  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.
