SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 919-945, April 2024.
Abstract. A general a posteriori error analysis applies to five lowest-order finite element methods for two fourth-order semilinear problems with trilinear nonlinearity and a general source. A quasi-optimal smoother extends the source term to the discrete trial space and, more important, modifies the trilinear term in the stream-function vorticity formulation of the incompressible two-dimensional Navier–Stokes equations and the von Kármán equations. This enables the first efficient and reliable a posteriori error estimates for the two-dimensional Navier–Stokes equations in the stream-function vorticity formulation for Morley, two discontinuous Galerkin, [math] interior penalty, and weakly overpenalized symmetric interior penalty discretizations with piecewise quadratic polynomials.

中文翻译：

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二次非线性四阶半线性问题的后验误差控制

SIAM 数值分析杂志，第 62 卷，第 2 期，第 919-945 页，2024 年 4 月
。摘要。一般后验误差分析适用于具有三线性非线性和一般源的两个四阶半线性问题的五种最低阶有限元方法。准最优平滑器将源项扩展到离散试验空间，更重要的是，修改了不可压缩二维纳维-斯托克斯方程和冯卡门方程的流函数涡度公式中的三线性项。这使得能够对莫利流函数涡度公式中的二维纳维-斯托克斯方程、两个不连续伽辽金、[数学]内部惩罚以及分段二次的弱过度惩罚对称内部惩罚离散化进行第一个有效且可靠的后验误差估计多项式。