H1− Galerkin mixed finite element method using tensor product of cubic B-splines for two-dimensional Kuramoto-Sivashinsky equation

Abstract

The two-dimensional $\left(2D\right)$ Kuramoto-Sivashinsky equation offers a robust framework for studying complex, chaotic, and nonlinear dynamics in various mathematical and physical contexts. Analyzing this model also provides insights into higher-dimensional spatio-temporal chaotic systems that are relevant to many fields. This article aims to solve the scalar form of the two-dimensional Kuramoto-Sivashinsky equation using the $H^1-$ mixed Galerkin finite element method. By introducing an intermediate variable, the equation is transformed into a coupled system. This system is then approximated using the $H^1-$ mixed Galerkin finite element method, with the tensor product of the cubic B-spline in x and y directions serving as the test and trial functions in both the semi-discrete and fully discrete schemes. In this approach, triangularization is avoided, thereby reducing the size of the stiffness matrix. In the fully discrete scheme, the time derivative is approximated using the backward Euler method. The suitable linearization method is used to simplify the nonlinear term in both schemes. The theoretical analysis yields optimal order error estimates for the scalar unknown and its flux in the $L^2$, $L^{\infty }$, and $H^1$ norms, demonstrating the accuracy, efficiency, and stability of the proposed method. Additionally, three test problems are numerically analyzed to validate these theoretical results. The chaotic behavior of the equation is analyzed, in relation to the viscosity coefficient γ, and is also numerically investigated using the proposed method.