Some Estimates for Virtual Element Methods

Susanne C. Brenner; Qingguang Guan; Li-Yeng Sung

doi:10.1515/cmam-2017-0008

Publicly Available Published by De Gruyter June 17, 2017

Some Estimates for Virtual Element Methods

Susanne C. Brenner , Qingguang Guan and Li-Yeng Sung

From the journal Computational Methods in Applied Mathematics

https://doi.org/10.1515/cmam-2017-0008

Abstract

We present novel techniques for obtaining the basic estimates of virtual element methods in terms of the shape regularity of polygonal/polyhedral meshes. We also derive new error estimates for the Poisson problem in two and three dimensions.

Keywords: Virtual Elements; Polygonal/Polyhedral Meshes; Poisson Problem

MSC 2010: 65N30

1 Introduction

A salient feature of virtual element methods [4, 5, 15, 1, 14, 22, 6, 11, 2, 9, 8, 3, 16, 7] is that they can be implemented on polygonal/polyhedral meshes. The stability analysis and error analysis for these methods require the extensions of well-known finite element results to general shape functions and general meshes. This is a delicate task since the notion of affine-equivalent elements is no longer available. (See the treatment of the original virtual element method in [10].) The first goal of our paper is to extend some basic finite element estimates to the virtual elements introduced in [1], under the shape regularity assumptions that can be found for example in [4, 1, 8]. The main tool is a discrete norm for virtual element functions that plays the role of the L2 norm in the analysis of standard Lagrange finite element functions and which can be controlled by standard shape regularity arguments. The second goal is to apply these results to the Poisson problem in two and three dimensions and derive new error estimates for certain computable piecewise polynomial approximations generated by the virtual element methods.

The rest of the paper is organized as follows: We consider two-dimensional virtual elements in Section 2, where we obtain basic estimates needed for the error analysis of virtual element methods. Novel techniques for exploiting the shape regularity assumptions are developed in Sections 2.4–2.7. The error analysis for the two-dimensional Poisson problem is then carried out in Section 3. The results in these sections are extended to polygonal meshes that satisfy relaxed shape regularity assumptions in Section 4 and to the Poisson problem in three dimensions in Section 5. The paper ends with some concluding remarks in Section 6.

We will follow the standard notation for differential operators, function spaces and norms that can be found for example in [17, 13], and also the notation in [8] for virtual elements.

2 Local Virtual Element Spaces in Two Dimensions

Let D be a polygon in ℝ2 with diameter hD. For a nonnegative integer k, ℙk is the space of polynomials of degree ≤k and we follow the convention that ℙ-1={0}. The space ℙk⁢(D) is the restriction of ℙk to D. The space of continuous functions on ∂⁡D (resp. D¯) is denoted by C⁢(∂⁡D) (resp. C⁢(D¯)). Let ℰD be the set of the edges of D and let ℙk⁢(e) be the restriction of ℙk to e∈ℰD. The space ℙk⁢(∂⁡D) of continuous piecewise polynomials of degree ≤k on ∂⁡D is defined by

ℙk⁢(∂⁡D)={v∈C⁢(∂⁡D):v|e∈ℙk⁢(e)⁢ for all ⁢e∈ℰD}.

2.1 Shape Regularity Assumptions and Consequences

We assume (cf. [4, 1, 8]) there exists ρ∈(0,1) such that

(2.1)|e|≥ρ⁢hD for any edge e∈ℰD,

and

(2.2)D⁢ is star-shaped with respect to a disc 𝔅 whose radius is ⁢ρ⁢hD.

We shall refer to the center of 𝔅 as the star-center of D.

Conditions (2.1) and (2.2) imply that

(2.3){the minimum angle of the simplicial triangulation 𝒯D of D determined by thestar-center of D and the vertices of D is controlled by ρ

(cf. Figure 1). In particular, the number of vertices of D is also controlled by ρ.

Figure 1

Shape regularity of a polygon.

We will use the notation A≲B to represent the inequality A≤(constant)⁢B, where the positive constant depends only on k and the parameter ρ, and it increases with k and 1/ρ. The notation A≈B is equivalent to A≲B and B≲A.

2.1.1 Trace Inequality

It follows from (2.3) that we have the (scaled) trace inequality

(2.4)hD-1⁢∥ζ∥L2⁢(∂⁡D)2≲hD-2⁢∥ζ∥L2⁢(D)2+|ζ|H1⁢(D)2 for all ⁢ζ∈H1⁢(D).

2.1.2 Discrete Estimates

In view of (2.3), we also have the following discrete estimates:

(2.5)∥p∥L2⁢(∂⁡D)≲hD-12⁢∥p∥L2⁢(D)for all ⁢p∈ℙk,

(2.6)|p|H1⁢(D)≲hD-1⁢∥p∥L2⁢(D)for all ⁢p∈ℙk,

(2.7)hD⁢∥∇⁡p∥L2⁢(∂⁡D)2+hD2⁢∥Δ⁢p∥L2⁢(D)2≲∥∇⁡p∥L2⁢(D)2for all ⁢p∈ℙk.

2.1.3 Sobolev Inequality

It follows from (2.2) that

(2.8)∥ζ∥L∞⁢(D)≲hD-1⁢∥ζ∥L2⁢(D)+|ζ|H1⁢(D)+hD⁢|ζ|H2⁢(D) for all ⁢ζ∈H2⁢(D).

Details can be found in [13, Lemma 4.3.4].

2.1.4 Bramble–Hilbert Estimates

Condition (2.2) implies that we have the following Bramble–Hilbert estimates [12]:

(2.9)infq∈ℙℓ⁡|ζ-q|Hm⁢(D)≲hDℓ+1-m⁢|ζ|Hℓ+1⁢(D) for all ⁢ζ∈Hℓ+1⁢(D),ℓ=0,…,k, and ⁢m≤ℓ.

Details can be found in [13, Lemma 4.3.8].

2.1.5 Poincaré–Friedrichs Inequalities

The following inequalities are direct consequences of (2.4) and (2.9) (with m=ℓ=0):

(2.10)hD-1⁢∥ζ∥L2⁢(D)≲hD-2⁢|∫Dζ⁢𝑑x|+|ζ|H1⁢(D) ⁢for all ⁢ζ∈H1⁢(D),

(2.11)hD-1⁢∥ζ∥L2⁢(D)≲hD-1⁢|∫∂⁡Dζ⁢𝑑s|+|ζ|H1⁢(D) ⁢for all ⁢ζ∈H1⁢(D).

2.2 The Projection Πk,D∇

In view of the Poincaré–Friedrichs inequality (2.11), the Sobolev space H1⁢(D) is a Hilbert space under the inner product ((⋅,⋅)) defined by

(2.12)((ζ,η))=∫D∇⁡ζ⋅∇⁡η⁢d⁢x+(∫∂⁡Dζ⁢𝑑s)⁢(∫∂⁡Dη⁢𝑑s).

The projection operator from H1⁢(D) onto ℙk⁢(D) with respect to ((⋅,⋅)) is denoted by Πk,D∇, i.e., Πk,D∇⁢ζ∈ℙk⁢(D) satisfies

(2.13)((Πk,D∇⁢ζ,q))=((ζ,q)) for all ⁢q∈ℙk⁢(D).

Letting q=1 in (2.13), we see that

(2.14)∫∂⁡DΠk,D∇⁢ζ⁢𝑑s=∫∂⁡Dζ⁢𝑑s,

and then (2.13) implies

∫D∇⁡(Πk,D∇⁢ζ)⋅∇⁡q⁢d⁢x=∫D∇⁡ζ⋅∇⁡q⁢d⁢x

(2.15)=∫∂⁡Dζ⁢(n⋅∇⁡q)⁢𝑑s-∫Dζ⁢(Δ⁢q)⁢𝑑x for all ⁢q∈ℙk⁢(D).

For k≥2, we can also use the alternative inner product (cf. (2.10))

(2.16)(((ζ,η)))=∫D∇⁡ζ⋅∇⁡η⁢d⁢x+(∫Dζ⁢𝑑x)⁢(∫Dη⁢𝑑x)

to define Πk,D∇, in which case (2.14) is replaced by

(2.17)∫DΠk,D∇⁢ζ⁢𝑑x=∫Dζ⁢𝑑x.

Remark 2.1.

It is clear that

(2.18)Πk,D∇⁢p=p for all ⁢p∈ℙk⁢(D).

2.3 The Space 𝒬k⁢(D)

We begin with the following well-posedness result for the Poisson problem.

Lemma 2.2.

Given any g∈Pk⁢(∂⁡D) and f∈Pk⁢(D), there exists a unique function v∈H1⁢(D) such that (i)v=g on ∂⁡D and (ii)

∫D∇⁡v⋅∇⁡w⁢d⁢x=∫Df⁢w⁢𝑑x for all ⁢w∈H01⁢(D).

Proof.

Let g~∈H1⁢(D) be a ℙk Lagrange finite element function with respect to the triangulation 𝒯D such that g~=g on ∂⁡D, and let ϕ∈H01⁢(D) be defined by

(2.19)∫D∇⁡ϕ⋅∇⁡w⁢d⁢x=∫Df⁢w⁢𝑑x-∫D∇⁡g~⋅∇⁡w⁢d⁢x for all ⁢w∈H01⁢(D).

Then the unique v∈H01⁢(D) with properties (i) and (ii) is given by ϕ+g~. ∎

Remark 2.3.

Note that the finite element function g~ belongs to H(3/2)-ϵ⁢(D) (for any ϵ>0) and hence the right-hand side of (2.19) defines a bounded linear functional on H0(1/2)+ϵ⁢(D) (for any ϵ>0). It then follows from elliptic regularity [18] that ϕ belongs to H(3/2)-ϵ⁢(D), and therefore ϕ belongs to C⁢(D¯) by the Sobolev embedding theorem. Consequently, v=ϕ+g~ also belongs to C⁢(D¯).

For k≥1, the space 𝒬k⁢(D)⊂H1⁢(D) is defined by the following conditions: v∈H1⁢(D) belongs to 𝒬k⁢(D) if and only if (i) the trace of v on ∂⁡D belongs to ℙk⁢(∂⁡D), (ii) there exists a polynomial qv(=-Δ⁢v)∈ℙk⁢(D) such that

(2.20)∫D∇⁡v⋅∇⁡w⁢d⁢x=∫Dqv⁢w⁢𝑑x for all ⁢w∈H01⁢(D),

and (iii) we have

(2.21)Πk,D0⁢v-Πk,D∇⁢v∈ℙk-2⁢(D),

where Πk,D0 is the orthogonal projection from L2⁢(D) onto ℙk⁢(D).

Remark 2.4.

It is clear that ℙk⁢(D) is a subspace of 𝒬k⁢(D). Note that (2.21) implies Πk,D0=Πk,D∇ if k=1. Moreover, if we use the alternative inner product (2.16) to define Πk,D∇, then Πk,D0=Πk,D∇ also for k=2.

Lemma 2.5.

We have (i)dim⁡Qk⁢(D)=dim⁡Pk⁢(∂⁡D)+dim⁡Pk-2⁢(D) and (ii)v∈Qk⁢(D) is uniquely determined by v|∂⁡D and Πk-2,D0⁢v.

Proof.

Let 𝒬~k(D)={v∈H1(D):v|∂⁡D∈ℙk(∂D) and Δv∈ℙk(D)}. The linear map v↦(v|∂⁡D,Δ⁢v) from 𝒬~k⁢(D) to ℙk⁢(∂⁡D)×ℙk⁢(D) is an isomorphism by Lemma 2.2. Furthermore, the linear map

v↦(v|∂⁡D,Πk-2,D0⁢v+(Πk,D0-Πk-2,D0)⁢(v-Πk,D∇⁢v))

is also an isomorphism from 𝒬~k⁢(D) to ℙk⁢(∂⁡D)×ℙk⁢(D) because

v⁢ belonging to the null space

⟹v|∂⁡D=0⁢ and ⁢Πk-2,D0⁢v=0

⟹Πk,D∇⁢v=0(by (2.14) and (2.15), or (2.15) and (2.17))

⟹Πk,D0⁢v=0

⟹∇⁡v=𝟎(by (2.20))

⟹v=0(by (2.14) or (2.17))

Since (2.21) is equivalent to

(2.22)(Πk,D0-Πk-2,D0)⁢(v-Πk,D∇⁢v)=(Πk,D0-Πk-2,D0)⁢(Πk,D0⁢v-Πk,D∇⁢v)=0,

the lemma follows immediately. ∎

Remark 2.6.

It follows from Lemma 2.5 that we can take the degrees of freedom of 𝒬k⁢(D) to be (i) the values at the vertices of D, (ii) the moments of order ≤k-2 on the edges of D, and (iii) and the moments of order ≤k-2 on D. Using these degrees of freedom for v∈𝒬k⁢(D), we can compute Πk,D∇⁢v through (2.14) and (2.15) (or (2.15) and (2.17) if k≥2) and then Πk,D0⁢v through

Πk,D0⁢v=Πk-2,D0⁢v+(Πk,D0-Πk-2,D0)⁢Πk,D∇⁢v.

Remark 2.7.

It follows from Remark 2.3 that 𝒬k⁢(D)⊂C⁢(D¯).

The following result shows that functions in 𝒬k⁢(D) enjoy a minimum energy principle.

Lemma 2.8.

The inequality

|v|H1⁢(D)≤|ζ|H1⁢(D)

holds for any v∈Qk⁢(D) and ζ∈H1⁢(D) such that ζ-v=0 on ∂⁡D and Πk,D0⁢(ζ-v)=0.

Proof.

It follows from (2.20) that

∫D∇⁡v⋅∇⁡(ζ-v)⁡d⁢x=∫Dqv⁢(ζ-v)⁢𝑑x=0,

and hence

|ζ|H1⁢(D)2=|ζ-v|H1⁢(D)2+|v|H1⁢(D)2.∎

2.4 The (Semi-)Norm |||⋅|||k,D

The semi-norm |||⋅|||k,D on H1⁢(D) is defined by

(2.23)|||ζ|||k,D2=hD⁢∑e∈ℰD∥Πk,e0⁢ζ∥L2⁢(e)2+∥Πk-2,D0⁢ζ∥L2⁢(D)2,

where Πk,e0 is the orthogonal projection from L2⁢(e) onto ℙk⁢(e). It will play the role of ∥⋅∥L2⁢(D) in the analysis of virtual elements.

Remark 2.9.

It is clear that

(2.24)|||ζ|||k,D2≤hD⁢∥ζ∥L2⁢(∂⁡D)2+∥Πk-2,D0⁢ζ∥L2⁢(D)2

and that |||⋅|||k,D is a computable norm on 𝒬k⁢(D). Note also that

(2.25)|||ζ|||k,D2=hD⁢∥ζ∥L2⁢(∂⁡D)2+∥Πk-2,D0⁢ζ∥L2⁢(D)2 for all ⁢v∈𝒬k⁢(D).

Let 𝒩⁢(Πk,D∇)={v∈𝒬k⁢(D):Πk,D∇⁢v=0} be the null space of the projection Πk,D∇ restricted to 𝒬k⁢(D). Below we will show that the norm |||⋅|||k,D is equivalent to the norm

hD⁢∥⋅∥L2⁢(∂⁡D) on ⁢𝒩⁢(Πk,D∇).

The following lemma on polynomials is needed for this purpose.

Lemma 2.10.

Given any p∈Pk-2 (k≥2), there exists q∈Pk such that Δ⁢q=p and

∥q∥L2⁢(B)≤C⁢∥p∥L2⁢(B),

where B is the unit disc and the positive constant C depends only on k.

Proof.

Since Δ maps ℙk onto ℙk-2, there exists an operator Δ†:ℙk-2→ℙk such that Δ⁢Δ† is the identity operator on ℙk-2. The lemma follows from the observation that both ∥p∥L2⁢(B) and ∥Δ†⁢p∥L2⁢(B) are norms on ℙk-2. ∎

Lemma 2.11.

We have

(2.26)|||v|||k,D≈hD⁢∥v∥L2⁢(∂⁡D) for all ⁢v∈𝒩⁢(Πk,D∇).

Proof.

Since Πk-2,D0⁢v=0 for k=1, we can assume k≥2. It also suffices to consider the case where hD=1.

Let v∈𝒩⁢(Πk,D∇) be arbitrary. It follows from (2.15) that

(2.27)∫Dv⁢(Δ⁢q)⁢𝑑x=∫∂⁡Dv⁢(n⋅∇⁡q)⁢𝑑s for all ⁢q∈ℙk⁢(D).

Let 𝔅⊂D be the disc with radius ρ stipulated in Section 2.1 and let 𝔅~ be the unit disc concentric with 𝔅. Then we have D⊂𝔅~ and

(2.28)∥p∥L2⁢(𝔅)≈∥p∥L2⁢(𝔅~) for all ⁢p∈ℙk

by the equivalence of norms on finite-dimensional vector spaces.

Given any p∈ℙk-2, there exists q∈ℙk such that Δ⁢q=p and

(2.29)∥q∥L2⁢(D)≤∥q∥L2⁢(𝔅~)≲∥p∥L2⁢(𝔅~)≲∥p∥L2⁢(𝔅)≤∥p∥L2⁢(D)

by Lemma 2.10 and (2.28).

Putting (2.6), (2.7), (2.27), and (2.29) together, we find

∫Dv⁢p⁢𝑑x≤∥v∥L2⁢(∂⁡D)⁢∥∇⁡q∥L2⁢(∂⁡D)≲∥v∥L2⁢(∂⁡D)⁢∥∇⁡q∥L2⁢(D)

≲∥v∥L2⁢(∂⁡D)⁢∥q∥L2⁢(D)≲∥v∥L2⁢(∂⁡D)⁢∥p∥L2⁢(D) for all ⁢p∈ℙk-2,

and hence

(2.30)∥Πk-2,D0⁢v∥L2⁢(D)=maxp∈ℙk-2⁡(∫Dv⁢p⁢𝑑x)/∥p∥L2⁢(D)≲∥v∥L2⁢(∂⁡D).

Estimate (2.26) follows from (2.25) and (2.30). ∎

The next result is needed for another characterization of |||⋅|||k,D on 𝒩⁢(Πk,D∇).

Lemma 2.12.

If v∈N⁢(Πk,D∇), then v must vanish at some point on ∂⁡D.

Proof.

This is trivial if Πk,D∇ is defined by (2.14) because we have ∫∂⁡Dv⁢𝑑s=0.

If k≥2 and Πk,D∇ is defined by (2.17), then ∫Dv⁢𝑑x=0 for all v∈𝒩⁢(Πk,D∇), and hence (2.27) implies

(2.31)∫∂⁡Dv⁢(n⋅∇⁡q)⁢𝑑s=0 for all ⁢v∈𝒩⁢(Πk,D∇)⁢ and ⁢q∈ℙ2.

Let x* be the center of the disc 𝔅⊂D from Section 2.1. Then D is star-shaped with respect to 𝔅 and the quadratic polynomial q⁢(x)=|x-x*|2 satisfies

(2.32)n⋅∇⁡q>0 on ⁢∂⁡D⁢ (except the corners).

The lemma follows from (2.31) and (2.32). ∎

Lemma 2.13.

We have

(2.33)|||v|||k,D≈hD3/2⁢∥∂⁡v/∂⁡s∥L2⁢(∂⁡D) for all ⁢v∈𝒩⁢(Πk,D∇),

where ∂/∂⁡s denotes a tangential derivative along ∂⁡D.

Proof.

First we observe that

(2.34)∥∂⁡v/∂⁡s∥L2⁢(∂⁡D)≲hD-1⁢∑e∈ℰD∥v∥L2⁢(∂⁡D) for all ⁢v∈𝒬k⁢(D)

by a standard inverse estimate for polynomials in one variable [17, 13]. On the other hand, the opposite estimate

(2.35)hD-1⁢∥v∥L2⁢(∂⁡D)≲∥∂⁡v/∂⁡s∥L2⁢(∂⁡D) for all ⁢v∈𝒩⁢(Πk,D∇)

holds by Lemma 2.12 and a direct calculation.

The equivalence (2.33) follows from (2.25), Lemma 2.11, (2.34), and (2.35). ∎

2.5 Estimates for Πk,D∇

There is an obvious stability estimate

(2.36)|Πk,D∇⁢ζ|H1⁢(D)≤|ζ|H1⁢(D) for all ⁢ζ∈H1⁢(D)

that follows from (2.15).

There is also a stability estimate for Πk,D∇ in ∥⋅∥L2⁢(D) in terms of the semi-norm |||⋅|||k,D.

Lemma 2.14.

We have

(2.37)∥Πk,D∇⁢ζ∥L2⁢(D)≲|||ζ|||k,D for all ⁢ζ∈H1⁢(D).

Proof.

It suffices to establish (2.37) in the case where hD=1.

It follows from (2.7) and (2.15) that

∫D∇⁡(Πk,D∇⁢ζ)⋅∇⁡(Πk,D∇⁢ζ)⁡d⁢x=∫∂⁡Dζ⁢n⋅∇⁡(Πk,D∇⁢ζ)⁡d⁢s-∫Dζ⁢Δ⁢(Πk,D∇⁢ζ)⁢𝑑x

≲∑e∈ℰD∥Πk-1,e0⁢ζ∥L2⁢(e)⁢∥∇⁡Πk,D∇⁢ζ∥L2⁢(e)+∥Πk-2,D0⁢ζ∥L2⁢(D)⁢∥Δ⁢(Πk,D∇⁢ζ)∥L2⁢(D)

≲(∑e∈ℰD∥Πk,e0⁢ζ∥L2⁢(e)2+∥Πk-2,D0⁢ζ∥L2⁢(D)2)12⁢|Πk,D∇⁢ζ|H1⁢(D),

and hence

(2.38)|Πk,D∇⁢ζ|H1⁢(D)2≲∑e∈ℰD∥Πk,e0⁢ζ∥L2⁢(e)2+∥Πk-2,D0⁢ζ∥L2⁢(D)2.

Moreover, (2.14) implies

(2.39)|∫∂⁡DΠk,D∇⁢ζ⁢𝑑s|=|∑e∈ℰD∫eΠ0,e0⁢ζ⁢𝑑s|≲(∑e∈ℰD∥Πk,e0⁢ζ∥L2⁢(e)2)12,

and

(2.40)|∫DΠk,D∇⁢ζ⁢𝑑x|=|∫DΠ0,D0⁢ζ⁢𝑑x|≲∥Πk-2,D0⁢ζ∥L2⁢(D)

if k≥2 and Πk,D∇ is defined by (2.17).

Estimate (2.37) follows from (2.10)–(2.11) and (2.38)–(2.40). ∎

Lemma 2.15.

We have

(2.41)∥ζ-Πk,D∇⁢ζ∥L2⁢(D)≲hDℓ+1⁢|ζ|Hℓ+1⁢(D)for all ⁢ζ∈Hℓ⁢(D), 0≤ℓ≤k,

(2.42)|ζ-Πk,D∇⁢ζ|H1⁢(D)≲hDℓ⁢|ζ|Hℓ+1⁢(D)for all ⁢ζ∈Hℓ⁢(D), 1≤ℓ≤k.

Proof.

In view of the stability estimates (2.36) and (2.37), estimates (2.41)–(2.42) follow from (2.4), (2.9), (2.24), and the fact that q-Πk,D∇⁢q=0 for all q∈ℙℓ⁢(D). ∎

Remark 2.16.

Estimates for Πk,D∇ in other Sobolev norms can be found in [19, 20].

2.6 Estimates for Πk,D0

The obvious stability estimate

∥Πk,D0⁢ζ∥L2⁢(D)≤∥ζ∥L2⁢(D) for all ⁢ζ∈L2⁢(D)

together with (2.9) and the fact that q-Πk,D0⁢q=0 for all q∈ℙℓ⁢(D) implies

(2.43)∥ζ-Πk,D0⁢ζ∥L2⁢(D)≤∥ζ-Πℓ,00⁢ζ∥L2⁢(D)≲hDℓ+1⁢|ζ|Hℓ+1⁢(D) for all ⁢ζ∈Hℓ+1⁢(D), 0≤ℓ≤k.

There is also a stability estimate for Πk,D0 in |⋅|H1⁢(D).

Lemma 2.17.

We have

(2.44)|Πk,D0⁢ζ|H1⁢(D)≲|ζ|H1⁢(D) for all ⁢ζ∈H1⁢(D).

Proof.

This is a consequence of (2.6), (2.36), (2.41), and (2.43):

|Πk,D0⁢ζ|H1⁢(D)≤|Πk,D0⁢ζ-Πk,D∇⁢ζ|H1⁢(D)+|Πk,D∇⁢ζ|H1⁢(D)

≲hD-1⁢∥Πk,D0⁢ζ-Πk,D∇⁢ζ∥L2⁢(D)+|ζ|H1⁢(D)

≲hD-1⁢(∥Πk,D0⁢ζ-ζ∥L2⁢(D)+∥ζ-Πk,D∇⁢ζ∥L2⁢(D))+|ζ|H1⁢(D)

≲|ζ|H1⁢(D),

as desired. ∎

Estimates (2.9), (2.44) and the fact that q-Πk,D0⁢q=0 for all q∈ℙℓ⁢(D) imply

(2.45)|ζ-Πk,D0⁢ζ|H1⁢(D)≲hDℓ⁢|ζ|Hℓ+1⁢(D) for all ⁢ζ∈Hℓ+1⁢(D), 1≤ℓ≤k.

The following lemma contains another useful estimate.

Lemma 2.18.

We have

∥Πk,D0⁢v∥L2⁢(D)≲|||v|||k,D for all ⁢v∈𝒬k⁢(D).

Proof.

Let v∈𝒬k⁢(D) be arbitrary. It follows from (2.22) that

∥Πk,D0⁢v∥L2⁢(D)2=∥Πk-2,D0⁢v∥L2⁢(D)2+∥(Πk,D0-Πk-2,D0)⁢v∥L2⁢(D)2

=∥Πk-2,D0⁢v∥L2⁢(D)2+∥(Πk,D0-Πk-2,D0)⁢Πk,D∇⁢v∥L2⁢(D)2

≤∥Πk-2,D0⁢v∥L2⁢(D)2+∥Πk,D∇⁢v∥L2⁢(D)2,

which together with (2.23) and (2.37) implies the lemma. ∎

2.7 Some Inequalities Involving |||⋅|||k,D

The role of the L2 norm in the following inverse inequality for 𝒬k⁢(D) is assumed by |||⋅|||k,D.

Lemma 2.19.

We have

(2.46)|v|H1⁢(D)≲hD-1⁢|||v|||k,D for all ⁢v∈𝒬k⁢(D).

Proof.

It suffices to prove (2.46) when hD=1.

Let ϕ≥0 be a smooth function supported on the disc 𝔅 with radius ρ (cf. Section 2.1) such that

∫Dϕ⁢𝑑x=1.

Let 𝔅~ be the unit disc concentric with 𝔅. Then D is a subset of 𝔅~ and we have, by the equivalence of norms on finite-dimensional vector spaces and scaling,

(2.47)∥p∥L2⁢(D)2≤∥p∥L2⁢(𝔅~)2≤C*⁢∫𝔅p2⁢ϕ⁢𝑑x for all ⁢p∈ℙk,

where the positive constant C* only depends on ρ and k.

Let w∈H1⁢(D) be the ℙk Lagrange finite element function with respect to the triangulation of 𝒯D such that (i) w=v on ∂⁡D and (ii) w=0 at all nodes interior to D. It follows from (2.3) and scaling that

(2.48)∥w∥L2⁢(D)≈∥w∥L2⁢(∂⁡D)=∥v∥L2⁢(∂⁡D)≈|w|H1⁢(D).

Let ζ=w+p⁢ϕ for p∈ℙk⁢(D) such that

∫D(ζ-v)⁢q⁢𝑑x=0 for all ⁢q∈ℙk⁢(D),

or equivalently

(2.49)∫Dp⁢q⁢ϕ⁢𝑑x=∫D(v-w)⁢q⁢𝑑x=∫D(Πk,D0⁢v-w)⁢q⁢𝑑x for all ⁢q∈ℙk⁢(D).

We have

(2.50)|v|H1⁢(D)≲|ζ|H1⁢(D)

by Lemma 2.8, and

(2.51)∥Πk,D0⁢v∥L2⁢(D)2≲∥v∥L2⁢(∂⁡D)2+∥Πk-2,D0⁢v∥L2⁢(D)2

by Lemma 2.18.

Putting (2.47), (2.48), (2.49), and (2.51) together, we arrive at the estimate

(2.52)∥p∥L2⁢(D)2≲∥v∥L2⁢(∂⁡D)2+∥Πk-2,D0⁢v∥L2⁢(D)2.

It then follows from (2.6), (2.48) and (2.52) that

|ζ|H1⁢(D)2≤2⁢|w|H1⁢(D)2+2⁢|p⁢ϕ|H1⁢(D)2

≲∥w∥L2⁢(∂⁡D)2+∥p∥L2⁢(D)2

(2.53)≲∥v∥L2⁢(∂⁡D)2+∥Πk-2,D0⁢v∥L2⁢(D)2.

Finally, estimate (2.46) follows from (2.50) and (2.53). ∎

Corollary 2.20.

We have

∥v∥L2⁢(D)≲|||v|||k,D for all ⁢v∈𝒬k⁢(D).

Proof.

This follows directly from Lemma 2.14, Lemma 2.15 and Lemma 2.19:

∥v∥L2⁢(D)≤∥v-Πk,D∇⁢v∥L2⁢(Ω)+∥Πk,D∇⁢v∥L2⁢(D)≲hD⁢|v|H1⁢(D)+|||v|||k,D≲|||v|||k,D.∎

Remark 2.21.

We say that two polygons have the same shape (or are similar) if one can be mapped to the other by a rigid motion followed by dilation. For each equivalence class of similar polygons, we have

∥v∥L2⁢(D)≈|||v|||k,D for all ⁢v∈𝒬k⁢(D)

by the equivalence of norms on finite-dimensional vector spaces and scaling, where the hidden constants depend on k and the shape of the polygon. But it is not clear that the constant in the estimate

|||v|||k,D≲∥v∥L2⁢(D) v∈𝒬k⁢(D)

can also be controlled by k and ρ.

The role of the L2 norm in the following Friedrichs’ inequality is also assumed by |||⋅|||k,D.

Lemma 2.22.

We have

(2.54)|||v-v¯|||k,D≲hD⁢|v|H1⁢(D) for all ⁢v∈H1⁢(D),

where v¯=(∫Dv⁢𝑑x)/|D| or (∫∂⁡Dv⁢𝑑s)/|∂⁡D|.

Proof.

It suffices to derive (2.54) for the case where hD=1 by using (2.4), (2.10), (2.11), and (2.24) as follows:

|||v-v¯|||k,D2≤∥v-v¯∥L2⁢(∂⁡D)2+∥Πk-2,D0⁢(v-v¯)∥L2⁢(D)2≲∥v-v¯∥H1⁢(D)2≲|v|H1⁢(D)2.∎

2.8 Interpolation Operator

The interpolation operator Ik,D:H2⁢(D)→𝒬k⁢(D) is defined by the condition that ζ and Ik,D⁢ζ have the same degrees of freedom (cf. Remark 2.6). It is clear that

Ik,D⁢q=q for all ⁢q∈ℙk⁢(D).

Lemma 2.23.

We have, for 1≤ℓ≤k,

(2.55)∥ζ-Ik,D⁢ζ∥L2⁢(D)+∥ζ-Πk,D∇⁢Ik,D⁢ζ∥L2⁢(D)≲hDℓ+1⁢|ζ|Hℓ+1⁢(D)for all ⁢ζ∈Hℓ+1⁢(D),

(2.56)|ζ-Ik,D⁢ζ|H1⁢(D)+|ζ-Πk,D∇⁢Ik,D⁢ζ|H1⁢(D)≲hDℓ⁢|ζ|Hℓ+1⁢(D)for all ⁢ζ∈Hℓ+1⁢(D).

Proof.

It suffices to consider the case where hD=1.

First we observe that

∥Ik,D⁢ζ∥L2⁢(D)+∥Πk,D∇⁢Ik,D⁢ζ∥L2⁢(D)≲|||Ik,D⁢ζ|||k,D

by Lemma 2.14 and Corollary 2.20, and

|Ik,D⁢ζ|H1⁢(D)+|Πk,D∇⁢Ik,D⁢ζ|H1⁢(D)≲|Ik,D⁢ζ|H1⁢(D)≲|||Ik,D⁢ζ|||k,D

by (2.36) and Lemma 2.19. Moreover, we have

|||Ik,D⁢ζ|||k,D2≤∥Ik,D⁢ζ∥L2⁢(∂⁡D)2+∥Πk-2,D0⁢(Ik,D⁢ζ)∥L2⁢(D)2≲∥ζ∥L∞⁢(∂⁡D)2+∥ζ∥L2⁢(D)2≲∥ζ∥Hℓ+1⁢(D)2

by (2.8) and (2.24).

Estimates (2.55)–(2.56) then follow from the Bramble–Hilbert estimates (2.9) and from the fact that q-Ik,D⁢q=0=q-Πk,D∇⁢Ik,D⁢q for all q∈ℙℓ⁢(D). ∎

The proof of the following result is similar.

Lemma 2.24.

We have

(2.57)∥Ik,D⁢ζ-Π1,D0⁢Ik,D⁢ζ∥L2⁢(D)≲hD2⁢|ζ|H2⁢(D) for all ⁢ζ∈H2⁢(D).

3 The Poisson Problem in Two Dimensions

Let Ω be a bounded polygonal domain in ℝ2, f∈L2⁢(Ω) and u∈H01⁢(Ω) such that

(3.1)a⁢(u,v)=(f,v) for all ⁢v∈H01⁢(Ω),

where

a⁢(u,v)=∫Ω∇⁡u⋅∇⁡v⁢d⁢x and (f,v)=∫Ωf⁢v⁢𝑑x.

The Poisson problem (3.1) can be solved numerically by virtual element methods.

3.1 The Virtual Element Space 𝒬hk

Let 𝒯h be a conforming partition of Ω by polygonal subdomains, i.e., the intersection of two distinct subdomains is either the empty set, the set of common vertices or the set of common edges. We assume that all polygons D∈𝒯h satisfy the shape regularity assumptions in Section 2.1.

We take the virtual element space 𝒬hk to be {v∈H01(Ω):v|D∈𝒬k(D) for all D∈𝒯h} and denote by 𝒫hk the space of (discontinuous) piecewise polynomials of degree ≤k with respect to 𝒯h. The operators

Πk,h∇:H1⁢(Ω)→𝒫hk,Πk,h0:L2⁢(Ω)→𝒫hk,Ik,h:H2⁢(Ω)∩H01⁢(Ω)→𝒬hk

are defined in terms of their local counterparts:

(Πk,h∇⁢ζ)|D=Πk,D∇⁢(ζ|D)for all ⁢ζ∈H1⁢(Ω),

(Πk,h0⁢ζ)|D=Πk,D0⁢(ζ|D)for all ⁢ζ∈L2⁢(Ω),

(Ik,h⁢ζ)|D=Ik,D⁢(ζ|D)for all ⁢ζ∈H2⁢(Ω)∩H01⁢(Ω).

Let the semi-norm |⋅|h,1 on H1⁢(Ω)+𝒫hk be defined by

|v|h,12=∑D∈𝒯h|v|H1⁢(D)2.

Then we have

|v|h,1=|v|H1⁢(Ω) for all ⁢v∈H1⁢(Ω)

and, in view of (2.15),

(3.2)|u-Πk,h∇⁢u|h,1=infw∈𝒫hk⁡|u-w|h,1 for all ⁢u∈H1⁢(Ω)+𝒫hk.

The local estimates (2.41), (2.42), (2.45), and (2.55)–(2.57) immediately imply the following global results, where h=maxD∈𝒯h⁡hD.

Lemma 3.1.

The following estimates are valid for ζ∈Hℓ+1⁢(Ω)∩H01⁢(Ω) and 1≤ℓ≤k:

(3.3)∥ζ-Ik,h⁢ζ∥L2⁢(Ω)+∥ζ-Πk,h∇⁢Ik,h⁢ζ∥L2⁢(Ω)≲hℓ+1⁢|ζ|Hℓ+1⁢(Ω),

(3.4)|ζ-Ik,h⁢ζ|H1⁢(Ω)+|ζ-Πk,h∇⁢ζ|h,1+|ζ-Πk,h∇⁢Ik,h⁢ζ|h,1≲hℓ⁢|ζ|Hℓ+1⁢(Ω).

Moreover, we have

(3.5)∥Ik,h⁢ζ-Π1,h0⁢Ik,h⁢ζ∥L2⁢(Ω)≲h2⁢|ζ|H2⁢(Ω) for all ⁢ζ∈H2⁢(Ω)∩H01⁢(Ω),

and

(3.6)|ζ-Πk,h0⁢ζ|h,1≲hℓ⁢|ζ|Hℓ+1⁢(Ω),

(3.7)∥ζ-Πk,h0⁢ζ∥L2⁢(Ω)≤∥ζ-Πℓ,h0⁢ζ∥L2⁢(Ω)≲hℓ+1⁢|ζ|Hℓ+1⁢(Ω)

for 0≤ℓ≤k and ζ∈Hℓ+1⁢(Ω).

3.2 Discrete Problem

The discrete problem is to find uh∈𝒬hk such that

(3.8)ah⁢(uh,v)=(f,Ξh⁢v) for all ⁢v∈𝒬hk,

where Ξh is a linear operator that maps 𝒬hk into 𝒫hk to be chosen in Section 3.2.3,

(3.9)ah⁢(w,v)=∑D∈𝒯h[aD⁢(Πk,D∇⁢w,Πk,D∇⁢v)+SD⁢(w-Πk,D∇⁢w,v-Πk,D∇⁢v)],

(3.10)aD⁢(w,v)=∫D∇⁡w⋅∇⁡v⁢d⁢x,

(3.11)SD⁢(w,v)=hD-1⁢∑e∈ℰD(Πk,e0⁢w,Πk,e0⁢v)L2⁢(e)+hD-2⁢(Πk-2,D0⁢w,Πk-2,D0⁢v)L2⁢(D).

Note that

(3.12)aD⁢(ζ-Πk,D∇⁢ζ,q)=0 for all ⁢ζ∈H1⁢(D)⁢ and ⁢q∈ℙk⁢(D)

by (2.15), and the restriction of SD⁢(⋅,⋅) to 𝒬k⁢(D)×𝒬k⁢(D) can be computed in terms of the degrees of freedom of 𝒬k⁢(D). Moreover, we have

SD⁢(ζ,ζ)=hD-2⁢|||ζ|||k,D2 for all ⁢v∈H1⁢(D).

3.2.1 Other Choices for SD⁢(⋅,⋅)

There are other choices for the bilinear form SD⁢(⋅,⋅). The analysis of the virtual element methods remains valid as long as

SD⁢(v,v)≈hD-2⁢|||v|||k,D2 for all ⁢v∈𝒩⁢(Πk,D∇).

For example, in view of Lemma 2.11, we can take (cf. [10])

(3.13)SD⁢(w,v)=hD-1⁢∑e∈ℰD(Πk,e0⁢w,Πk,e0⁢v)L2⁢(e),

and according to Lemma 2.13, another choice [24, 10] is

(3.14)SD⁢(w,v)=hD⁢∑e∈ℰD(∂⁡w/∂⁡s,∂⁡v/∂⁡s)L2⁢(e),

where ∂/∂⁡s denotes a tangential derivative.

3.2.2 Well-Posedness

Since we have 𝒩⁢(Πk,D∇)={v-Πk,D∇⁢v:v∈𝒬k⁢(D)}, the well-posedness of (3.8) is established through the following result.

Lemma 3.2.

We have

SD⁢(w,w)≈|w|H1⁢(D)2 for all ⁢v∈𝒩⁢(Πk,D∇).

Proof.

On the one hand, it follows from Lemma 2.19 and (3.11) that

|w|H1⁢(D)2≲hD-2⁢|||w|||k,D2=SD⁢(w,w) for all ⁢w∈𝒩⁢(Πk,D∇).

On the other hand, we have, by Lemma 2.22 and (3.11),

(3.15)SD⁢(w,w)=hD-2⁢|||w|||k,D2≲|w|H1⁢(D)2 for all ⁢w∈𝒩⁢(Πk,D∇)

since (w,1)L2⁢(∂⁡D)=0 by (2.14). (For k≥2 and Πk,D∇ defined by (2.17), we can use (w,1)L2⁢(D)=0.) ∎

It follows from (3.10), (3.12) and Lemma 3.2 that

aD⁢(Πk,D∇⁢v,Πk,D∇⁢v)+SD⁢(v-Πk,D∇⁢v,v-Πk,D∇⁢v)≈aD⁢(Πk,D∇⁢v,Πk,D∇⁢v)+aD⁢(v-Πk,D∇⁢v,v-Πk,D∇⁢v)

=aD⁢(v,v)

for all v∈𝒬k⁢(D), and hence

(3.16)ah⁢(v,v)≈a⁢(v,v) for all ⁢v∈𝒬hk.

Therefore, the discrete problem (3.8) is uniquely solvable.

Remark 3.3.

Estimate (3.15) and the Cauchy–Schwarz inequality imply that

(3.17)SD⁢(v-Πk,D∇⁢v,w-Πk,D∇⁢w)≲|v-Πk,D∇⁢v|H1⁢(D)⁢|w-Πk,D∇⁢w|H1⁢(D) for all ⁢v,w∈𝒬k⁢(D).

3.2.3 Choices for Ξh

We will choose Ξh according to the following recipe:

(3.18)Ξh={Π1,h0if k=1,2,Πk-2,h0if k≥3.

The following result is useful for the error analysis in |⋅|H1⁢(Ω).

Lemma 3.4.

For 1≤ℓ≤k, we have

(3.19)(f,w-Ξh⁢w)≲hℓ⁢|f|Hℓ-1⁢(Ω)⁢|w|H1⁢(Ω) for all ⁢f∈Hℓ-1⁢(Ω),w∈𝒬hk.

Proof.

In view of the relation

(f,w-Ξh⁢w)=(f-Πk-2,h0⁢f,w-Ξh⁢w)

≤∥f-Πk-2,h0⁢f∥L2⁢(Ω)⁢∥w-Ξh⁢w∥L2⁢(Ω)

≤∥f-Πℓ-2,h0⁢f∥L2⁢(Ω)⁢∥w-Π0,h0⁢w∥L2⁢(Ω),

estimate (3.19) follows from (3.7). ∎

The following result is useful for the error analysis in ∥⋅∥L2⁢(Ω).

Lemma 3.5.

For 1≤ℓ≤k, we have

(3.20)(f,Ik,h⁢ζ-Ξh⁢Ik,h⁢ζ)≲hℓ+1⁢|f|Hℓ-1⁢(Ω)⁢|ζ|H2⁢(Ω) for all ⁢f∈Hℓ-1⁢(Ω),ζ∈H2⁢(Ω).

Proof.

In view of the relation

(f,Ik,h⁢ζ-Ξh⁢Ik,h⁢ζ)=(f-Πk-2,h0⁢f,Ik,h⁢ζ-Ξh⁢Ik,h⁢ζ)

≤∥f-Πk-2,h0⁢f∥L2⁢(Ω)⁢∥Ik,h⁢ζ-Ξh⁢Ik,h⁢ζ∥L2⁢(Ω)

≤∥f-Πℓ-2,h0⁢f∥L2⁢(Ω)⁢∥Ik,h⁢ζ-Π1,h0⁢Ik,h⁢ζ∥L2⁢(Ω),

estimate (3.20) follows from (3.5) and (3.7). Note that this is the reason why Ξh is chosen to be Π1,h0 for k=2 instead of Πk-2,h0=Π0,h0. ∎

3.3 Error Estimates in the Energy Norm

We begin with an abstract estimate for u-uh.

Theorem 3.6.

Assuming f∈Hℓ-1⁢(Ω) for some ℓ between 1 and k, there exists a positive constant C depending only on k and ρ such that

(3.21)|u-uh|H1⁢(Ω)≤C⁢(infv∈𝒬hk⁡|u-v|H1⁢(Ω)+infw∈𝒫hk⁡|u-w|h,1+hℓ⁢|f|Hℓ-1⁢(Ω)).

Proof.

Given any v∈𝒬hk, we have by (3.16),

(3.22)|v-uh|H1⁢(Ω)≲maxz∈𝒬hk⁡ah⁢(v-uh,z)|z|H1⁢(Ω),

and, in view of (3.8),

ah⁢(v-uh,z)=ah⁢(v,z)-(f,Ξh⁢z).

Furthermore, we have by (3.1), (3.9) and (3.12),

ah⁢(v,z)=∑D∈𝒯h∫D∇⁡(Πk,D∇⁢v)⋅∇⁡(Πk,D∇⁢z)⁡d⁢x+∑D∈𝒯hSD⁢(v-Πk,D∇⁢v,z-Πk,D∇⁢z)

=∑D∈𝒯h∫D∇⁡(Πk,D∇⁢v)⋅∇⁡z⁢d⁢x+∑D∈𝒯hSD⁢(v-Πk,D∇⁢v,z-Πk,D∇⁢z)

=∑D∈𝒯h∫D∇⁡(Πk,D∇⁢v-u)⋅∇⁡z⁢d⁢x+(f,z)+∑D∈𝒯hSD⁢(v-Πk,D∇⁢v,z-Πk,D∇⁢z),

and hence, by (2.36), (2.56), (3.17), Lemma 3.4, and the triangle inequality,

ah⁢(v-uh,z)=∑D∈𝒯h∫D∇⁡(Πk,D∇⁢(v-u)+(Πk,D∇⁢u-u))⋅∇⁡z⁢d⁢x+(f,z-Ξh⁢z)+∑D∈𝒯hSD⁢(v-Πk,D∇⁢v,z-Πk,D∇⁢z)

(3.23)≲(|v-u|H1⁢(Ω)+|u-Πk,h∇⁢u|h,1+hℓ⁢|f|Hℓ-1⁢(Ω))⁢|z|H1⁢(Ω).

Estimate (3.21) follows from (3.2), (3.22), (3.23), and the triangle inequality. ∎

We also have an abstract error estimate for the computable Πk,h∇⁢uh.

Corollary 3.7.

Assuming f∈Hℓ-1⁢(Ω) for some ℓ between 1 and k, there exists a positive constant C depending only on k and ρ such that

(3.24)|u-Πk,h∇⁢uh|h,1≤C⁢(infv∈𝒬hk⁡|u-v|H1⁢(Ω)+infw∈𝒫hk⁡|u-w|h,1+hℓ⁢|f|Hℓ-1⁢(Ω)).

Proof.

Estimate (3.24) follows immediately from (2.36), (3.2), (3.21), and the relation

|u-Πk,h∇⁢uh|h,1≤|u-Πk,h∇⁢u|h,1+|Πk,h∇⁢(u-uh)|h,1≤|u-Πk,h∇⁢u|h,1+|u-uh|H1⁢(Ω).∎

Remark 3.8.

It follows from (3.21) and (3.24) that

infv∈𝒬hk⁡|u-v|H1⁢(Ω)+infw∈𝒫hk⁡|u-w|h,1≤|u-uh|H1⁢(Ω)+|u-Πk,h∇⁢uh|H1⁢(Ω)

≤C⁢(infv∈𝒬hk⁡|u-v|H1⁢(Ω)+infw∈𝒫hk⁡|u-w|h,1+hℓ⁢|f|Hℓ-1⁢(Ω)),

and hence the virtual element method produces best approximate solutions up to the perturbation error due to Ξh.

We can derive concrete energy error estimates under additional regularity assumptions.

Theorem 3.9.

Assuming u∈Hℓ+1⁢(Ω) for some ℓ between 1 and k, there exists a positive constant C depending only on k and ρ such that

|u-uh|H1⁢(Ω)+|u-Πk,h∇⁢uh|h,1≤C⁢hℓ⁢|u|Hℓ+1⁢(Ω).

Proof.

Take v=Ik,h⁢u and w=Πk,h∇⁢u in Theorem 3.6 and Corollary 3.7, and then apply (3.4). ∎

We have a similar error estimate for Πk,h0⁢uh.

Corollary 3.10.

Assuming u∈Hℓ+1⁢(Ω) for some ℓ between 1 and k, there exists a positive constant C depending only on k and ρ such that

(3.25)|u-Πk,h0⁢uh|h,1≤C⁢hℓ⁢|u|Hℓ+1⁢(Ω).

Proof.

Estimate (3.25) follows from (2.44), (3.6), Theorem 3.9, and the relation

|u-Πk,h0⁢uh|h,1≤|u-Πk,h0⁢u|h,1+|Πk,h0⁢(u-uh)|h,1

≲|u-Πk,h0⁢u|h,1+|u-uh|H1⁢(Ω).∎

3.4 L2 Error Estimates

We will derive L2 error estimates under the assumption that Ω is convex.

We begin with a consistency estimate.

Lemma 3.11.

Assuming that u∈Hℓ+1⁢(Ω) for some ℓ between 1 and k, we have

(3.26)a⁢(u-uh,Ik,h⁢ζ)≲hℓ+1⁢|u|Hℓ+1⁢(Ω)⁢|ζ|H2⁢(Ω) for all ⁢ζ∈H2⁢(Ω)∩H01⁢(Ω).

Proof.

We have, by (3.1), (3.8), (3.9), and (3.12),

a⁢(u-uh,Ik,h⁢ζ)=∑D∈𝒯haD⁢(u-uh,Ik,D⁢ζ)

=∑D∈𝒯haD⁢(u-uh,Ik,D⁢ζ-Πk,D∇⁢Ik,D⁢ζ)+∑D∈𝒯haD⁢(u-uh,Πk,D∇⁢Ik,D⁢ζ)

=∑D∈𝒯haD⁢(u-uh,Ik,D⁢ζ-Πk,D∇⁢Ik,D⁢ζ)+∑D∈𝒯haD⁢(Πk,D∇⁢u,Ik,D⁢ζ)-ah⁢(uh,Ik,h⁢ζ)

+∑D∈𝒯hSD⁢(uh-Πk,D∇⁢uh,Ik,D⁢ζ-Πk,D∇⁢Ik,D⁢ζ)

and

∑D∈𝒯haD⁢(Πk,D∇⁢u,Ik,D⁢ζ)-ah⁢(uh,Ik,h⁢ζ)=∑D∈𝒯haD⁢(Πk,D∇⁢u-u,Ik,D⁢ζ)+(f,Ik,h⁢ζ-Ξh⁢Ik,h⁢ζ)

=∑D∈𝒯haD⁢(Πk,D∇⁢u-u,Ik,D⁢ζ-Πk,D∇⁢Ik,D⁢ζ)+(f,Ik,h⁢ζ-Ξh⁢Ik,h⁢ζ),

which together imply (3.26) because of (3.4), (3.17), Lemma 3.5, and Theorem 3.9. ∎

Theorem 3.12.

Assuming u∈Hℓ+1⁢(Ω) for some ℓ between 1 and k, there exists a positive constant C depending only on Ω, k and ρ such that

(3.27)∥u-uh∥L2⁢(Ω)≤C⁢hℓ+1⁢|u|Hℓ+1⁢(Ω).

Proof.

Let ζ∈H01⁢(Ω) be defined by

a⁢(v,ζ)=(v,u-uh) for all ⁢v∈H01⁢(Ω).

Then we have, by elliptic regularity [23],

(3.28)∥ζ∥H2⁢(Ω)≤CΩ⁢∥u-uh∥L2⁢(Ω)

and

(3.29)∥u-uh∥L2⁢(Ω)2=a⁢(u-uh,ζ)=a⁢(u-uh,ζ-Ik,h⁢ζ)+a⁢(u-uh,Ik,h⁢ζ).

The first term on the right-hand side of (3.29) satisfies

(3.30)a⁢(u-uh,ζ-Ik,h⁢ζ)≤|u-uh|H1⁢(Ω)⁢|ζ-Ik,h⁢ζ|H1⁢(Ω)≲h⁢|u-uh|H1⁢(Ω)⁢|ζ|H2⁢(Ω)

by (3.4), and thus (3.27) follows from Theorem 3.9, Lemma 3.11 and (3.28)–(3.30). ∎

We have a similar L2 error estimate for Πk,h0⁢uh.

Corollary 3.13.

Assuming u∈Hℓ+1⁢(Ω) for some ℓ between 1 and k, there exists a positive constant C depending only on Ω, k and ρ such that

(3.31)∥u-Πk,h0⁢uh∥L2⁢(Ω)≤C⁢hℓ+1⁢|u|Hℓ+1⁢(Ω).

Proof.

From (3.7), Theorem 3.12 and the relation

∥u-Πk,h0⁢uh∥L2⁢(Ω)≤∥u-Πk,h0⁢u∥L2⁢(Ω)+∥Πk,h0⁢(u-uh)∥L2⁢(Ω)

≤∥u-Πk,h0⁢u∥L2⁢(Ω)+∥u-uh∥L2⁢(Ω),

estimate (3.31) follows. ∎

There is also a similar L2 error estimate for Πk,h∇⁢uh.

Theorem 3.14.

Assuming u∈Hℓ+1⁢(Ω) for some ℓ between 1 and k, there exists a positive constant C depending only on Ω, k and ρ such that

(3.32)∥u-Πk,h∇⁢uh∥L2⁢(Ω)≤C⁢hℓ+1⁢|u|Hℓ+1⁢(Ω).

Proof.

We have

(3.33)∥u-Πk,h∇⁢uh∥L2⁢(Ω)≤∥u-Πk,h∇⁢Ik,h⁢u∥L2⁢(Ω)+∥Πk,h∇⁢(Ik,h⁢u-uh)∥L2⁢(Ω)

and, in view of (2.4), (2.24), and (2.37),

∥Πk,h∇⁢(Ik,h⁢u-uh)∥L2⁢(Ω)2≲∑D∈𝒯h|||Ik,D⁢u-uh|||k,D2

≲∑D∈𝒯h(hD⁢∥Ik,D⁢u-uh∥L2⁢(∂⁡D)2+∥Πk-2,D0⁢(Ik,D⁢u-uh)∥L2⁢(D)2)

≲∑D∈𝒯h(∥Ik,D⁢u-uh∥L2⁢(D)2+hD2⁢|Ik,D⁢u-uh|H1⁢(D)2)

(3.34)≲∥u-uh∥L2⁢(Ω)2+∥u-Ik,h⁢u∥L2⁢(Ω)2+h2⁢|u-Ik,h⁢u|H1⁢(Ω)2+h2⁢|u-uh|H1⁢(Ω)2.

Estimate (3.32) follows from (3.3), (3.4), Theorem 3.9, Theorem 3.12, and (3.33)–(3.34). ∎

4 Relaxed Shape Regularity Assumptions

The results in Section 2 and Section 3 are also valid under the following relaxed shape regularity assumptions on the polygonal mesh (cf. Figures 2 and 3): Each polygonal subdomain D in 𝒯h is the union of N disjoint polygons D1,…,DN such that (i) the polygons D1,…,DN satisfy the shape regularity assumptions in Section 2.1, and (ii) Dj and Dj+1 share a common edge. We will refer to D1,…,DN as the components of D.

First we observe that the simplicial triangulation 𝒯D of D determined by the star-centers of D1,…,DN (cf. Figure 2) satisfies a minimum angle condition because of (2.3). Moreover, condition (2.1) implies that

ρ≤diam⁡Dj+1diam⁡Dj≤ρ-1

because Dj and Dj+1 share a common edge. Hence we have

diam⁡Djdiam⁡Dk≤ρ-|j-k| for ⁢1≤j,k≤N.

Figure 2

A subdomain with four components, that satisfies the relaxed shape regularity assumptions.

Figure 3

Two components that share a common edge.

It follows that the estimates in Sections 2.1.1–2.1.3 can be extended to D, where

(4.1)hD=max⁡{diam⁡D1,…,diam⁡DN}

is the maximum of the diameters of the components of D, and the hidden constants in (2.4)–(2.8) depend on the parameter ρ, the polynomial degree k and the number of components N.

Next we consider the Bramble–Hilbert estimates. The following observation will be useful: There exists a disc Bj with radius rj centered at the midpoint of the common edge of Dj and Dj+1 such that (i) Bj is a subset of the polygon formed by the union of Dj and Dj+1 (cf. Figure 3), and (ii) Dj and Dj+1 are subsets of the disc B~j, where B~j shares the same center with Bj and has radius M⁢rj. Here M is a positive integer that only depends on ρ.

It follows from the equivalence of norms on finite-dimensional vector spaces and scaling that for any m≤k we have

(4.2)|z|Hm⁢(Bj)≤C♭⁢|z|Hm⁢(Bj∩Dj)for all ⁢z∈ℙk,

(4.3)|z|Hm⁢(B~j)≤C♯⁢|z|Hm⁢(Bj∩Dj+1)for all ⁢z∈ℙk,

where the positive constant C♭ depends only on k and the positive constant C♯ depends only on k and M (and hence only on k and ρ).

The proof of the following result is based on the ideas in [21, Section 7].

Lemma 4.1.

Let D be a polygonal domain (with N components) that satisfies the relaxed shape regularity assumptions. Then we have

(4.4)infq∈ℙℓ⁡|ζ-q|Hm⁢(D)≤C⁢hDℓ+1-m⁢|ζ|Hℓ+1⁢(D) for all ⁢ζ∈Hℓ+1⁢(D),ℓ=0,1,2,…,k,

and m≤ℓ, where hD is given by (4.1) and the positive constant C depends only on ρ, k and N.

Proof.

Let G′ be the polygon formed by the union of D1,…,Dj, let G be the polygon formed by the union of G′ and Dj+1, and let m≤ℓ≤k and ζ∈Hℓ+1⁢(G).

Suppose that

(4.5)infq∈ℙℓ⁡|ζ-q|Hm⁢(G′)≤CG′⁢hG′ℓ+1-m⁢|ζ|Hℓ+1⁢(G′),

where hG′=max⁡{diam⁡D1,…,diam⁡Dj}. Then we can find q∈ℙℓ such that

(4.6)|ζ-q|Hm⁢(G′)≤2⁢CG′⁢hG′ℓ+1-m⁢|ζ|Hℓ+1⁢(G′).

By the Bramble–Hilbert estimates for the disc Bj, we can find a polynomial p1∈ℙℓ such that

(4.7)|ζ-p1|Hm⁢(Bj)≤C*⁢rjℓ+1-m⁢|ζ|Hℓ+1⁢(Bj)≤C*⁢(diam⁡Dj)ℓ+1-m⁢|ζ|Hℓ+1⁢(Bj),

where the positive constant C* depends only on k.

It follows from (4.2), (4.6), (4.7), and the triangle inequality that

|ζ-q|Hm⁢(G′∪Bj)≤|ζ-q|Hm⁢(G′)+|ζ-p1|Hm⁢(Bj)+|p1-q|Hm⁢(Bj)

≤|ζ-q|Hm⁢(G′)+|ζ-p1|Hm⁢(Bj)+C♭⁢|p1-q|Hm⁢(Bj∩Dj)

≤|ζ-q|Hm⁢(G′)+|ζ-p1|Hm⁢(Bj)+C♭⁢(|p1-ζ|Hm⁢(Bj)+|q-ζ|Hm⁢(Dj))

≤(1+C♭)⁢|ζ-q|Hm⁢(G′)+(1+C♭)⁢|ζ-p1|Hm⁢(Bj)

(4.8)≤(1+C♭)⁢(2⁢CG′+C*)⁢hG′ℓ+1-m⁢|ζ|Hℓ+1⁢(G′∪Bj).

By the Bramble–Hilbert estimate for Dj+1 (cf. (2.9)), we can find p2∈ℙℓ such that

(4.9)|ζ-p2|Hm⁢(Dj+1)≤C†⁢(diam⁡Dj+1)ℓ+1-m⁢|ζ|Hℓ+1⁢(Dj+1),

where the positive constant C† depends only on ρ and k.

Combining (4.3), (4.8), (4.9), and the triangle inequality, we find

|ζ-q|Hm⁢(G)≤|ζ-q|Hm⁢(G′∪Bj)+|ζ-p2|Hm⁢(Dj+1)+|p2-q|Hm⁢(Dj+1)

≤|ζ-q|Hm⁢(G′∪Bj)+|ζ-p2|Hm⁢(Dj+1)+|p2-q|Hm⁢(B~j)

≤|ζ-q|Hm⁢(G′∪Bj)+|ζ-p2|Hm⁢(Dj+1)+C♯⁢|p2-q|Hm⁢(Bj∩Dj+1)

≤(1+C♯)⁢|ζ-q|Hm⁢(G′∪Bj)+(1+C♯)⁢|ζ-p2|Hm⁢(Dj+1)

≤(1+C♯)⁢[(1+C♭)⁢(2⁢CG′+C*)+C†]⁢hGℓ+m-1⁢|ζ|Hℓ+1⁢(G),

where hG=max⁡{hG′,diam⁡Dj+1}=max⁡{diam⁡D1,…,diam⁡Dj+1}. Hence we have

(4.10)infq∈ℙℓ⁡|ζ-q|Hm⁢(G)≤CG⁢hGℓ+m-1⁢|ζ|Hℓ+1⁢(G),

where

(4.11)CG=(1+C♯)⁢[(1+C♭)⁢(2⁢CG′+C*)+C†].

Estimate (4.4) now follows from the recursive estimates defined by (4.5), (4.10) and (4.11), together with the (initial) estimate

infq∈ℙℓ⁡|ζ-q|Hm⁢(D1)≤C†⁢(diam⁡D1)ℓ+1-m⁢|ζ|Hℓ+1⁢(D1),

which proves the lemma. ∎

It follows from Lemma 4.1 that the estimates in Section 2.1.5 also hold under the general shape regularity assumptions. The rest of the results in Section 2 and Section 3 can then be established by the same arguments, provided that the operator Πk,D∇ is defined by (2.14) if we use the definition of SD⁢(⋅,⋅) in (3.14). The only difference is that now the constants also depend on the maximum number of components of the subdomains of 𝒯h.

5 Virtual Element Methods for the Poisson Problem in Three Dimensions

In this section, we discuss the extensions of the results in Section 2 and Section 3 to three dimensions. We begin with the shape regularity assumptions on a polyhedron D.

5.1 Shape Regularity Assumptions

We assume that (i) there exists ρ∈(0,1) such that D is star-shaped with respect to a ball 𝔅 whose radius is ρ⁢hD, (ii) the diameter of the faces of D are ≥ρ⁢hD, (iii) the faces of D are similar to a fixed number of reference polygons that satisfy the regularity assumptions in Section 2.1 with the same ρ.

Remark 5.1.

Note that the analog of assumption (iii) is automatically satisfied in two dimensions since all line segments are similar to the unit interval.

We can form a simplicial triangulation 𝒯∂⁡D of ∂⁡D by using the vertices of D and the star-centers of the faces of D, and then form a triangulation 𝒯D of D by tetrahedrons generated by the triangles in ∂⁡D and the star-center of D. Under the regularity assumptions (i)–(iii), the shape regularity of 𝒯D is determined by ρ, and the estimates in Sections 2.1.1–2.1.5 remain valid.

Remark 5.2.

We can further relax the shape regularity assumptions as in Section 4.

5.2 The Local Virtual Element Space 𝒬k⁢(D)

Let D be a polyhedron that satisfies the regularity assumptions in Section 5.1. We can define the inner product ((⋅,⋅)) by (2.12) where the infinitesimal arc-length ds is replaced by the infinitesimal surface area dS. Then the projection operator Πk,D∇:H1⁢(D)→ℙk⁢(D) is defined by (2.13) (equivalently by (2.14)–(2.15)). For k≥2, we can also use the inner product (((⋅,⋅))) given by (2.16) to define Πk,D∇.

The set of the edges of D is again denoted by ℰD and the set of the faces of D is denoted by ℱD. The space 𝒬k⁢(∂⁡D) of continuous piecewise (two-dimensional) virtual element functions of order ≤k on ∂⁡D is defined by

𝒬k⁢(∂⁡D)={v∈C⁢(∂⁡D):v|F∈𝒬k⁢(F)⁢ for all ⁢F∈ℱD}.

In view of Remark 2.21, an important consequence of the shape regularity assumption (iii) in Section 5.1 is that

(5.1)∥v∥L2⁢(F)≈|||v|||k,F for all ⁢v∈𝒬k⁢(F)⁢ and ⁢F∈ℱD.

Here and below all hidden constants will depend on k, ρ and the references polygons for ℱD. In two dimensions a function in ℙk⁢(∂⁡D) can be extended to D as a finite element function. We will use the following extension result for the three-dimensional case.

Lemma 5.3.

Given g∈Qk⁢(∂⁡D), let g~ be defined by (i)g~=g on ∂⁡D, (ii)g~=0 at the star-center x* of D, and (iii)g~ is linear on any line segment that connects x* to a point on ∂⁡D. Then we have g~∈H1⁢(D) and

(5.2)hD-3⁢∥g~∥L2⁢(D)2≈hD-2⁢∥g∥L2⁢(∂⁡D)2≈hD-1⁢|g~|H1⁢(D)2.

Proof.

Observe that g~∈C⁢(D¯) since 𝒬k⁢(∂⁡D)⊂C⁢(∂⁡D) (cf. Remark 2.7). Therefore, it only remains to show that g~ belongs to H1⁢(G) on the polyhedron G formed by x* and a face F of D (cf. Figure 4), and to establish the analog of (5.2) on G.

We may also assume that hD=1, x* is the origin and the polygon F⊂ℝ2 is placed at a distance τ (≈1) from the origin and parallel to the plane x3=0. Then the polyhedron G is the set

{x∈ℝ3:0<x3<τ,(τ⁢x1/x3,τ⁢x2/x3)∈F},

and

g~⁢(x)=(x3/τ)⁢g⁢(τ⁢x1/x3,τ⁢x2/x3) for ⁢x∈G.

Let ϕn be C∞ functions on F that converge to g in H1⁢(F), and let ϕ~n be the corresponding extension of ϕn to G. Then the function ϕ~n on G is given by the formula

ϕ~n⁢(x)=(x3/τ)⁢ϕn⁢(τ⁢x1/x3,τ⁢x2/x3) for ⁢x∈G.

We find by the change of variables y=(τ⁢x1/x3,τ⁢x2/x3) that

∫Gϕ~n2⁢𝑑x=∫0τ(x3/τ)4⁢[∫Fϕn2⁢𝑑y]⁢𝑑x3,

∫G(∂⁡ϕ~n∂⁡xj)2⁢𝑑x=∫0τ(x3/τ)2⁢[∫F(∂⁡ϕn∂⁡yj)2⁢𝑑y]⁢𝑑x3 for ⁢j=1,2,

∫G(∂⁡ϕ~n∂⁡x3)2⁢𝑑x=∫0ττ-2⁢(x3/τ)2⁢[∫F(ϕn-y⋅∇⁡ϕn)2⁢𝑑y]⁢𝑑x3.

Therefore, we have ϕ~n∈H1⁢(G),

(5.3)∥ϕ~n∥L2⁢(G)≈∥ϕn∥L2⁢(F) and |ϕ~n|H1⁢(G)≈∥ϕn∥H1⁢(F),

ϕ~n converges to g~ in L2⁢(G), and ∇⁡ϕ~n converges in [L2⁢(G)]3.

Hence g~ belongs to H1⁢(G) and ∇⁡ϕ~n converges to ∇⁡g~ in [L2⁢(G)]3. The estimates in (5.3) together with Lemma 2.19 and (5.1) then imply

∥g~∥L2⁢(G)≈∥g∥L2⁢(F)≈|g~|H1⁢(G),

as desired. ∎

Figure 4

The polyhedron G.

Using Lemma 5.3, we can prove the following analog of Lemma 2.2 by identical arguments.

Lemma 5.4.

Given any g∈Qk⁢(∂⁡D) and f∈Pk⁢(D), there exists a unique function v∈H1⁢(D) such that (i)v=g on ∂⁡D and (ii)

∫D∇⁡v⋅∇⁡w⁢d⁢x=∫Df⁢w⁢𝑑x for all ⁢w∈H01⁢(D).

We can now define the local virtual element space 𝒬k⁢(D)⊂H1⁢(D) as follows: v∈H1⁢(D) belongs to 𝒬k⁢(D) if and only if the trace of v on ∂⁡D belongs to 𝒬k⁢(∂⁡D) and the conditions (2.20) and (2.21) are satisfied. Lemma 2.5 remains valid provided we replace ℙk⁢(∂⁡D) by 𝒬k⁢(∂⁡D), and Lemma 2.8 stays the same.

The degrees of freedom for 𝒬k⁢(D) consist of (i) the values at the vertices of D, (ii) the moments of order ≤k-2 on the edges of D, (iii) the moments of order ≤k-2 on the faces of D, and (iv) the moments of order ≤k-2 on D.

Note that for ζ∈𝒬k⁢(D) the polynomial Πk,D∇⁢ζ∈ℙk⁢(D) can be computed by using the degrees of freedom of ζ and (2.15) because on each face all moments up to order k are available (cf. Remark 2.6).

5.3 The (Semi-)Norm |||⋅|||k,D and Estimates for Πk,D∇ and Πk,D0

The analog of (2.23) is given by

(5.4)|||v|||k,D2=hD2⁢∑e∈ℰD∥Πk,e0⁢v∥L2⁢(e)2+hD⁢∑F∈ℱD∥Πk-2,F0⁢v∥L2⁢(F)2+∥Πk-2,D0⁢v∥L2⁢(D)2,

where Πk,F0 is the orthogonal projection from L2⁢(F) onto ℙk⁢(F).

It follows from (2.23), (5.1) and (5.4) that

|||v|||k,D2≈hD⁢∥v∥L2⁢(∂⁡D)2+∥Πk-2,D0⁢v∥L2⁢(D)2 for all ⁢v∈𝒬k⁢(D).

Lemma 2.10, Lemma 2.11 and Lemmas 2.14, 2.15, 2.17, and 2.18 can then be extended to three dimensions by identical arguments.

5.4 Inverse Inequality, Friedrichs’ Inequality and Estimates for Ik,D

We can establish the inverse inequality (2.46) in three dimensions provided we take the function w∈H1⁢(D) in the proof of Lemma 2.19 to be the extension of v|∂⁡D from Lemma 5.3, which gives us the estimates in (2.48). The rest of the arguments remains the same. The estimates in Corollary 2.20 and Lemma 2.22 then follow immediately.

The interpolation operator Ik,D:H2⁢(D)→𝒬k⁢(D) is again defined by the condition that ζ and Ik,D⁢ζ have the same degrees of freedom. The estimates in Section 2.8 can be established by identical arguments.

5.5 Virtual Element Methods for the Poisson Problem

Let Ω⊂ℝ3 be a bounded polyhedral domain, let 𝒯h be a conforming partition of Ω by polyhedral subdomains, and let 𝒬hk={v∈H01⁢(Ω):v|D∈𝒬k⁢(D)⁢ for all ⁢D∈𝒯h}. We assume that the polyhedral subdomains in 𝒯h satisfy the shape regularity assumptions in Section 5.1, and hence the estimates in Section 3.1 and Section 3.2.3 are valid.

The discrete problem for (3.1) is defined by (3.8), (3.9) and (3.18), where aD⁢(⋅,⋅) is defined by (3.10) and SD⁢(⋅,⋅) is given by the following analogs of (3.11) and (3.13):

SD⁢(w,v)=∑e∈ℰD(Πk,e⁢w,Πk,e⁢v)L2⁢(e)+hD-1⁢∑F∈ℱD(Πk-2,F0⁢w,Πk-2,F0⁢v)L2⁢(F)+hD-2⁢(Πk-2,D0⁢w,Πk-2,D0⁢v)L2⁢(D)

and

SD⁢(w,v)=∑e∈ℰD(Πk,e⁢w,Πk,e⁢v)L2⁢(e)+hD-1⁢∑F∈ℱD(Πk-2,F0⁢w,Πk-2,F0⁢v)L2⁢(F).

Again the restriction of SD⁢(⋅,⋅) to 𝒬k⁢(D)×𝒬k⁢(D) can be computed in terms of the degrees of freedom of 𝒬k⁢(D). The proof of Lemma 3.2 stays the same and the discrete problem is well-posed.

The results in Section 3.3 and Section 3.4 can be extended to three dimensions through identical arguments. The only difference is that now all constants depend on k, ρ and the reference polygons for ℱD.

6 Concluding Remarks

We have developed new techniques for obtaining estimates for virtual elements in terms of the shape regularity of polyhedral meshes, and also error estimates for the computable piecewise polynomials Πk,D∇⁢uh and Πk,D0⁢uh generated by the virtual element methods for the Poisson problem. We note that the results in three dimensions require the condition that the faces of the polyhedrons in the mesh are similar to a fixed set of reference polygons.

For simplicity, we have only presented results for integer order Sobolev spaces. But they can also be extended to fractional order Sobolev spaces. In particular, the interpolation operator Ik,h is well-defined on the Sobolev space Hs⁢(Ω) as long as s>d/2, and it is known that the solution of the Poisson problem on a polyhedral domain belongs to such Sobolev spaces in two and three dimensions [23, 18].

With minor modifications the results in this paper can be extended to the original virtual element methods in [4] for two-dimensional problems where the definition of 𝒬k⁢(D) is given by

𝒬k⁢(D)={v∈H1⁢(D):v|∂⁡D∈ℙk⁢(∂⁡D)⁢ and ⁢Δ⁢v∈ℙk-2⁢(D)}.

Such virtual element methods have been analyzed in [10] by a different approach where polygonal meshes with arbitrarily short edges are also considered.

The local estimates in Section 2 (and their three-dimensional analogs) are relevant for general second elliptic boundary value problems [8] and nonconforming virtual elements [3]. We also expect that the new techniques can be extended to virtual element methods for higher order problems [15, 16].

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-16-20273

Funding statement: The first and third authors were supported in part by the National Science Foundation under grant no. DMS-16-20273. They would also like to acknowledge the support provided by the Hausdorff Research Institute of Mathematics at Universität Bonn during their visit in Spring 2017.

References

[1] B. Ahmad, A. Alsaedi, F. Brezzi, L. D. Marini and A. Russo, Equivalent projectors for virtual element methods, Comput. Math. Appl. 66 (2013), 376–391. 10.1016/j.camwa.2013.05.015Search in Google Scholar

[2] P. F. Antonietti, L. Beirão da Veiga, D. Mora and M. Verani, A stream virtual element formulation of the Stokes problem on polygonal meshes, SIAM J. Numer. Anal. 52 (2014), no. 1, 386–404. 10.1137/13091141XSearch in Google Scholar

[3] B. Ayuso de Dios, K. Lipnikov and G. Manzini, The nonconforming virtual element method, ESAIM Math. Model. Numer. Anal. 50 (2016), 879–904. 10.1051/m2an/2015090Search in Google Scholar

[4] L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013), 199–214. 10.1142/S0218202512500492Search in Google Scholar

[5] L. Beirão da Veiga, F. Brezzi and L. D. Marini, Virtual elements for linear elasticity problems, SIAM J. Numer. Anal. 51 (2013), 794–812. 10.1137/120874746Search in Google Scholar

[6] L. Beirão da Veiga, F. Brezzi, L. D. Marini and A. Russo, The Hitchhiker’s guide to the virtual element method, Math. Models Methods Appl. Sci. 24 (2014), 1541–1573. 10.1142/S021820251440003XSearch in Google Scholar

[7] L. Beirão da Veiga, F. Brezzi, L. D. Marini and A. Russo, Mixed virtual element methods for general second order elliptic problems on polygonal meshes, ESAIM Math. Model. Numer. Anal. 50 (2016), 727–747. 10.1051/m2an/2015067Search in Google Scholar

[8] L. Beirão da Veiga, F. Brezzi, L. D. Marini and A. Russo, Virtual element method for general second-order elliptic problems on polygonal meshes, Math. Models Methods Appl. Sci. 26 (2016), 729–750. 10.1142/S0218202516500160Search in Google Scholar

[9] L. Beirão da Veiga, C. Lovadina and D. Mora, A virtual element method for elastic and inelastic problems on polytope meshes, Comput. Methods Appl. Mech. Engrg. 295 (2015), 327–346. 10.1016/j.cma.2015.07.013Search in Google Scholar

[10] L. Beirão da Veiga, C. Lovadina and A. Russo, Stability analysis for the virtual element method, preprint (2016), https://arxiv.org/abs/1607.05988v1. 10.1142/S021820251750052XSearch in Google Scholar

[11] L. Beirão da Veiga and G. Manzini, A virtual element method with arbitrary regularity, IMA J. Numer. Anal. 34 (2014), 759–781. 10.1093/imanum/drt018Search in Google Scholar

[12] J. H. Bramble and S. R. Hilbert, Estimation of linear functionals on Sobolev spaces with applications to Fourier transforms and spline interpolation, SIAM J. Numer. Anal. 7 (1970), 113–124. 10.1137/0707006Search in Google Scholar

[13] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 3rd ed., Springer, New York, 2008. 10.1007/978-0-387-75934-0Search in Google Scholar

[14] F. Brezzi, R. S. Falk and L. D. Marini, Basic principles of mixed virtual element methods, ESAIM Math. Model. Numer. Anal. 48 (2014), 1227–1240. 10.1051/m2an/2013138Search in Google Scholar

[15] F. Brezzi and L. D. Marini, Virtual element methods for plate bending problems, Comput. Methods Appl. Mech. Engrg. 253 (2013), 455–462. 10.1016/j.cma.2012.09.012Search in Google Scholar

[16] C. Chinosi and L. D. Marini, Virtual element method for fourth order problems: L2-estimates, Comput. Math. Appl. 72 (2016), 1959–1967. 10.1016/j.camwa.2016.02.001Search in Google Scholar

[17] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. 10.1115/1.3424474Search in Google Scholar

[18] M. Dauge, Elliptic Boundary Value Problems on Corner Domains, Lecture Notes in Math. 1341, Springer, Berlin, 1988. 10.1007/BFb0086682Search in Google Scholar

[19] D. A. Di Pietro and J. Droniou, A hybrid high-order method for Leray–Lions elliptic equations on general meshes, Math. Comp. (2016), 10.1090/mcom/3180. 10.1090/mcom/3180Search in Google Scholar

[20] D. A. Di Pietro and J. Droniou, Ws,p-approximation properties of elliptic projectors on polynomial spaces, with application to the error analysis of a hybrid high-order discretisation of Leray–Lions problems, M3AS (2017), 10.1142/S0218202517500191. 10.1142/S0218202517500191Search in Google Scholar

[21] T. Dupont and R. Scott, Polynomial approximation of functions in Sobolev spaces, Math. Comp. 34 (1980), 441–463. 10.1090/S0025-5718-1980-0559195-7Search in Google Scholar

[22] A. L. Gain, C. Talischi and G. H. Paulino, On the virtual element method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes, Comput. Methods Appl. Mech. Engrg. 282 (2014), 132–160. 10.1016/j.cma.2014.05.005Search in Google Scholar

[23] P. Grisvard, Elliptic Problems in Non Smooth Domains, Pitman, Boston, 1985. Search in Google Scholar

[24] P. Wriggers, W. T. Rust and B. D. Reddy, A virtual element method for contact, Comput. Mech. 58 (2016), 1039–1050. 10.1007/s00466-016-1331-xSearch in Google Scholar

Received: 2017-4-4

Revised: 2017-4-6

Accepted: 2017-4-7

Published Online: 2017-6-17

Published in Print: 2017-10-1