Superconvergence of the Q_k+1,k-Q_k,k+1 divergence-free finite element

2011/12/16 by Yunqing Huang, Huang, Yunqing, Shangyou Zhang +1
Engineering · #65M60 #65N30 #76D07 #Advanced Numerical Methods in Computational Mathematics #Elasticity and Material Modeling #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.1112.3705

openalex publication_date 2011/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

By the standard theory, the stable Q_k+1,k-Q_k,k+1/Q_k^dc' divergence-free element converges with the optimal order of approximation for the Stokes equations, but only order k for the velocity in H¹-norm and the pressure in L²-norm. This is due to one polynomial degree less in y direction for the first component of velocity, a Q_k+1,k polynomial. In this manuscript, we will show a superconvergence of the divergence free element that the order of convergence is truly k+1, for both velocity and pressure. Numerical tests are provided confirming the sharpness of the theory.

Citations

• Stability of Finite Elements under Divergence Constraints

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