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Stability of the SUPG finite element method for transient advection-diffusion problems

机译：SUPG有限元方法对瞬态对流扩散问题的稳定性

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摘要

Implicit time integration coupled with SUPG discretization in space leads to additional terms that provide consistency and improve the phase accuracy for convection dominated flows. Recently, it has been suggested that for small Courant numbers these terms may dominate the streamline diffusion term, ostensibly causing destabilization of the SUPG method. While consistent with a straightforward finite element stability analysis, this contention is not supported by computational experiments and contradicts earlier Von-Neumann stability analyses of the semidiscrete SUPG equations. This prompts us to re-examine finite element stability of the fully discrete SUPG equations. A careful analysis of the additional terms reveals that, regardless of the time step size, they are always dominated by the consistent mass matrix. Consequently, SUPG cannot be destabilized for small Courant numbers. Numerical results that illustrate our conclusions are reported.
机译：隐式时间积分与空间中的SUPG离散化相结合会产生其他术语，这些术语可提供一致性并提高对流主导流的相位精度。最近，有人提出，对于小的库兰特数，这些术语可能主导流线扩散术语，表面上引起SUPG方法的不稳定。尽管与简单的有限元稳定性分析是一致的，但是该争论不受计算实验的支持，并且与半离散SUPG方程的早期Von-Neumann稳定性分析相矛盾。这促使我们重新检查完全离散的SUPG方程的有限元稳定性。对附加项的仔细分析表明，无论时间步长如何，它们始终受一致的质量矩阵支配。因此，对于小的Courant数，SUPG不会不稳定。数值结果表明了我们的结论。

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