coercivity and boundedness under proper penalty parameter choices.
• Stability and error analyses confirm IIPG's effectiveness for challenging problems, such as the complex Ginzburg–Landau equation, with optimal convergence rates.

The Incomplete Interior Penalty Galerkin (IIPG) method is a member of the interior penalty family of Discontinuous Galerkin (DG) finite element methods for the spatial discretization of partial differential equations, most notably employed in reaction–diffusion systems such as the complex Ginzburg–Landau equation. Positioned between the Symmetric (SIPG) and Nonsymmetric (NIPG) schemes, IIPG offers a favorable compromise in computational efficiency and stability, especially for nonlinear, stiff problems. The method modifies the unified interior-penalty formulation by omitting symmetric consistency terms on faces, leading to distinct properties in coercivity, stability, and practical implementation.

1. Mesh, Jumps, Averages, and the DG Norm

Let $\Omega\subset\mathbb{R}^d$ denote a bounded Lipschitz domain with a conforming, shape-regular triangulation $\mathcal{T}_h=\{K\}$ and interior faces $\mathcal{E}_h$. The local finite element space is

$$V_h = \{ v \in L^2(\Omega): \, v|_K \in P_r(K) \; \forall K \in \mathcal{T}_h \},$$

where $P_r$ denotes polynomials of degree at most $r$. For an interior face $e = \partial K^+ \cap \partial K^-$ with outward normals $n^+, n^-$, standard DG notation defines the jump $[w] = w^- n^- + w^+ n^+$ and average $\{w\} = \frac{1}{2}(w^- + w^+)$ for scalar $w$.

The broken DG norm on $V_h$ is

$$\|w\|_{DG}^2 := \sum_{K\in\mathcal{T}_h} \|\nabla w\|^2_{L^2(K)} + \sum_{e\in\mathcal{E}_h} \frac{\sigma}{h_e} \|[w]\|^2_{L^2(e)},$$

where $h_e \approx \operatorname{diam}(e)$ and $\sigma>0$ is the penalty parameter.

2. Formulation of the IIPG Bilinear Form

The unified interior-penalty DG form for the Laplacian is

$$a_h(u, v) = \sum_{K} \int_K \nabla u \cdot \nabla v\,dx - \sum_e \int_e \{\nabla u\}\cdot [v]\,ds + \theta \sum_e \int_e \{\nabla v\}\cdot [u]\,ds + \sum_e \int_e \frac{\sigma}{h_e}[u]\cdot[v]\,ds,$$

with $\theta\in\{1,0,-1\}$ selecting SIPG ($\theta=1$), IIPG ($\theta=0$), or NIPG ($\theta=-1$). The IIPG scheme omits the consistency term present in SIPG: $$a_h^{IIPG}(u,v) = \sum_{K} \int_K \nabla u \cdot \nabla v\,dx - \sum_e \int_e \{\nabla u\}\cdot [v]\,ds + \sum_e \int_e \frac{\sigma}{h_e}[u]\cdot[v]\,ds.$$ This modification results in fewer face integrals, streamlining assembly without significant loss in stability or accuracy for well-chosen $\sigma$.

3. Consistency, Coercivity, and Boundedness

Consistency: For sufficiently smooth $u\in H^2(\Omega)$, $v\in V_h$, face integration and local integration by parts yield

$a_h^{IIPG}(u,v) = \sum_K \int_K (-\Delta u) v\,dx,$

demonstrating that the method is formally consistent for elliptic operators.

Coercivity: There exists a minimal penalty $\sigma_0>0$, depending on the mesh regularity and polynomial degree $r$, such that for $\sigma\geq\sigma_0$,

$$a_h^{IIPG}(v,v) \geq \beta \|v\|_{DG}^2 - C_0 \|v\|_{L^2(\Omega)}^2,$$

with $\beta>0$. Application of the discrete Poincaré inequality and selection of sufficiently large $\sigma$ yields the fully coercive estimate

$a_h^{IIPG}(v,v) \geq \frac{\beta}{2} \|v\|_{DG}^2.$

Boundedness: The bilinear form is also bounded: $|a_h^{IIPG}(u,v)| \leq C_b \|u\|_{DG} \|v\|_{DG}.$ Coercivity and boundedness are foundational for stability and convergence results.

4. Stability Analysis in Nonlinear Dynamics

For the time-discretized real/imaginary decomposition of the complex Ginzburg–Landau PDE using a $\theta$-scheme, the IIPG operator defines the spatial discretization. The stability analysis proceeds by testing each real and imaginary component, summing, and applying discrete Grönwall’s lemma. This results in the global energy recurrence: $$A^n - A^{n-1} + \Delta t\, \beta \left( \|u_{h1}^n\|_{DG}^2 + \|u_{h2}^n\|_{DG}^2 \right) \leq C\Delta t ( \|u_{h1}^n\|_{L^2}^2 + \|u_{h2}^n\|_{L^2}^2 ) + C'\Delta t,$$ with $$A^n := \frac{1}{2}( \|u_{h1}^n\|_{L^2}^2 + \|u_{h2}^n\|_{L^2}^2 )$$. Recursive application establishes unconditional stability: $$\|u_h^n\|_{L^2(\Omega)}^2 + \Delta t \sum_{j=1}^n \|u_h^j\|_{DG}^2 \leq C(T)\left( \|u_h^0\|_{L^2}^2 + 1 \right),$$ for $n\leq N$. The IIPG method is thus stable in $L^\infty(0,T;L^2(\Omega)) \cap L^2(0,T;DG)$ norms, even in the presence of nonlinearity.

5. Error Estimates and Convergence

Although detailed error estimates are not the primary focus, standard DG elliptic projection theory yields that, given $u\in H^{s+1}(\Omega)$ for $1\leq s\leq r$ and $\sigma\gg p^2/h$,

$$\|u(t) - u_h(t)\|_{DG} \leq C(\sigma) h^s p^{-s} \|u\|_{H^{s+1}},$$

$$\|u(t) - u_h(t)\|_{L^2} \leq C' h^{s+1} p^{-(s+1)} \|u\|_{H^{s+1}},$$

uniformly in $t$. For implicit (backward Euler) time stepping, overall first-order accuracy in $\Delta t$ is achieved. This supports the use of IIPG in simulations where both spatial and temporal accuracy requirements are nontrivial.

6. Comparative Properties: IIPG, SIPG, and NIPG

A summary of the three interior penalty variants:

Scheme	Symmetry	Coercivity	Stability (nonlinear Landau)
SIPG	Fully symmetric	Strong (SPD for large $\sigma$)	Most robust (L$^2$ norm $\lesssim$ 1)
IIPG	Penalty block only	Coercive if $\sigma\geq\sigma_0$, $\beta>0$	Modest norm growth ($\lesssim$ 2–3)
NIPG	Fully nonsymmetric	Coercivity constant $\beta$ can be very small	Becomes unstable as $h{\downarrow}$

Numerical experiments for the nonlinear Landau model (with $\sigma\approx 10^3$, $a=b\approx 0.82$) demonstrate the SIPG method maintains a bounded $L^2$-norm, IIPG remains stable with modest norm growth, while NIPG can exhibit rapid norm blow-up for mesh refinement. For very large penalty ($\sigma\approx 10^8$), performance across all schemes becomes nearly indistinguishable.

7. Implementation Guidance and Parameter Selection

Practical recommendations stem directly from the balance between computational efficiency and stability:

• When the highest robustness is needed, SIPG is favored, particularly for moderate penalty parameters.
• The IIPG method is beneficial when computational resources are constrained, as it necessitates assembling one fewer face term than SIPG but provides superior stability relative to NIPG.
• Provided $\sigma$ is chosen such that $\sigma\geq O(p^2 h^{-1})$, IIPG maintains optimal coercivity and achieves the same $h^r$ rate of convergence as SIPG.

Key scholars for foundational theory include Arnold, Brezzi, Cockburn & Marini (2002) for coercivity, boundedness, and error estimates. The stability analysis leverages discrete functional inequalities and energy methods as articulated in the analysis for the complex Ginzburg–Landau system (Kostas, 8 Nov 2025).

A plausible implication is that, for large-scale nonlinear PDE simulation, IIPG represents an attractive trade-off, yielding computational savings over fully symmetric SIPG, while avoiding the stability pitfalls of NIPG, especially when the penalty parameter is set sufficiently high.