Stability estimates for Interior Penalty D.G. Methods for the Nonlinear Dynamics of the complex Ginzburg Landau equation
Dimitrios Kostas
TL;DR
This work addresses stability of interior-penalty DG discretizations for the complex Landau equation, deriving stability estimates and performing a comprehensive comparison of SIPG, NIPG, and IIPG schemes. It establishes existence/uniqueness and develops discrete energy estimates that yield stability uniform in mesh size and time step, with Grönwall-type bounds. Numerically, SIPG shows the strongest robustness to nonlinearities, while IIPG typically outperforms NIPG; at very large penalty values, all three schemes converge to comparable stability behavior. The results offer practical guidance for selecting DG schemes in nonlinear reaction–diffusion systems and motivate future extensions to space–time DG formulations and adaptive strategies.
Abstract
This study investigates the complex Landau equation, a reaction diffusion system with applications in nonlinear optics and fluid dynamics. The equation's nonlinear imaginary component introduces rich dynamics and significant computational challenges. We address these challenges using Discontinuous Galerkin (DG) finite element methods. A rigorous stability analysis and a comparative study are performed on three distinct DG schemes : Symmetric Interior Penalty Galerkin (SIPG), Nonsymmetric Interior Penalty Galerkin (NIPG), and Incomplete Interior Penalty Galerkin (IIPG). These methods are compared in terms of their stability and computational efficiency. Our numerical analysis and computational results demonstrate that all three discontinuous Galerkin (DG) schemes are stable. However, the Symmetric Interior Penalty Galerkin (SIPG) scheme proves to be the most robust, as its norm remains bounded even in the presence of nonlinear terms a property not shared by the others. A comparison between the Incomplete Interior Penalty Galerkin (IIPG) and Nonsymmetric Interior Penalty Galerkin (NIPG) schemes shows that IIPG has superior stability properties. For high values of the penalty parameter, all methods exhibit similar stability behavior. Our results highlight the suitability of DG methods for simulating complex nonlinear reaction-diffusion systems and provide a practical framework for selecting the most efficient scheme for a given problem.