Some error analysis on virtual element methods
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Abstract
Some error analyses on virtual element methods (VEMs) including inverse inequalities, norm equivalence, and interpolation error estimates are developed for polygonal meshes, each element of which admits a virtual quasiuniform triangulation. This submesh regularity covers the usual ones used for theoretical analysis of VEMs, and the proofs are presented by means of standard technical tools in finite element methods.
Keywords
Virtual elements Inverse inequality Norm equivalence Interpolation error estimateMathematics Subject Classification
65N30 65N121 Introduction
Since the pioneer work in [2, 3, 4], virtual element methods (VEMs) have been widely used to approximate various partial differential equations in recent years. Compared with the standard finite element methods (cf. [10, 17]), such methods have several significant advantages: (1) they are natively adapted to polygonal/polyhedral meshes, leading to great convenience in mesh generation for problems with complex geometries. For example, in [16] a simple and efficient interfacefitted polyhedral mesh algorithm is developed and VEM has been successfully applied to the elliptic interface problem. (2) They are suitable for attacking highorder elliptic problems. For instance, it is very difficult to construct the usual \(H^2\)conforming finite element method for fourthorder elliptic problems, hence many nonconforming elements were devised to overcome the difficulty (see [25]). It is, however, very straightforward to construct \(H^2\)conforming virtual element methods for this type of problems (cf. [13]). Until now, both conforming and nonconforming VEMs for elliptic problems have been developed with elaborated details (cf. [2, 3, 8, 13, 15, 16, 20]).
 C1.

There exists a real number \(\gamma >0\) such that, for each element \(K\in \mathcal {T}_h\), it is starshaped with respect to a disk of radius \(\rho _K\ge \gamma h_K\), where \(h_K\) is the diameter of K.
 C2.

There exists a real number \(\gamma _1>0\) such that, for each element \(K\in \mathcal {T}_h\), the distance between any two vertices of K is \(\ge \gamma _1 h_K\).
Hence, when applying the generalized scaling argument to derive the estimate (1) for virtual element spaces, we require to show the solution of the Poisson equation defined over K depends on the shape of K continuously, since the local space \(V_K\) is defined with the help of the Laplacian operator (for details see [2, 3] or Sect. 2). In fact, such results can be obtained rigorously in a very subtle and technical way (cf. [19]).
Similarly, we remark that we should use the trace inequality or the Sobolev embedding inequality over K carefully, since the generic constant depends on the geometric nature of K implicitly.
 A1.

For each \(K\in \mathcal T_h\), there exists a “virtual triangulation” \(\mathcal T_K\) of K such that \(\mathcal T_K\) is uniformly shape regular and quasiuniform. The corresponding mesh size of \(\mathcal T_K\) is proportional to \(h_K\). Each edge of K is a side of a certain triangle in \(\mathcal T_K\).
For triangular meshes, one can use an affine transformation to map an arbitrary triangle to a socalled reference triangle and then work on the reference triangle. Results established on the reference triangle can be pulled back to the original triangle by estimating the Jacobian of the affine map. For polygons, scaling can be still used but not the affine transformation. Therefore we cannot work on a reference polygon which does not exist for a family of polygons with general shapes. Instead we decompose a polygon K into shape regular triangles and use the scaling argument in each triangle.
Throughout this paper, we will always assume the mesh \(\mathcal T_h\) satisfies the conditions A1, and the generic constant hidden in the symbol \(\lesssim \) depends only on the parameters involving the shape regularity and quasiuniformity of the auxiliary triangulation \(\mathcal T_K\) given in A1. Moreover, for any two quantities a and b, “\(a\eqsim b\)” indicates “\(a\lesssim b\lesssim a\)”. We will also use the standard notations and symbols for Sobolev spaces and their norms/seminorms; the reader is referred to [1] for details.
 1.
Inverse inequality: \(v_{1,K}\lesssim h_K^{1}\Vert v\Vert _{0,K}\).
 2.
Norm equivalence: \(h_K \Vert \varvec{\chi } (v)\Vert _{l^2} \lesssim \Vert v\Vert _{0,K} \lesssim h_K \Vert \varvec{\chi } (v)\Vert _{l^2},\) where \(\varvec{\chi }(v)\) is the vector formed by the degrees of freedom of v.
 3.Stability estimate of the VEM formulation:where \(\Pi _k^{\nabla },\Pi _k^0\) are \(H^1, L^2\)projection to the polynomial space \(\mathbb {P}_k(K)\), respectively.$$\begin{aligned} \Vert \nabla v\Vert _{0,K}^2&\eqsim \left\ \nabla \Pi _k^{\nabla } v\right\ _{0,K}^2 + \left\ \varvec{\chi } \left( v  \Pi _k^{\nabla } v\right) \right\ _{l^2}^2,\\ \Vert \nabla v\Vert _{0,K}^2&\eqsim \left\ \nabla \Pi _k^{\nabla } v\right\ _{0,K}^2 + \left\ \varvec{\chi }_{\partial K} \left( v  \Pi _k^0 v\right) \right\ _{l^2}^2, \end{aligned}$$
 4.Interpolation error estimate: if \(I_K u\in V_K\) denotes the canonical interpolant defined by d.o.f. of u, then$$\begin{aligned} \Vert u  I_Ku\Vert _{0,K} + h_{K}u  I_Ku_{1,K} \lesssim h_K^{k+1}\Vert u\Vert _{k+1,K} \quad \forall u\in H^{k+1}(K). \end{aligned}$$
The rest of the paper is organized as follows. The virtual element method is introduced in Sect. 2. Inverse inequalities, norm equivalence, and interpolation error estimates for several types of VEM spaces are derived with technical details in Sects. 3–5, respectively.
2 Virtual element methods
A two dimensional domain \(\Omega \) is decomposed into a polygonal mesh \(\mathcal T_h\) so that each element in \(\mathcal T_h\) is a simple polygon and a generic element is denoted by K. We work under the two dimensional setting for a clear illustration, and the generalization to higher dimensions shall be commented afterwards.
2.1 Assumptions on the polygon mesh
As mentioned in the introduction, we shall carry out the analysis based on the assumption A1, for which some more discussions are given as follows. Recall that a triangle is shape regular if there exists a constant \(\kappa \) such that the ratio of the diameter of this triangle to the radius of its inscribed circle is bounded by \(\kappa \). It is also equivalent to the condition that the minimum angle is bounded below by a positive constant \(\theta \). A triangulation \(\mathcal T\) is quasiuniform if any two triangles in the triangulation are of comparable sizes. Namely there exists a constant \(\sigma \), such that \(\max _{\tau \in \mathcal T}h_{\tau } \le \sigma \min _{\tau \in \mathcal T}h_{\tau }\). The term “uniform” means the constants \(\kappa , \theta \) and \(\sigma \) are independent of K.
By assumption A1, the number of triangles of each ‘virtual triangulation’ \(\mathcal T_K\) is uniformly bounded by a number L and the size of each triangle is comparable to that of the polygon, i.e. \(h_{K}\lesssim h_{\tau }\le h_K, \; \forall \tau \in \mathcal T_K\). The constants in our inequalities depend on the shape regularity constant \(\kappa \) (or equivalently \(\theta \)) and the quasiuniformity constant \(\sigma \) (or equivalently L).
Assumption A1 is introduced so that the estimates for finite elements on triangles can be used. If K is assumed to be starshaped and each edge is of comparable size, e.g. assumption C2, then a virtual triangulation can be obtained by connecting vertices of K to the center of the star. In contrast, A1 allows the union of starshaped regions to form irregular polygons.
Note that such virtual triangulations can be created with additional artificial vertices in the interior of K but not on \(\partial K\).
2.2 Spaces in virtual element methods

\(\chi _a\): the values at the vertices of K;
 \(\chi _e^{k2}\): the moments on edges up to degree \(k2\)$$\begin{aligned} \chi _e (v) = e^{1}(m, v)_{e} \quad \forall m\in {\mathbb {M}}_{k2}(e), \forall \text { edge } e\subset \partial K; \end{aligned}$$
 \(\chi _K^{l}\): the moments on element K up to degree l$$\begin{aligned} \chi _{K}(v) = K^{1}(m, v)_K \quad \forall m\in {\mathbb {M}}_{l}(K). \end{aligned}$$
Remark 2.1
The operator \(\Delta \) used in the definition of VEM space (4) can be replaced by other operators as long as the space \(V_{k,l}(K)\) contains a polynomial space with appropriate degree, which ensures the approximation property. For example, when K is triangulated to form a triangulation \(\mathcal T_K\), we can introduce a standard kth order Lagrange element space \(S_{k}(\mathcal T_K)\) on \(\mathcal T_K\) and impose \(\Delta _h v \in {\mathbb {P}}_{l}(K)\) where \(\Delta _h\) is the standard Galerkin discretization of \(\Delta \) related to \(S_{k}(\mathcal T_K)\). From this point of view, VEM is similar to a certain kind of subgrid upscaling.
For the pure diffusion problem, the choice of \(V_h\) is enough to produce numerical solutions with optimal accuracy. However, when dealing with second order elliptic equations with lowerorder terms (e.g., reactiondiffusion problems), the use of the function spaces \(W_h\) and \(\widetilde{V}_h\) are more efficient (see [2]).
2.3 Approximate stiffness matrix
A conforming virtual finite element space \(V_h^0 : = V_h \cap H_0^1(\Omega )\) is chosen to discretize (3). We cannot, however, compute the Galerkin projection of u to \(V_h^0\) since the traditional way of computing \(a(u_h, v_h)\) using numerical quadrature requires pointwise information of functions and their gradient inside each element. In virtual element methods, only d.o.f is enough to assemble an approximated stiffness matrix.
Lemma 2.2
The scaling factor \(h_K\) is not presented in the form in [9] but can be easily obtained by the following scaling argument. The transformation \(\hat{\varvec{x}} = (\varvec{x} \varvec{x}_c)/h_{K}\) is applied on \(\varvec{x}\in K\), so that \(\hat{K}\), the image of K, is contained in the unit disk. The transformed triangulation \(\mathcal T_{\hat{K}}\) is still shape regular so that we can apply results in [9]. Then the constant \(h_K\) can be obtained by scaling back to K. As pointed out in [9], the generic constant depends only on the shape regularity not the quasiuniformity of the triangulation \(\mathcal T_K\).
2.4 Stabilization
 kconsistency: for \(p_k\in {\mathbb {P}}_k(K)\)$$\begin{aligned} S_{K}(p_k, v) = 0 \quad \forall v\in V_h. \end{aligned}$$
 stability:$$\begin{aligned} S_{K}(\tilde{u}, \tilde{u}) \eqsim (\nabla \tilde{u}, \nabla \tilde{u})_K \quad \forall \tilde{u}\in \left( I  \Pi _k^{\nabla }\right) V_h. \end{aligned}$$
3 Inverse inequalities
Note that if the definition of virtual element spaces is modified by using the discrete Laplacian operator (cf. Remark 2.1), then the inverse inequality is trivially true as now the function in VEM space is a finite element function on the virtual triangulation.
We first establish an inverse inequality for polynomial spaces on polygons.
Lemma 3.1
Proof
 1.
\(Q_K v_{\partial K} = v_{\partial K}\);
 2.
\((Q_K v, \phi )_K = (v, \phi )_K\) for all \(\phi \in S_{k}^{0}(\mathcal T_K)\).
Lemma 3.2
Proof
To develop various estimates for a function in VEM spaces, we shall separate it into two functions, related to the moment and the trace of the function, respectively.
Lemma 3.3
 1.
\(v_1\in H^1(K), v_1 _{\partial K} = v_{\partial K}, \Delta v_1 = 0\) in K,
 2.
\(v_2 \in H_0^1(K), \Delta v_2 = \Delta v\) in K.
Proof
For the harmonic part, we have the following inequality.
Lemma 3.4
Proof
We now estimate the second part in the decomposition.
Lemma 3.5
Proof
Now, we summarize our main result in this section as follows.
Theorem 3.6
Proof
As an application of the inverse inequality, we prove the \(L^2\)stability of the projection operators \(Q_K\) and \(\Pi _k^{\nabla }\) restricted to VEM spaces.
Corollary 3.7
Proof
Simply apply the inverse inequality to bound \(h_K\Vert \nabla v\Vert _{0,K}\lesssim \Vert v\Vert _{0,K}\) in Lemma 3.2 to get the desired result. \(\square \)
Corollary 3.8
Proof
4 Norm equivalence
We shall prove the norm equivalence between \(L^2\)norm of a VEM function and \(l^2\)norm of the corresponding vector representation using d.o.f. In light of this result, we are able to derive two stabilization methods used in VEM formulation.
4.1 Norm equivalence of polynomial spaces on a polygon
We begin with a norm equivalence of polynomial spaces on polygons.
Lemma 4.1
Proof
4.2 Norm equivalence for VEM spaces
In this subsection, we are going to prove the norm equivalence of the \(L^2\)norm of VEM functions to the \(l^2\)norm of their corresponding d.o.f. vectors.
Lemma 4.2
Proof
The d.o.f.s are grouped into two categories: \(\varvec{\chi }_{\partial K}(\cdot )\) are d.o.f.s associated with the boundary of K, and \(\varvec{\chi }_{K}(\cdot )\) are moments in K.
The proof of the estimate of the upper bound turns out to be technical. Again we shall use the \(H^1\) decomposition presented in Lemma 3.3.
Lemma 4.3
Proof
Restricting \(\phi _i\) to the boundary, one can use the scaling argument for each edge and conclude \(\Vert \phi _i\Vert _{\infty ,\partial K}\lesssim 1\). As \(\phi _i\) is harmonic, by the maximum principle, \(\Vert \phi _i\Vert _{\infty ,K} \le \Vert \phi _i\Vert _{\infty ,\partial K}\lesssim 1\). Then \(\Vert \phi _i\Vert _{0,K}\lesssim h_K\) follows. \(\square \)
Lemma 4.4
Proof
Finally the proof is completed by using the Poincaré–Friedrichs inequality \(\Vert v_2\Vert _{0,K}\lesssim h_K\Vert \nabla v_2\Vert _{0,K}\) for \(v_2\in H_0^1(K)\). \(\square \)
In summary, the following theorem holds.
Theorem 4.5
For functions in space \(V_k(K)\), Theorem 4.5 can be applied directly. For space \(W_k(K)\subset V_{k,k}(K)\), if Theorem 4.5 is applied to functions in \(V_{k,k}(K)\), additional moments in \(\varvec{\chi }_{K}^{k}\backslash \varvec{\chi }_{K}^{k2}\) are involved. Henceforth we shall show that no additional moments are required for \(W_k(K)\).
Corollary 4.6
Proof
The lower bound \(h_K \Vert \varvec{\chi } (v)\Vert _{l^2} \lesssim \Vert v\Vert _{0,K}\) is trivial, since \(W_k(K)\) is a subspace of \(V_{k,k}(K)\), and the d.o.f.s in \(V_{k,k}(K)\), comparing with that of \(W_k(K)\), contain additional moments with weights \(\varvec{\chi }_{K}^{k}\backslash \varvec{\chi }_{K}^{k2}\). To prove the upper bound, it suffices to bound these additional moments by the other degrees of freedom.
4.3 Norm equivalence of VEM formulation
With Theorem 4.5, we can obtain the following stability result.
Theorem 4.7
Proof
Corollary 4.8
Proof
As both \(\Pi _k^{\nabla }\) and \(\Pi _k^0\) preserve polynomial of degree k, \((I  \Pi _k^0) v = (I  \Pi _k^0) (I  \Pi _k^{\nabla })v\) and \((I  \Pi _k^{\nabla }) v = (I  \Pi _k^{\nabla })(I  \Pi _k^0)v\).
Remark 4.9
5 Interpolation error estimates

\(v_{\pi }\in {\mathbb {P}}_k(K)\): the \(L^2\) projection of v to the polynomial space;

\(v_c\in S_k(\mathcal T_K)\): the standard nodal interpolant to finite element space \(S_k(\mathcal T_K)\) based on the auxiliary triangulation \(\mathcal T_K\) of K;
 \(v_I\in V_k(K)\) defined as the solution of the local problem$$\begin{aligned} \Delta v_I = \Delta v_{\pi } \text { in } K, \quad v_I = v_c \text { on } \partial K. \end{aligned}$$
 \(I_Kv\in V_k(K)\) defined by d.o.f., i.e.,$$\begin{aligned} I_K v = v_c \text { on } \partial K, \quad (I_K v, p)_K = (v, p)_K, \; \forall p\in {\mathbb {P}}_{k2}(K). \end{aligned}$$
 \(I_K^Wv\in W_k(K)\) defined by d.o.f., i.e.,$$\begin{aligned} I_K^W v = v_c \text { on } \partial K, \quad (I_K^W v, p)_K = (v, p)_K, \; \forall p\in {\mathbb {P}}_{k2}(K). \end{aligned}$$
Remark 5.1
Error estimate for \(v_{\pi }\) is usually presented for a starshaped domain but can be generalized to a domain which is a union of star shaped subdomains (see [21]). Under Assumption A1, the polygon K satisfies the previous condition, so the estimate (26) holds for \(w_K=v_{\pi }\).
The following error estimate can be found in [23, Proposition 4.2]. For completeness, we present a shorter proof by comparing \(v_I\) with \(v_c\).
Lemma 5.2
Proof
By the triangle inequality, it suffices to estimate the difference \(v_I v_c \in H_0^1(K)\). By the Poincaré–Friedrichs inequality \(\Vert v\Vert _{0,K} \le h_K \Vert \nabla v\Vert _{0,K}\) for \(v\in H_0^1(K)\), it suffices to bound the \(H^1\)seminorm of \(v_I v_c\).
Now we estimate \(v  I_K v\) by comparing \(I_K v\) with \(v_I\).
Theorem 5.3
Proof
Next, we present the interpolation error estimate of \( v  I_K^W v\) by comparing \(I_K^Wv\) with \(I_Kv\).
Theorem 5.4
Proof
Again by the triangle inequality and the obtained error estimate for \(v  I_Kv\), it suffices to estimate \(I_K^W v I_K v\in H_0^1(K)\). A crucial observation is that both interpolants, although in different VEM spaces, share the same d.o.f., i.e., \(\varvec{\chi }(I_K^W v) = \varvec{\chi }(I_K v)\). Therefore \(\Pi _k^{\nabla }I_K^W v = \Pi _k^{\nabla } I_K v = \Pi _k^{\nabla } v.\)
Remark 5.5
Notice that the norm equivalence to \(I_K^W v I_K v\) cannot be applied directly since they are in different spaces. Here we use the relations \(\Pi _k^{\nabla }I_K^W v = \Pi _k^{\nabla } I_K v = \Pi _k^{\nabla }v\) and \(\varvec{\chi }(I_K^W v) = \varvec{\chi }(I_K v)\) as a bridge to switch the estimate for \(I_K^Wv\) to that of \(I_Kv\).
6 Conclusion and future work
In this paper we have established the inverse inequality, norm equivalence between the norm of a virtual element function and its degrees of freedom, and interpolation error estimates for several VEM spaces on a polygon which admits a virtual quasiuniform triangulation, i.e., Assumption A1.
We note that A1 rules out polygons with high aspect ratio. Equivalently the constant is not robust to the aspect ratio of K. For example, a rectangle K with two sides \(h_{\max }\) and \(h_{\min }\). It can be decomposed into union of shape regular rectangles but the number depends on the aspect ratio \(h_{\max }/h_{\min }\). In numerical simulation, however, VEM is also robust to the aspect ratio of the elements. In a forthcoming paper, we will examine anisotropic error analysis of VEM based on certain maximum angle conditions.
We present our proofs in two dimensions but it is possible to extend the techniques to three dimensions. The outline is given as follows. Given a polyhedral region K, we need to assume A1 holds for each face \(F\subset \partial K\) and are able to prove results restricted to each face. Then we assume A1 holds for K and prove results as for the 2D case. It is our ongoing study to develop the details in this case.
Notes
Acknowledgements
We thank the referees for valuable suggestions and comments which improved an early version of the paper. The first author was supported by the National Science Foundation (NSF) DMS1418934 and in part by the Sea Poly Project of Beijing Overseas Talents. The second author was partially supported by NSFC (Grant No. 11571237).
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