SIPG: Symmetric Interior Penalty Galerkin

• SIPG is a discontinuous Galerkin method that uses symmetric penalty terms to weakly enforce continuity in the discretization of elliptic and parabolic PDEs.
• It relies on carefully scaled penalty parameters and inverse inequalities to guarantee stability and optimal convergence across irregular, polytopic meshes.
• The method utilizes total-degree polynomial bases defined in the physical frame to reduce computational degrees-of-freedom, enabling efficient high-order approximations.

The Symmetric Interior Penalty Galerkin (SIPG) method is a discontinuous Galerkin (DG) finite element scheme for elliptic and parabolic PDEs, notable for its symmetric imposition of interior penalties to weakly enforce inter-element continuity. SIPG is widely used to achieve provably stable, optimally convergent spatial discretizations on general meshes, including polytopic, polygonal/polyhedral domains and for high-order ($hp$ and spline-based) approximations. The method is characterized by the design of its discrete bilinear forms, the scaling and selection of penalty parameters, and by distinctive error and stability estimates that remain robust even in the presence of mesh degeneration or high polynomial degree.

1. Mathematical Formulation and Discrete Bilinear Forms

Let $\Omega\subset\mathbb{R}^d$ be a bounded domain, discretized into elements $\mathcal{T}$, with each element $T$ possibly polygonal or polyhedral. The strong form for a prototypical fourth-order elliptic problem, such as the biharmonic equation, reads $\Delta^2 u = f$ in $\Omega$, with boundary conditions $u=0$, $\partial_n u = 0$ on $\partial\Omega$ (Dong, 2018).

The SIPG method approximates $u$ in the broken Sobolev space $$V_h = \{v\in L^2(\Omega) : v|_T\in \mathcal{P}_{p_T}(T)\}$$, using polynomials of total degree $p_T\geq 2$ on each $T$. The SIPG discrete bilinear form for two functions $u_h, v_h\in V_h$ is:

$$\begin{aligned} B(u_h,v_h) &= \sum_{T\in\mathcal{T}} \int_T \Delta u_h\,\Delta v_h\,dx \ &- \sum_{F\in\mathcal{E}} \int_F \big\{ \nabla(\Delta u_h) \big\} \cdot [v_h]\,ds - \sum_{F\in\mathcal{E}} \int_F \big\{ \nabla(\Delta v_h) \big\} \cdot [u_h]\,ds \ &- \sum_{F\in\mathcal{E}} \int_F \big\{ \Delta u_h \big\} [\nabla v_h]\,ds - \sum_{F\in\mathcal{E}} \int_F \big\{ \Delta v_h \big\} [\nabla u_h]\,ds \ &+ \sum_{F\in\mathcal{E}} \int_F (\sigma [u_h]\cdot[v_h] + \tau [\nabla u_h]\cdot[\nabla v_h])\,ds \end{aligned}$$

where $[\cdot]$ and $\{\cdot\}$ denote the DG jump and average across face $F$, and $\sigma$, $\tau$ are face-wise penalty parameters.

This general structure is mirrored for SIPG discretizations of second-order (diffusion), mixed, or lower-order elliptic and parabolic problems, with corresponding forms for the fluxes, gradient averages, and jump penalties (Karasözen et al., 2015, Zhang et al., 2018, Charrier et al., 2016).

2. Penalty Parameter Selection and Inverse Inequalities

Stability and optimal error bounds in SIPG are critically dependent on the face penalty parameters. For each face $F$, the recommended scaling is

$$\sigma|_F = C_\sigma \max_{i\in\{+,-\}} \left( C_{\rm INV}(p_i,T_i,F)\, \frac{p_i^2 |F|}{|T_i|} \left( C_{\rm inv,2}\, \frac{p_i^4}{h_i^2} \right) \right), \qquad \tau|_F = C_\tau \max_{i\in\{+,-\}} \left( C_{\rm INV}(p_i,T_i,F)\, \frac{p_i^2 |F|}{|T_i|} \right)$$

(Dong, 2018)

Here, $C_{\rm INV}(p,T,F)$ is the constant from the sharp trace inverse inequality

$$\|v\|_{L^2(F)}^2 \le C_{\rm INV}(p,T,F)\,p^2\frac{|F|}{|T|} \|v\|_{L^2(T)}^2$$

which is chosen to be robust even as $|F|\to0$ or the number of faces grows arbitrarily large. For harmonic polynomial bases, a further refined inverse inequality is proved:

$$\| \nabla v \|_{L^2(T)}^2 \le \left( C_s\,\frac{(p+1)(p+d)}{h_T} \right)^2\, \| v \|_{L^2(T)}^2, \quad v\in\mathcal{H}_p(T),\ \Delta v = 0$$

This improved bound achieves robustness to the mesh face count and degeneration, contrasting with the standard inverse estimate $p^4/h_T^2$ (Dong, 2018).

3. Stability Theory: Coercivity, Continuity, and Error Estimates

Stability analysis for SIPG relies on verification of coercivity and continuity in mesh-dependent norms. For the biharmonic SIPG scheme,

$B(v,v) \ge C_{\rm coer} \| v \|_{DG}^2$

is established for $p=2,3$ using harmonic polynomial inverse bounds, and

$B(w,v) \le C_{\rm cont} \| w \|_{DG} \| v \|_{DG}$

follows via Cauchy–Schwarz and the same inverse techniques.

The a-priori error estimate for the DG-norm is

$$\| u - u_h \|_{DG} \le C\, h^{\min\{ p+1, l \} - 2} p^{l-2} \| u \|_{H^l(\Omega)}$$

for $u|_T \in H^l(T),\ l > 3 + d/2$, provided the mesh either has uniformly bounded face number for general $p\ge 2$, or arbitrarily many faces for $p=2,3$ (Dong, 2018).

4. Basis Construction and Physical Frame, DOF Scaling

A distinguishing advantage of this SIPG framework is the use of total-degree polynomial bases $\mathcal{P}_p(T)$ defined directly in the physical coordinates, obviating the need for reference-to-physical mappings, which become highly complex on arbitrary polygons/polyhedra. The local basis on $T$ is

$\mathcal{P}_p(T) = \{ \text{all polynomials of total degree} \le p\ \text{in physical %%%%29%%%%} \}$

This approach reduces local degrees-of-freedom from $\sim (p+1)^d$ (tensor-product) to $\sim p^d/d!$, which is particularly efficient for high-order methods. Numerical results demonstrate faster convergence per degree-of-freedom and effective applicability on meshes with arbitrarily many faces per element (Dong, 2018).

5. Performance and Robustness: Numerical Results

Extensive numerical tests confirm that the SIPG scheme maintains stable condition numbers, spectrum clustering, and optimal convergence properties:

• $h$-refinement: For $p=2$, DG-norm errors behave as $O(h^{p-1})$; $H^1$-seminorm errors as $O(h^p)$; $L^2$-errors as $O(h^{p+1})$, with $L^2$ slightly suboptimal for $p=2$.
• $p$-refinement: Exponential convergence in DG-norm with $p$ for analytic solutions.
• Robustness to mesh degeneration: Stable performance on meshes with hundreds of faces per element or arbitrarily small faces; conditioning and spectral properties remain favorable.

6. Connections to Other SIPG Applications

The methodology and penalty parameter selection presented here have direct analogues in SIPG schemes for second-order elliptic and parabolic PDEs (Karasözen et al., 2015, Zhang et al., 2018, Charrier et al., 2016), isogeometric analysis (Takacs, 2019, Moore, 2020), and problems featuring highly irregular or polytopic meshes. Face-wise penalty scaling using robust inverse inequalities is now recognized as essential for stability and optimality in SIPG formulations on general mesh topologies.

7. Limitations, Practical Trade-offs, and Implementation Notes

The SIPG framework described is provably robust as $|F|\to 0$ or the number of faces grows without bound, for $p=2,3$ polynomial bases. For higher $p$, the analysis requires further refinement in polynomial approximation and inverse inequalities. The use of total-degree polynomial bases in the physical frame yields significant computational savings, but complicates the application of quadrature, basis construction, and assembly algorithms relative to classical reference-element approaches.

In implementation, it is critical to:

• Employ face-wise trace inverse constants $C_{\rm INV}(p,T,F)$ computed directly for each face, especially on agglomerated or degenerate meshes.
• Select penalty constants $C_\sigma$, $C_\tau$ sufficiently large for coercivity yet moderate to avoid ill-conditioning.
• Use covering arguments and simplicial sub-embeddings to support inverse inequalities for general element shapes.

Mesh regularity assumptions are minimal; only the existence of a simplicial sub-embedding with base $F$ and controlled height is required to guarantee sharp inverse estimates.

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The SIPG methodology examined here affords a stable, high-order, flexible DG approach to fourth-order elliptic problems of biharmonic type on fully general computational meshes, distinguished by its robust penalty scaling, physical-frame polynomial bases, and analytic error bounds. It sets the paradigm for discontinuous Galerkin discretizations on polytopic meshes with arbitrary element-face counts, and forms a blueprint for the design and analysis of SIPG schemes in a diverse array of high-order DG settings (Dong, 2018).