The focus of this paper is on a implicit Backward-Euler (BE) scheme with the mixed finite element method (FEM) for the two-dimentional general Rosenau-RLW equation. In which the bilinear element is used to approximate the exact solution v and the variable $p=-\mathrm{\Delta }v$, and the zero-order nédéléc's finite element to the variable $\overrightarrow{q}=\sqrt{\varphi }\mathrm{\nabla }v$, respectively. An important point is that with the proposed scheme we are able to bound the numerical solution in $H^1$-norm, so that we can efficiently solve the nonlinear term. Moreover, the stability, existence and uniqueness of approximate solution are demonstrated. Based on the combination of the interpolation and projection technique, the superconvergence estimates of order $\mathcal{O}(h^2+\tau )$ for v in $H^1$-norm and the introducing variable $\overrightarrow{q}$ in $L^2$-norm, and optimal error estimate of order $\mathcal{O}(h+\tau )$ for introducing variable p in $H^1$-norm is proved. Finally, numerical example is done to certify our theoretical results.

本文的重点是对二维一般 Rosenau-RLW 方程使用混合有限元方法 (FEM) 的隐式 Backward-Euler (BE) 方案。其中双线性元素用于逼近精确解v和变量$磷=-\mathrm{\Delta }v$，以及零阶 nédéléc 的有限元到变量 $\overrightarrow{q}=\sqrt{\phi }\mathrm{\nabla }v$， 分别。重要的一点是，使用所提出的方案，我们能够将数值解限制在$H^1$-norm，以便我们可以有效地求解非线性项。此外，证明了近似解的稳定性、存在性和唯一性。基于插值和投影技术相结合的阶次超收敛估计$\mathcal{哦}(H^2+\tau )$对于v输入$H^1$-范数和引入变量 $\overrightarrow{q}$ 在 ${升}^2$-范数，以及阶数的最优误差估计 $\mathcal{哦}(H+\tau )$用于引入变量p在$H^1$——范数得到证明。最后,通过数值例子验证了我们的理论结果。