Nonconforming Linear Element Method for a Generalized Tensor-Valued Stokes Equation with Application to the Triharmonic Equation

A nonconforming linear element method is developed for a three-dimensional generalized tensor-valued Stokes equation associated with the Hessian complex in this paper.

A discrete Helmholtz decomposition for the piecewise constant space of traceless tensors is established, ensuring the well-posedness of the nonconforming method, and optimal error estimates are derived.

Building on this, a low-order decoupled finite element method for the three-dimensional triharmonic equation is constructed by combining the Morley-Wang-Xu element methods for the biharmonic subproblems with the proposed nonconforming linear element method.

Numerical experiments confirm the theoretical convergence rates.