The Lowest Order Differentiable Finite Element on Rectangular Grids

Jun Hu, Yunqing Huang, Shangyou Zhang
2011 SIAM Journal on Numerical Analysis  
A macro type of biquadratic C 1 finite elements is constructed on rectangle grids. This is a rectangular version of the C 1 Powell-Sabin element, a C 1 -P 2 element on triangular grids. Here, each rectangle of the base grid is refined into four subrectangles. As in the case of the Powell-Sabin element, we have more constraints than the number of degrees of freedom on each macroelement. However, the extra constraints are consistent. It is shown further that the constructed finite element space
more » ... the full C 1 -Q 2 space on the grid. It is also shown that the finite element space is a tensor product space of one-dimensional C 1 -P 2 spaces, where the nodal basis is supported on four intervals. The B-spline function of P 2 is supported on three intervals. The Girault-Scott operator is extended to the element. The application and the convergence of the finite element to the biharmonic equation are presented. Numerical tests are provided.
doi:10.1137/100806497 fatcat:nio3hbqhz5c3bktz7zwz4bdxmy