In this paper, two new third order nonconforming finite element methods (NFEMs) on the rectangular grid are proposed for the biharmonic problem. Such finite elements are constructed by enriching 9 high-order polynomial bubbles to the P₄ polynomial space on each rectangle, so that the extra degrees of freedom can enforce the required sub-continuity of the finite element on the four edges. We prove that the methods are well-defined, have unique solution and converge at Oh³ in the H² norm and Oh⁵ in L² norm. The numerical results show that the new elements are very efficient.
Li et al. (Mon,) studied this question.