A hybrid high-order method for a class of quasi-Newtonian Stokes equations on general meshes

Abstract

In this paper, we introduce a hybrid high-order (HHO) discrete scheme for numerically solving a class of incompressible quasi-Newtonian Stokes equations in ${\mathbb{R}}^2$. The presented HHO method depends on hybrid discrete velocity unknowns at cells and edges, and pressure unknowns at cells. Benefiting from the hybridization of unknowns, the computation cost can be reduced by the technique of static condensation and the solvability of the static condensation algebra system is proved. Furthermore, we study the HHO scheme by polynomials of arbitrary degrees k (k ≥ 1) on the general meshes and geometries. The unique solvability of the discrete scheme is proved. Additionally, the optimal a priori error estimates for the velocity gradient and pressure approximations are obtained. Finally, we provide several numerical results to verify the good performance of the proposed HHO scheme and confirm the optimal approximation properties on a variety of meshes and geometries.

Introduction

Let $\Omega \subset {\mathbb{R}}^2$ be an open, bounded and convex polygonal domain with the Lipschitz boundary ∂Ω. We consider the following steady quasi-Newtonian Stokes equations: Given body force f ∈ L²(Ω)², find a velocity field u and pressure field p such that

$$-\mathbf{\text{div}}\left(\mu \right(|{\nabla }_s\mathit{u}\left|\right){\nabla }_s\mathit{u})+\nabla p=\mathit{f}\text{in}\Omega ,$$

$$\text{div}\mathit{u}=0\text{in}\Omega ,$$

$$\mathit{u}=\mathit{0}\text{on}\partial \Omega ,$$

where div stands for the usual divergence operator div acting along each row of tensor, $\mathit{u}={(u_1,u_2)}^T$ is the velocity and p is the pressure. The notation ∇_s denotes the symmetric part of the gradient operator with ${\nabla }_s\mathit{u}=(\nabla \mathit{u}+\nabla {\mathit{u}}^T)/2$ and |∇_su| is the shear rate, where | · | stands for Frobenius norm of ${\mathbb{R}}^{2\times 2}$ tensors. Additionally, $\mu :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is the nonlinear function of shear dependent viscosity which is assumed to be continuous and satisfies

$$m_{\mu }(\xi -\eta )\le \mu \left(\xi \right)\xi -\mu \left(\eta \right)\eta \le M_{\mu }(\xi -\eta )\forall \xi \ge \eta \ge 0.$$

Here, m_μ and M_μ are two positive constants related to μ.

In the past few decades, more and more researchers have paid attention to the study of efficient numerical methods for non-Newtonian flows and related problems, which the viscosity is not constant and varies upon the imposed rate of deformation. One way to describe non-Newtonian fluids is to plot the viscosity measurements versus the imposed shear rate, and then, to fit the obtained curve with a simple template viscosity function, adjusting some parameters [1]. It could be viewed as a first step inside the non-Newtonian fluids, which is the main idea of the quasi-Newtonian fluids models. So it is highly significant to study the quasi-Newtonian problem. One of the simplified quasi-Newtonian models is the incompressible quasi-Newtonian Stokes fluid flow [2], [3], [4], [5]. Some numerical methods have already been well developed, such as the conforming and nonconforming finite element method [6], [7], [8], the mixed finite element method [3], [9], [10], the dual-mixed finite element method [11], [12], the discontinuous Galerkin (DG) method [13], the local discontinuous Galerkin (LDG) method [14], the virtual element method (VEM) [15], the hybrid discontinuous Galerkin (HDG) method [16] and the weak Gakerkin (WG) method [17].

Recently, the hybrid high-order (HHO) methods are an active field of research and have been applied to many equations and models. This method is originally derived for the scalar diffusion and linear elasticity problems [18], [19], which supports general meshes possibly including non-matching interfaces and polyhedral elements. With a great development, this framework has been successfully extended to the discretization of advection-dominated transport equation [20], Cahn-Hilliard equation [21], Stokes problem [22] and non-linear Leray-Lions problem [23], [24]. It is defined hybrid because both cell- and edge-based discrete unknowns are used, like hybrid finite element (HFE) and HDG. The proposed method enjoys the following good virtues ([23]): (i) the construction is dimension-independent; (ii) are wellsuited for hp-adaptivity; (iii) it is efficiently parallelisable and it has reduced computational cost. Compared with the DG method, the most important point is that the HHO method does not need the penalty constant and reduces the computational cost by the compactness techniques. Links between HDG, VEM and HHO methods have been pointed out in [25], [26], [27]. Moreover, another remarkable feature of HHO method is that the corresponding discrete schemes make the energy error estimates of order $(k+1)$ for smooth solutions. In the previous works [23], [24], Daniele and Jérôme developed and analyzed the HHO method for steady nonlinear Leray-Lions problems; The main contributions are the optimal W^s,p-approximation properties for elliptic projectors on local polynomial spaces and a priori error estimates. Based on the works [19], [23], [24], Michele et al. analyzed a novel HHO discretization of a class of nonlinear elasticity models in the small deformation regime. Then, inspired from [19], Di Pietro et al. devised arbitrary-order nonconforming methods for the viscosity-dependent Stokes equations in [22] and obtained the pressure-independent energy-error estimate for the velocity of order $(k+1)$. To the best of our knowledge, there is no literature on the HHO method to solve the quasi-Newtonian Stokes models. Note that it is the key to satisfy (1.2) and keep the continuity of the nonlinear function μ. From [4, Lemma 2.1], when μ satisfies (1.2), there exist positive constants C₁ and C₂ such that, for all $\underline{\mathit{\tau }},\underline{\mathit{\omega }}\in {\mathbb{R}}^{2\times 2}$

$$\left|\mu \right(|\underline{\mathit{\tau }}\left|\right)\underline{\mathit{\tau }}-\mu \left(\right|\underline{\mathit{\omega }}\left|\right)\underline{\mathit{\omega }}|\le C_1|\underline{\mathit{\tau }}-\underline{\mathit{\omega }}|,$$

$$\left(\mu \right(|\underline{\mathit{\tau }}\left|\right)\underline{\mathit{\tau }}-\mu \left(\right|\underline{\mathit{\omega }}\left|\right)\underline{\mathit{\omega }}):(\underline{\mathit{\tau }}-\underline{\mathit{\omega }})\ge C_2{|\underline{\mathit{\tau }}-\underline{\mathit{\omega }}|}^2.$$

Referring to [1], we recall two classical examples of μ. The power law [28]

$$\mu \left(\xi \right)=K{\xi }^{(-2+n)}\forall \xi \in {\mathbb{R}}^{+},$$

where n > 0 and K ≥ 0 are given real constants, called the power index and the consistency, respectively. The power law is simple and useful to describe the fluid behavior within the shear rate range to which the coefficient fits. Another is the Carreau’s law, which was first proposed by Carreau et al. in [29],

$$\mu \left(\xi \right)={\mu }_{\infty }+({\mu }_0-{\mu }_{\infty }){(1+\lambda {\xi }^2)}^{(-2+n)/2}\forall \xi \in {\mathbb{R}}^{+},$$

where n > 0 and μ₀ ≥ 0, μ_∞ ≥ 0 and λ ≥ 0 are given real constants, satisfying μ₀ ≥ μ_∞ when n ≤ 2 and μ₀ ≤ μ_∞ when n ≥ 2. The Carreau’s law viscosity models have been used to the blood flows [30] and fluid machines [31]. Notice that when $n=2,$ both the power law and Carreau’s law in the model problem (1.1a)–(1.1c) reduce to constants.

In this paper, we adopt the HHO method to solve the quasi-Newtonian Stokes flows (1.1a)–(1.1c). We use the k-order polynomial basis functions for the approximation of velocity and pressure in both interior of cells and edges. By using the properties (1.2) and (1.3), we derive the discrete inf-sup condition and give the solvability of the HHO discretization scheme. Furthermore, the error estimates of order $(k+1)$ in the H¹-like norm for velocity and L²-norm for pressure are obtained. Besides, in our numerical experiments, we verify the convergence of the HHO method for polynomials degrees k on kinds of general meshes. And we illustrate the numerical and exact solutions.

The rest of the paper is organized as follows. In Section 2, we introduce some notations, a weak formulation of the nonlinear quasi-Newtonian problems (1.1a)–(1.1c), mesh settings and some key results on broken polynomials space. In Section 3, we give the local reconstructions and propose the HHO numerical discrete scheme. In Section 4, we deduce the solvability analysis of the discrete scheme, while the a priori error estimates for velocity and pressure are detailed in Section 5. The analysis of static condensation is presented in Section 6. Section 7 contains ample numerical examples, which illustrate our theoretical results. Finally, we end the paper by some conclusions in Section 8.