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This paper demonstrates efficient methods for the biharmonic equation using nonconforming finite elements, indicating enhanced convergence rates.
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.
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Li et al. (2025) studied this question.
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