Pointwise divergence free velocity field approximations of the Stokes system are gaining popularity due to their necessity in precise modelling of physical flow phenomena. Several methods have been designed to satisfy this requirement; however, these typically come at a greater cost when compared with standard conforming methods, for example, because of the complex implementation and development of specialized finite element bases. Motivated by the desire to mitigate these issues for 2D simulations, we present a $C^0$-interior penalty Galerkin (IPG) discretization of the Stokes system in the stream function formulation. In order to preserve a spatially varying viscosity this approach does not yield the standard and well known biharmonic problem. We further employ the so-called robust interior penalty Galerkin (RIPG) method; stability and convergence analysis of the proposed scheme is undertaken. The former, which involves deriving a bound on the interior penalty parameter is particularly useful to address the $\mathcal{O}(h^{-4})$ growth in the condition number of the discretized operator. Numerical experiments confirming the optimal convergence of the proposed method are undertaken. Comparisons with thermally driven buoyancy mantle convection model benchmarks are presented.

翻译：Stokes系统点态散度自由速度场逼近因精确模拟物理流动现象的必要性而日益受到关注。为满足这一要求，研究者已设计出多种方法；然而，与标准协调方法相比，这些方法通常成本更高，例如由于需要复杂实现和开发专用有限元基函数。受缓解二维模拟中这些问题的需求驱动，我们提出了流函数形式下Stokes系统的$C^0$-内部惩罚伽辽金(IPG)离散格式。为保留空间变化粘度，该方法并未生成标准且熟知的双调和问题。我们进一步采用稳健内部惩罚伽辽金(RIPG)方法；对所提格式进行了稳定性和收敛性分析。前者涉及推导内部惩罚参数的界，这对解决离散算子条件数中$\mathcal{O}(h^{-4})$增长问题尤为有用。数值实验验证了所提方法的最优收敛性。同时呈现了与热驱动浮力地幔对流模型基准的对比结果。