TY - JOUR
T1 - On the $P_1$ Powell-Sabin Divergence-Free Finite Element for the Stokes Equations
AU - Shangyou Zhang
JO - Journal of Computational Mathematics
VL - 3
SP - 456
EP - 470
PY - 2008
DA - 2008/06
SN - 26
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/jcm/8636.html
KW - Powell Sabin triangles, Mixed finite elements, Stokes, Divergence-free element.
AB - <p style="text-align: justify;">The stability of the $P_1$-$P_0$ mixed-element &nbsp; is established on general Powell-Sabin triangular grids. &nbsp;The piecewise linear finite element solution approximating &nbsp; the velocity is divergence-free pointwise &nbsp; for the Stokes equations. &nbsp;The finite element solution approximating the pressure in &nbsp; the Stokes equations can be obtained as a byproduct if &nbsp; an iterative method is adopted for solving the discrete &nbsp; linear system of equations. &nbsp;Numerical tests are presented confirming the theory on the stability and the optimal order of convergence for the $P_1$ Powell-Sabin &nbsp; divergence-free finite element method.</p>