Elsevier

Computers & Mathematics with Applications

Volume 98, 15 September 2021, Pages 201-217
Computers & Mathematics with Applications

Low-order nonconforming brick elements for the 3D Brinkman model

https://doi.org/10.1016/j.camwa.2021.07.009Get rights and content

Abstract

This work gives a general framework for the convergence analysis of low-order nonconforming elements for the 3D Brinkman model over cubical meshes, including the theory of uniform convergence with respect to the given parameters and the high accuracy analysis in the Darcy region. We apply our theory to three elements known in other literature and a newly constructed edge-face-based element to show their optimal approximation ability. Numerical examples are also provided to confirm our theory.

Introduction

Let ΩR3 be a polyhedral domain with the boundary ∂Ω. We consider the following 3D Brinkman model with the unknown velocity u(x) and pressure p(x):div(νu)+αu+p=fin Ω,divu=gin Ω,u=0on Ω. Here ν(x)>0 is the viscosity parameter and α(x)0 is the dynamic viscosity parameter divided by the permeability. Both these parameters are dependent on the types of subregions in Ω: If x is in a subregion of porous medium, ν(x) will be small and α(x) will be relatively large, and so the model is of the Darcy type. Otherwise, if x is in a fluid subregion, we can find a large ν(x) and a small α(x), then this model behaves like a Stokes problem. The terms f(x) and g(x) are known, and g fulfills the solvability condition Ωgdx=0.

A comprehensive study on this issue was given by Xie et al. [15]. They pointed out that divergence-free H1-conforming Stokes elements are proper choices to achieve the uniform convergence. However, conforming elements suffer from the strong C0-continuity requirement, which leads to usually a complicated element structure, especially in low-order, three-dimensional and non-simplicial cases. An alternative strategy is to adopt H1-nonconforming elements with proper weak smoothness, whose structures are easier to handle. Two types of H1-nonconforming elements have been reported in the literature. The first type is H1-nonconforming but H(div)-conforming. One can refer such successful constructions [9], [15], [7], [3], [6] in the 2D case and [14], [7], [4], [5] in the 3D case. The other type is H(div)-nonconforming, considerably reducing the complexity of the first type caused by the strong continuity of the normal trace, especially when the element shape is non-simplicial and irregular [20], [21] or contains some kind of symmetry [17]. However, we must remark that all low-order elements of the first type can achieve a high accuracy in the Darcy region because of the H(div)-conforming property, but for the second type such a high accuracy might no longer hold.

In this work we focus on cubical meshes, and provide a general framework for the convergence analysis of low-order H(div)-nonconforming brick elements for the model (1.1). We show that if five hypotheses are satisfied for a finite element scheme, the uniform convergence property will be guaranteed with respect to ν and α. Note that a similar analysis has been given by Zhang et al. [17] for the same problem applied to a 3D nonconforming rotated Q1 type element. In comparison with the analysis in [17], our theory is quite different, mainly in that we adopt a completely different consistency error estimate strategy, although both analyses are based on special properties of the shape function space and the symmetry of the element shape. Indeed, our method extends our previous framework [20] from 2D to 3D, and not only recovers the same conclusions in [17], but can also be applied to elements of other types. One can refer Remark 2.3 for more details.

Based on the uniform convergence analysis and the tangent-normal switching strategy provided in [22], we also give an abstract high accuracy analysis in the Darcy region over divisionally uniform meshes, namely the meshes are generated by uniformly dividing each cell of a coarse mesh successively. This partly recovers the high accuracy of H(div)-conforming elements in the Darcy region, while simple structures of H(div)-nonconforming ones are preserved. Based on the five hypotheses for the purpose of uniform convergence, if some additional mild element conditions are satisfied, we show the convergence order is at least O(h3/2) for the velocity in L2-norm, and the accuracy for the pressure can be effectively improved as well via a simple post-processing technique. Moreover, the order O(h3/2) can be improved to O(h2) provided the mesh is precisely uniform and the pressure fulfills a special boundary condition.

We apply our theory to four different elements possessing all the above good properties. The first three are known in other literature, including the nonconforming rotated Q1 type element investigated in [17], the enriched nonconforming rotated Q1 element developed in [8], [18] and a subspace method of the first element, initially designed for the Stokes problem in [19]. To the best of our knowledge, the fourth element is newly constructed, with the degrees of freedom (DoFs) selected as edge midpoint values of edge tangential components and face normal integrals. This element has fewer global DoFs than the first two elements, and its global basis selection is standard, simpler than the third one over complicated domains. From our theoretical analysis and the numerical tests, we see all these four elements are competitive ones for Stokes, Darcy and Brinkman problems.

The remaining parts of this work are organized as follows. Section 2 deals with the abstract uniform convergence analysis for (1.1). Next we focus on the high accuracy in the Darcy region in Section 3. As an application of our theory, we analyze four elements in Section 4. Numerical examples are given in Section 5 and finally further discussions are provided in Section 6.

Standard notations in Sobolev spaces are employed in this work. For a domain DR3, we write n as the unit outward normal vector on ∂D. The polynomial space over D of degree equal to or lower than k is denoted by Pk(D) in a usual manner. For the Sobolev space Hm(D), the norms and semi-norms are indicated by m,D and ||m,D, respectively. The space H0m(D) is a subspace of Hm(D) with vanishing trace on ∂D. The subspace H0(div;D)H(div;D) is analogously defined by setting the normal trace on ∂Ω to be zero. We additionally write L2(D)=H0(D) with its inner-product denoted by (,)D. The functions in its subspace L02(D) are of zero integral. These notations of norms, semi-norms and inner-products are also valid for vector- and matrix-valued Sobolev spaces, where the subscript Ω will be omitted if the domain D=Ω. Moreover, C is a positive constant independent of the mesh size h, ν and α, and can be taken different values in different places.

Section snippets

Uniform convergence analysis for the Brinkman model

In this section we provide an abstract convergence analysis for the Brinkman problem (1.1) in the case that Ω can be partitioned into cubical cells, so that we can adopt simple brick elements for the approximation purpose. For the sake of convenience, we assume that the parameters ν>0 and α0 are constants.

We first write the weak form of (1.1) as follows: Find (u,p)[H01(Ω)]3×L02(Ω) satisfyingν(u,v)+α(u,v)(divv,p)=(f,v),v[H01(Ω)]3,(divu,q)=(g,q),qL02(Ω), where L02(Ω) is the subspace of L2

High accuracy analysis in the Darcy region

In this section we consider a region or a subregion of porous medium, also denoted by Ω. Then by taking a vanishing viscosity, the Brinkman model (1.1) degenerates to a Darcy equation. We shall not only investigate such a Darcy limit of (1.1) with u|Ω=0 by using the standard approximation spaces Vh and Ph fulfilling (H1)(H5), but also consider a more standard case in which the boundary condition should be properly modified. In both cases, the high accuracy conclusions in this Section are

Applied to nonconforming brick elements

We now apply our abstract convergence analysis developed in Sections 2 and 3 to four different brick elements. To this end, we first consider the reference cube Kˆ=[1,1]3 and introduce the corresponding elements over Kˆ, and then use an affine equivalent technique to determine the general construction. Based on the specific coordinate selection of Th given in Section 2, for each KTh, there exists an affine transformation FK:KˆK such thatFK(xˆ)=x=(a1,Kxˆ+b1,K,a2,Kyˆ+b2,K,a3,Kzˆ+b3,K)TK,xˆ=(x

Numerical examples

Numerical tests are provided below. We choose the solution domain Ω=[0,3]×[0,2]×[0,1]. In the first part we test the performances of all the four elements Vh(t)×Ph(t), t=1,2,3,4, for the Brinkman model (1.1). Two types of partitions {Th(i)}, i=1,2 are considered in this part consisting of n3 brick cells. In Th(1), all the cells are of the stochastic size, while the family {Th(2)} are divisionally uniform. More precisely, we take Th0(2) as the initial coarse mesh by settingTh0(2)={Ix×Iy×Iz:Ix{[0

Further discussions

We end this work by some further discussions. Let us give a brief comparison among several known nonconforming brick elements for the Brinkman model, including the four elements analyzed in this work and the DSC33 element constructed by Chen et al. [4], in terms of the efficiency and accuracy. All the elements are of the lowest order, indicating a simple structure and low computational costs. We list the numbers of local/global DoFs and convergence orders under different mesh conditions of all

Acknowledgements

This work is supported by NSF of Jiangsu Province (No. BK20200902) and NNSFC (No. 61772105), and “the Fundamental Research Funds for the Central Universities”. The authors would also like to thank the editors and the anonymous reviewers for their helpful comments and suggestions.

References (22)

  • X. Zhou et al.

    Simple nonconforming brick element for 3D Stokes equations

    Appl. Math. Lett.

    (2018)
  • X. Zhou et al.

    Nonconforming polynomial mixed finite element for the Brinkman problem over quadrilateral meshes

    Comput. Math. Appl.

    (2018)
  • D.N. Arnold et al.

    Finite element differential forms on cubical meshes

    Math. Comput.

    (2014)
  • D. Boffi et al.

    Mixed Finite Element Methods and Applications

    (2013)
  • S. Chen et al.

    Uniformly convergent H(div)-conforming rectangular elements for Darcy-Stokes problem

    Sci. China Math.

    (2013)
  • S. Chen et al.

    Uniformly convergent cubic nonconforming element for Darcy-Stokes problem

    J. Sci. Comput.

    (2017)
  • L. Dong et al.

    Uniformly convergent nonconforming tetrahedral element for Darcy-Stokes problem

    J. Comput. Math.

    (2018)
  • A. Gillette et al.

    Nonstandard finite element de Rham complexes on cubical meshes

    BIT Numer. Math.

    (2020)
  • J. Guzman et al.

    A family of nonconforming elements for the Brinkman problem

    IMA J. Numer. Anal.

    (2012)
  • Q. Lin et al.

    Superconvergence and extrapolation of nonconforming low order elements applied to the Poisson equation

    IMA J. Numer. Anal.

    (2005)
  • K.A. Mardal et al.

    A robust finite element method for Darcy-Stokes flow

    SIAM J. Numer. Anal.

    (2002)
  • Cited by (3)

    This project is supported by NSF of Jiangsu Province (No. BK20200902) and NNSFC (No. 61772105), and “the Fundamental Research Funds for the Central Universities”.

    View full text