Low-order nonconforming brick elements for the 3D Brinkman model☆
Introduction
Let be a polyhedral domain with the boundary ∂Ω. We consider the following 3D Brinkman model with the unknown velocity and pressure : Here is the viscosity parameter and 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, will be small and 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 and a small , then this model behaves like a Stokes problem. The terms and are known, and g fulfills the solvability condition .
A comprehensive study on this issue was given by Xie et al. [15]. They pointed out that divergence-free -conforming Stokes elements are proper choices to achieve the uniform convergence. However, conforming elements suffer from the strong -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 -nonconforming elements with proper weak smoothness, whose structures are easier to handle. Two types of -nonconforming elements have been reported in the literature. The first type is -nonconforming but -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 -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 -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 -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 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 -conforming elements in the Darcy region, while simple structures of -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 for the velocity in -norm, and the accuracy for the pressure can be effectively improved as well via a simple post-processing technique. Moreover, the order can be improved to 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 type element investigated in [17], the enriched nonconforming rotated 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 , 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 in a usual manner. For the Sobolev space , the norms and semi-norms are indicated by and , respectively. The space is a subspace of with vanishing trace on ∂D. The subspace is analogously defined by setting the normal trace on ∂Ω to be zero. We additionally write with its inner-product denoted by . The functions in its subspace 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 . 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 and are constants.
We first write the weak form of (1.1) as follows: Find satisfying where is the subspace of
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 by using the standard approximation spaces and 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 and introduce the corresponding elements over , and then use an affine equivalent technique to determine the general construction. Based on the specific coordinate selection of given in Section 2, for each , there exists an affine transformation such that
Numerical examples
Numerical tests are provided below. We choose the solution domain . In the first part we test the performances of all the four elements , , for the Brinkman model (1.1). Two types of partitions , are considered in this part consisting of brick cells. In , all the cells are of the stochastic size, while the family are divisionally uniform. More precisely, we take as the initial coarse mesh by setting
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.
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This project is supported by NSF of Jiangsu Province (No. BK20200902) and NNSFC (No. 61772105), and “the Fundamental Research Funds for the Central Universities”.