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An edge-bubble stabilized finite element method for fourth-order parabolic problems

机译：四阶抛物线问题的边缘气泡稳定有限元方法

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摘要

We develop an edge-bubble stabilized finite element method for fourth-order parabolic problems. The method begins with a non-conforming approach, in which C~0 basis functions are used to approximate the coarse scale of the bulk field. Continuity of function derivatives is enforced at element edges with Lagrange multipliers. The fine-scale bulk field is approximated with higher order edge-bubbles that are held fixed over time slabs, providing for static condensation and an elimination of the multipliers. The resulting formulation shares several common features with recent non-conforming approaches based on Nitsche's method, albeit with the important difference that stability terms follow automatically from the approximation to the fine scale. As an application, we consider the problem of plane Poiseuille flow for a second-gradient fluid. Convergence studies provided for the case of steady flow indicate synchronous rates of convergence in L~2 and H~1 error norms. Some new time-dependent results for the second-gradient theory are also provided.
机译：我们针对四阶抛物线问题开发了一种边缘气泡稳定有限元方法。该方法以不合格方法开始，在该方法中，使用C〜0基函数来近似体场的粗尺度。函数导数的连续性通过Lagrange乘子在元素边缘强制执行。精细体积场近似于随时间推移而固定的高阶边缘气泡，从而提供了静态冷凝并消除了乘数。所得的公式与基于Nitsche方法的最新不合格方法具有几个共同的特征，尽管其重要区别在于，稳定性项从近似值到精确范围都会自动遵循。作为一种应用，我们考虑了用于第二梯度流体的平面Poiseuille流动问题。针对稳定流的收敛研究表明，L〜2和H〜1误差范数中的同步收敛速率。第二梯度理论还提供了一些新的时变结果。

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