We present a novel and comparative analysis of finite element discretizations for a nonlinear Rosenau-Burgers model including a biharmonic term. We analyze both continuous and mixed finite element approaches, providing stability, existence, and uniqueness statements of the corresponding variational methods. We also obtain optimal error estimates of the semidiscrete scheme in corresponding B\^ochner spaces. Finally, we construct a fully discrete scheme through a backward Euler discretization of the time derivative, and prove well-posedness statements for this fully discrete scheme. Our findings show that the mixed approach removes some theoretical impediments to analysis and is numerically easier to implement. We provide numerical simulations for the mixed formulation approach using $C^0$ Taylor-Hood finite elements on several domains. Our numerical results confirm that the algorithm has optimal convergence in accordance with the observed theoretical results.

翻译：本文对包含双调和项的非线性Rosenau-Burgers模型提出了一种新颖的有限元离散比较分析。我们分析了连续和混合有限元两种方法，给出了相应变分方法的稳定性、存在性和唯一性论述。同时，我们在相应的Bochner空间中获得了半离散格式的最优误差估计。最后，通过时间导数的后向欧拉离散化构建了全离散格式，并证明了该全离散格式的适定性。研究结果表明，混合方法消除了某些理论分析障碍，且在数值实现上更为简便。我们采用$C^0$ Taylor-Hood有限元在多个区域上对混合公式进行了数值模拟。数值结果证实该算法具有与理论结果相一致的最优收敛性。