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

TL;DR

The paper tackles the 3D generalized tensor-valued Stokes equation arising from the Hessian complex by developing a low-order nonconforming linear element discretization that avoids vertex degrees of freedom through a $L^2$ traceless-tensor pressure and a discrete Helmholtz decomposition. It proves well-posedness and optimal error estimates for the discretization, and then leverages these results to construct a decoupled, low-order finite element method for the triharmonic equation by splitting it into two biharmonic problems and a generalized Stokes problem, solved with Morley-Wang-Xu elements for the biharmonics and the nonconforming Stokes discretization for the Stokes part. The approach provides a practical, efficient framework for high-order PDEs by using low-order, compatible finite elements and a robust decomposition to ensure stability. Numerical experiments on unit cubes and non-convex L-shaped domains confirm the predicted convergence rates and demonstrate the method’s robustness.

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.

Theorems & Definitions (45)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof