A hybrid $C^{0}$-interior penalty method for the nematic Helmholtz--Korteweg equation

Abstract

The nematic Helmholtz--Korteweg equation models the propagation of time-harmonic acoustic waves in nematic Korteweg fluids, such as nematic liquid crystals. The PDE augments the classical Helmholtz equation with two additional fourth-order terms, one of which is anisotropic in the direction of the nematic field. We refine the previous continuous analysis of Farrell et al. (2025) by using the Cordes condition and present a $C^0$-hybrid interior penalty discretization. The proposed discretization offers greater flexibility than $C^1$-conforming methods and is well-suited for applications in three dimensions and on curved domains. We prove stability of the method for any polynomial degree greater than or equal to two, independent of the spatial dimension, provided that the anisotropy is sufficiently small. Further, we show that the sequence of discrete solutions converges to the continuous solution under minimal regularity assumptions and derive convergence rates if the continuous solution has additional regularity. Finally, we illustrate the capabilities of the method through numerical examples.