A family of quadratic finite volume element schemes over triangular meshes for elliptic equations

doi:10.1016/j.camwa.2019.11.017

Computers & Mathematics with Applications

Volume 79, Issue 9, 1 May 2020, Pages 2473-2491

https://doi.org/10.1016/j.camwa.2019.11.017 Get rights and content

Abstract

In this paper, we construct and analyze a family of quadratic finite volume element schemes over triangular meshes for elliptic equations. This family of schemes cover some existing quadratic schemes. For these schemes, by element analysis, we find that each element matrix can be split as two parts : the first part is the element stiffness matrix of the standard quadratic finite element method, while the second part is a tensor product of two vectors. Thanks to this finding, we obtain a sufficient condition to ensure the existence, uniqueness and coercivity result of the finite volume element solution on triangular meshes. More interesting is that, the above condition has a simple and analytic expression, and only relies on the interior angles of each triangular element. Based on this result, a minimum angle condition, better than some existing ones, can be obtained. Moreover, based on the coercivity result, we prove that the finite volume element solution converges to the exact solution with an optimal convergence rate in $H^{1}$ norm. Finally, some numerical examples are provided to validate the theoretical findings.

Introduction

As a popular numerical tool for solving partial differential equations, the finite volume method (FVM) has a long history. It possesses the local conservation property and can deal with the complex geometry boundary, and so on. For these reasons, FVM has attracted a lot of attentions and has wide applications in scientific and engineering computations, see [1], [2], [3], [4], [5], [6], [7] for an incomplete references. Finite volume element method (FVEM) is one category of FVM. There are many researchers concentrate on linear FVEM, see [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21] and the references cited therein, which makes the development of the mathematical theory of linear FVEM almost as satisfactory as linear finite element method (FEM). Recently, under a mesh assumption, a unified theoretical analysis of any order FVEM over quadrilateral mesh is established and the references can be found in [22], [23], [24], [25].

However, for high order FVEM over triangular mesh, it is still a challenging work especially to establish the coercivity result. Based on the coercivity result, some researchers proposed special high order FVEM, or studied their convergence properties. For example, Vogel, Xu and Wittum [26] constructed a high order FVEM. By introducing some orthogonal conditions (on element boundary and in the interior of element), Wang and Li [27] proposed a high order FVEM such that the $L^{2}$ error has an optimal convergence rate. Recently Wang and Li [28] analyzed the superconvergence of the quadratic FVEM proposed in [27], and this quadratic scheme has been applied to solve a nonlinear elliptic problem [29].

In order to establish the coercivity result, most existing works adopted the element analysis and required a certain minimum angle condition of the triangular mesh. For instance, by assuming that the maximum angle of each triangular element is not greater than $90 °$ , and the ratio of the lengths of the two sides of the maximum angle belongs to $[\sqrt{2 ∕ 3}, \sqrt{3 ∕ 2}]$ , [30] obtained a coercivity result for the first quadratic FVEM. For some special quadratic FVEM, Liebau [31] required that the geometry of the triangulation is not too extreme. Xu and Zou [21] obtained the inf–sup condition for several quadratic FVEM and improved some existing results, for example, the minimum angle should be greater than or equal to $7.11 °$ for the scheme proposed in [32], $9.98 °$ for the scheme proposed in [31] and $20.95 °$ for the scheme proposed in [30]. A general framework for construction and analysis of higher-order finite volume methods was developed in [33], where three sufficient conditions for the coercivity were proposed. For a specific scheme, these conditions can be verified by a computer program. For example, by the method in [33], the minimum angle for the quadratic FVEM in [27], [28], [29] is $5.24 °$ . Later, the relationship of the uniform ellipticity, inf–sup condition and uniform local ellipticity of high order FVEM was investigated in [34]. Recently, Zou [35] proposed an unconditionally stable quadratic FVEM, regrettably, this scheme does not ensure the optimal convergence order in $L^{2}$ error norm, at least it seems so numerically. More studies about the hybrid FVEM and the Hermite FVEM can be found in [33], [36], [37].

Based on the development of quadratic FVEM over triangular mesh, one can see that most existing coercivity analysis is scheme-wise, i.e., it must be done one by one for each specific scheme. On the other hand, we observe that the quadratic FVEM proposed in [27] owns the optimal $L^{2}$ error order and superconvergence property, whose coercivity is based on a computer program proposed in [33]. So it is worth to improve the coercivity result of this scheme. Unfortunately, it is difficult to obtain the coercivity result with a simple and analytic expression through the existing techniques.

In this work, we propose and study a family (depend on a parameter $β$ ) of quadratic finite volume element schemes over triangular meshes. Precisely, we first convert the element bilinear form to a quadratic form with respect to a 6-by-6 singular element matrix. Unlike the previous works [16], [21], [31], [33], [37], here the computation of this element matrix is decomposed into two parts: the first part is the element stiffness matrix of the standard quadratic finite element method, where the finite element stiffness matrix is symmetric and singular; the second part is the tensor product of two vectors, where one vector relies on $β$ and the other vector relies on the three interior angles. Therefore, each entry of this element matrix only depends on the three interior angles of triangle and the parameter $β$ . Then, the analysis of this element matrix can be further transformed to that of a 5-by-5 symmetric matrix. As a direct consequence, we reach a sufficient condition to ensure the existence, uniqueness and coercivity of these schemes over triangular meshes. Based on the coercivity result, we obtain the optimal $H^{1}$ error estimate.

Compared with existing works, the present work has several contributions. Firstly, we provide a uniform framework to construct and analyze a family of quadratic FVE schemes with a parameter $β$ , covering some existing quadratic schemes. Secondly, by the element analysis, we obtain a sufficient condition with a simple and analytic expression, to guarantee the existence, uniqueness and coercivity result of the finite volume element solutions. Thirdly, compared to the previous works [16], [21], [27], [31], [33], [37], the above sufficient condition depends only on the scheme parameter $β$ and the interior angles of each triangular element. Therefore, it is easy to judge on any triangular mesh. In addition, in order to understand clearly the sufficient condition, two minimum angle conditions are derived and graphically shown in Fig. 4. In particular, the minimum angle condition for the quadratic scheme in [27] is improved from $5.24 °$ to $1.42 °$ .

The rest of this paper is organized as follows. A family of quadratic FVE schemes are constructed in Section 2. In Section 3 we present the coercivity result and discuss the minimum angle conditions to guarantee the existence and uniqueness of the finite volume element solutions. The proof of the coercivity result is given in Section 4 and the optimal $H^{1}$ error estimate is obtained in Section 5. Several numerical examples are provided in Section 6 to verify the theoretical results, and some concluding remarks are given in the last section.

Section snippets

A family of quadratic FVE schemes

We focus here on the isotropic steady state diffusion problem $- \nabla \cdot (κ \nabla u) = f in Ω,$ $u = 0 on \partial Ω,$ where $Ω \subset R^{2}$ is a bounded polygonal domain, $f \in L^{2} (Ω)$ and $κ$ is a piecewise smooth function that can be bounded above and below, i.e., there exist positive constants $κ_{min}$ and $κ_{max}$ , such that $κ_{min} \leq κ (x, y) \leq κ_{max}, for a.e. (x, y) \in Ω .$

Suppose that the primary mesh $T_{h} ≔ {K}$ is a triangular partition of $Ω$ , where the nonempty intersection of any two closed triangles is either a common edge or a common vertex and $h$ denotes the

The main results

For simplicity of exposition, we introduce some notations. For each triangular element $K \in T_{h}$ , let $θ_{i} = ∠ P_{i + 2} P_{i} P_{i + 1} (i = 1, 2, 3)$ be the three interior angles of $K$ . Denote $\tilde{β} = \frac{2}{3} (1 - 3 α) (β - α)$ and $ϱ_{K} = 4 - 36 \tilde{β} - 27 {\tilde{β}}^{2} (\sum_{i = 1}^{3} {cot}^{2} θ_{i} - 1),$ where $α$ and $β$ are defined by (6), (7), respectively. Next, let $Π$ be a linear mapping that maps $u_{h} \in U_{h}$ to $u_{h}^{*} ≔ Π u_{h} \in V_{h}$ , satisfying for each vertex $P_{i}$ , $u_{h}^{*} (P_{i}) = u_{h} (P_{i}), i = 1, 2, 3,$ and for each midpoint $M_{i}$ , $u_{h}^{*} (M_{i}) = \frac{u_{h} (P_{i}) + u_{h} (P_{i + 1})}{2} (1 - \frac{2}{\sqrt{3}}) + \frac{2}{\sqrt{3}} u_{h} (M_{i}), i = 1, 2, 3 .$ We remark that the above mapping was introduced

The proof of Theorem 1

In this section, the proof of Theorem 1 is performed through the element analysis, the main idea of which is presented in the following subsection.

$H^{1}$ error estimates

Once the coercivity result is obtained, the estimate of $H^{1}$ error is a standard routine. We have the following results.

Theorem 2

Assume that $T_{h}$ is shape regular and $κ$ is piecewise $W^{1, \infty}$ with respect to $T_{h}$ . Assume also that the exact solution $u \in H_{0}^{1} (Ω) \cap H^{3} (Ω)$ . Then, for any $β \in (0, 2 ∕ 3)$ , under the assumption $(A2)$ , we have ${| u - u_{h} |}_{1} ≲ h^{2} {‖ u ‖}_{3} .$

Proof

By the Green’s formula, we have $a_{h} (u, v_{h}) = (f, v_{h}), \forall v_{h} \in V_{h} .$ It follows that $a_{h} (u - u_{h}, v_{h}) = 0, \forall v_{h} \in V_{h} .$ Suppose $u_{I} \in U_{h}$ is the standard Lagrange interpolation of $u$ , satisfying for each

Numerical examples

In this section, we present some numerical examples to verify the theoretical results. Example 1, Example 2 are designed for continuous and discontinuous coefficients, respectively, while Example 3 is for a singular solution. In these examples, we choose $Ω = {[0, 1]}^{2}$ and employ four types of triangular meshes. The first mesh (Mesh I) is a uniform triangular one, see Fig. 5(a), and the coordinates of the vertices are given by $x_{i j} = (i - 1) h, y_{i j} = (j - 1) h, 1 \leq i, j \leq n + 1,$ where $h = 1 ∕ n$ is the mesh size. The third

Conclusions

We propose and study a family of quadratic FVE schemes over triangular meshes for elliptic equations. Some existing schemes in the literature can be viewed as special members of this family of schemes. The existence and uniqueness of these FVE schemes are obtained under a certain mesh geometry assumption $(A1)$ that has a simple and analytic expression. Based on this result, we prove the coercivity results under an even stronger assumption $(A2)$ . As a byproduct, the coercivity result in [27] is

Acknowledgments

The authors would like to thank the reviewers for their careful readings and useful suggestions. This work was partially supported by the National Natural Science Foundation of China (No. 11871009) and CAEP Foundation, China (No. CX2019028).

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Citation Excerpt :
In order to establish the coercivity result of the quadratic FVE schemes, the existing works only faced the difficulties generated by the mesh geometry. In this paper, we extend the result of [31] to the anisotropic diffusion problems, following the element analysis approach. Obviously, we have to overcome the difficulties caused by the mesh geometry, the diffusion tensor and both combined.
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A family of quadratic finite volume element schemes over triangular meshes for elliptic equations

Abstract

Introduction

Section snippets

A family of quadratic FVE schemes

The main results

The proof of Theorem 1

H1 error estimates

Numerical examples

Conclusions

Acknowledgments

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