Superconvergent and Divergence-Free Finite Element Methods for Stokes Equation

TL;DR

This work develops stabilization-free, divergence-free finite element methods for the Stokes equation by employing $H(\operatorname{div})$-conforming velocity spaces and discontinuous pressures, connected via a weak deviatoric gradient operator $\operatorname{dev}\,\mathrm{grad}_w$. A distributional discretization of the vector Laplacian is constructed with weak div stability, and a commuting property with appropriate interpolants is established to enable exact mass conservation on divergence-free subspaces. The resulting mixed formulations yield velocity and pressure superconvergence, with $\|\boldsymbol{\sigma}-\boldsymbol{\sigma}_h\|$ and $\|\operatorname{dev}\,\mathrm{grad}_w(I_{k,k}^{\operatorname{div}}\boldsymbol{u}-\boldsymbol{u}_h)\|$ scaling as $h^{k+1}$ and $\|Q_{\ell} p - p_h\|$ as $h^{k+1}$; a postprocessing step provides a divergence-free velocity $\boldsymbol{u}_h^{*}$ with $\|\boldsymbol{u}-\boldsymbol{u}_h^{*}\| \lesssim h^{k+2}$. The framework also connects to hybridization, stabilization-free virtual elements, and pseudostress-velocity-pressure formulations, with numerical experiments confirming the theoretical rates in both 2D and 3D.

Abstract

Superconvergent and divergence-free finite element methods for the Stokes equation are developed. The velocity and pressure are discretized using $H(\mathrm{div})$-conforming vector elements and discontinuous piecewise polynomials. The discrete formulation employs a weak deviatoric gradient operator built with tangential-normal continuous finite elements for traceless tensors, requiring no stabilization. Optimal and superconvergent error estimates are established. The method connects to nonconforming virtual element and pseudostress-velocity-pressure mixed formulations. Numerical experiments verify the theory.

Figures (1)

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Theorems & Definitions (37)

• Lemma 2.1
• Lemma 2.2
• proof
• Remark 2.3
• Lemma 2.4
• proof
• Theorem 2.5
• proof
• Lemma 2.6
• proof