A Coupled High-Order Continuous and Discontinuous Galerkin Finite Element Scheme for the Davey-Stewartson System (Communications in Computational Physics, 2025)
In this paper, we propose a coupled discontinuous Galerkin (DG) and continuous Galerkin (CG) scheme for solving the nonlinear evolution Davey-Stewartson(DS) system in dimensionless form. The DS system consists of two coupled nonlinear and complex structure partial differential equations. The wave’s amplitude in thefirst equation is solved by the high-efficiency local DG method, and the velocity in thesecond equation is obtained by a standard CG method. No matching conditions areneeded for the two finite element spaces since the normal component of the velocityis continuous across element boundaries. The main strengths of our approach are thatwe combine the advantage of DG and CG methods, using DG methods handling thenonlinear Schrödinger equation to obtain high parallelizability and high-order formal accuracy, using the continuous finite elements solving the velocity to maintain total energy conservation. We prove the energy-conserving properties of our scheme and error estimates in $L^2$-norm. However, the non-linearity terms bring a lot of trouble to theproof of error estimates. With the help of energy-conserving properties, we construct aseries of energy equations to obtain error estimates. Numerical tests for different typesof systems are presented to clarify the effectiveness of numerical methods.
Keywords: Davey-Stewartson system; local discontinuous Galerkin method; continuous Galerkin method; error estimates.