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

TL;DR

This paper introduces a nonconforming linear element method for a 3D tensor-valued Stokes equation, establishing well-posedness and optimal error estimates, and applies it to develop a low-order decoupled finite element method for the triharmonic equation, validated by numerical experiments.

Contribution

It develops a novel nonconforming linear element method for tensor-valued Stokes equations and applies it to construct a decoupled finite element method for the triharmonic equation.

Findings

Optimal error estimates are derived for the method.

Numerical experiments confirm the theoretical convergence rates.

The method ensures well-posedness through a discrete Helmholtz decomposition.

Abstract

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.