This paper aims to present the virtual element method (VEM) for solving the Davey–Stewartson equations with application in fluid mechanics. The VEM is a recent technology that can be regarded as a generalization of the standard finite element method (FEM) to general meshes without the need to integrate complex nonpolynomial functions on the elements. This method only utilizes degrees of freedom associated with the boundary, hence reducing computational complexity compared to the standard FEM. To obtain a full- discrete scheme we combine a semi-implicit scheme with the VEM for time and space variable discretizations, respectively. Furthermore, we obtain an error bound for the full-discrete scheme. The theoretical analysis demonstrates that the convergence rate in the L2 norm is O(h2+τ). Numerical examples confirm efficiency and applicability of the presented method and validate the theoretical outcomes.

中文翻译：

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使用虚拟元法对 Davey-Stewartson 方程进行数值模拟和误差估计


本文旨在介绍用于求解 Davey-Stewartson 方程的虚拟元方法 （VEM） 及其在流体力学中的应用。VEM 是一项最新的技术，可以看作是标准有限元法 （FEM） 对一般网格的推广，而无需在单元上集成复杂的非多项式函数。这种方法只利用与边界相关的自由度，因此与标准 FEM 相比，降低了计算复杂性。为了获得全离散方案，我们将半隐式方案与 VEM 分别用于时间和空间变量离散化。此外，我们获得了全离散方案的误差边界。理论分析表明，L2 范数中的收敛速率为 O（h2+τ）。数值算例证实了所提方法的效率和适用性，并验证了理论结果。