Some Estimates for Virtual Element Methods in Three Dimensions

2.1 The Virtual Element Spaces on Polygonal Faces

Let F be a polygon embedded in ℝ 3 , which is treated as a two-dimensional submanifold using local coordinates ( s , t ) on the face as shown in Figure 1. The boundary of the polygon is oriented in a cyclic order as a 1 , … , a n . Let e = a 1 ⁢ a 2 ¯ be the first edge, and let n e and t e be the normal vector and tangential vector, respectively. Then we can define a local coordinate system with a 1 being the original point by using these two vectors. In what follows, we use the subscript “F” to indicate the locally defined symbols.

The virtual element method for two-dimensional problems was first introduced and analyzed in [2], where the virtual element spaces are defined by

To present the degrees of freedom (DoFs), we introduce a scaled monomial 𝕄 r ⁢ ( D ) on a d-dimensional domain D:

where h D is the diameter of D, 𝐱 D is the centroid of D, and r is a non-negative integer. For the multi-index 𝐬 ∈ ℕ d , we follow the usual notation

Conventionally, 𝕄 r ⁢ ( D ) = { 0 } for r ≤ - 1 . For example, with the help of the local coordinates we can represent the scaled monomials of degree r ≤ 1 as

where ( s F , t F ) and h F are the barycenter and the diameter of F, respectively.

The local dual space is

where the DoFs are the functional vectors as follows:

1. χ a are the values at the vertices of F:

   χ a i ⁢ ( v ) = v ⁢ ( a i ) , a i ⁢  is a vertex of  ⁢ F .

2. χ e k - 2 are the moments on edges of F up to degree k - 2 :

   χ e ⁢ ( v ) = | e | - 1 ⁢ ( m e , v ) e , m e ∈ 𝕄 k - 2 ⁢ ( e ) , e ⊂ ∂ ⁡ F .

3. χ F k - 2 are the moments on F up to degree k - 2 :

   χ F ⁢ ( v ) = | F | - 1 ⁢ ( m F , v ) K , m F ∈ 𝕄 k - 2 ⁢ ( F ) .

Figure 1

Local coordinate system of a face or polygon embedded in ℝ 3 .

To ensure the computability of the L 2 projector Π k , F 0 , we first introduce the lifting virtual element space defined by

in which the functions are uniquely determined by the previous DoFs together with the additional moments on F:

To remove the redundant DoFs of the third type, Ahmad, Alsaedi, Brezzi, Marini and Russo [1] modified the lifting virtual element space to a local enhancement space

where for a VEM function v ∈ V ~ k ⁢ ( F ) the elliptic projection Π k , F ∇ ⁢ v is the solution of the following problem:

An integration by parts shows that the redundant moments are not involved in the computation of the right-hand side of the first equation in (2.1).

2.2 The Virtual Element Spaces on Polyhedral Elements

The three-dimensional virtual element spaces can be introduced as the two-dimensional case recursively, where the boundary spaces are given by the virtual element spaces on polygonal faces instead of the polynomial spaces [5]. Let K be a polyhedral element. We consider the first virtual element space

with the DoFs given as follows:

1. χ p are the values at the vertices of K:

   χ p i ⁢ ( v ) = v ⁢ ( p i ) , p i ⁢  is a vertex of  ⁢ K .

2. χ e k - 2 are the moments on edges of K up to degree k - 2 :

   χ e ⁢ ( v ) = | e | - 1 ⁢ ( m e , v ) e , m e ∈ 𝕄 k - 2 ⁢ ( e ) , e ⊂ ∂ ⁡ F , F ⊂ ∂ ⁡ K .

3. χ F k - 2 are the moments on faces of K up to degree k - 2 :

   χ F ⁢ ( v ) = | F | - 1 ⁢ ( m F , v ) F , m F ∈ 𝕄 k - 2 ⁢ ( F ) , F ⊂ ∂ ⁡ K .

4. χ K k - 2 are the moments on K up to degree k - 2 :

   χ K ⁢ ( v ) = | K | - 1 ⁢ ( m K , v ) K , m K ∈ 𝕄 k - 2 ⁢ ( K ) .

To ensure the computability of the L 2 projector Π k 0 , one can also introduce the enhanced space (see [5])

where the lifting space is

and the elliptic projection Π k ∇ ⁢ w for w ∈ V ~ k ⁢ ( K ) is defined by

Similarly, the integration by parts shows that the elliptic projection is uniquely determined by the previous DoFs.

4.1 Inverse Inequalities

Based on Lemma 3.3, we are able to develop inverse inequalities for the three-dimensional conforming VEMs.

4.2 Norm Equivalence

With the help of the inverse inequalities, we can prove the following norm equivalence as in [12, 14].

4.3 Interpolation Error Estimates

The interpolation operator I K : H 1 ⁢ ( K ) ∩ C 0 ⁢ ( K ) → V k ⁢ ( K ) or W k ⁢ ( K ) is defined by the condition that v and I K ⁢ v have the same degrees of freedom:

As a by-product of the last two theorems, we can obtain the optimal error estimates for the interpolation operator I K .

5.1 The Reaction-Diffusion Problems

Let Ω ⊂ ℝ 3 be a polyhedral domain and let Γ denote a subset of its boundary consisting of faces. We consider the following model problem:

where α ≥ 0 is a constant, f ∈ L 2 ⁢ ( Ω ) and g N ∈ L 2 ⁢ ( Γ ′ ) are the applied load and Neumann boundary data, respectively, and g D ∈ H 1 / 2 ⁢ ( Γ ) is the data function for the Dirichlet boundary condition.

Introduce the following spaces:

The continuous variational problem of (5.1) is to find u ∈ H Γ 1 ⁢ ( Ω ) such that

where

It is clear that the bilinear form a ⁢ ( ⋅ , ⋅ ) is bounded and coercive over V, and for a given polyhedral mesh 𝒯 h it can be decomposed as

where

5.2 Consistent and Non-consistent VEMs in the Lowest-Order Case

We only conduct the numerical experiment for the lowest-order case k = 1 . For the ease of the presentation, we assume that Γ = ∂ ⁡ Ω , and the Dirichlet boundary condition is homogeneous, i.e., g D = 0 .

In the context of VEMs, a computable approximate bilinear form a h K ⁢ ( ⋅ , ⋅ ) is often constructed in order to satisfy the following conditions [2]:

1. k-consistency: For all p ∈ ℙ 1 ⁢ ( K ) and for all v h ∈ U h ⁢ ( K ) ,

   (5.3) a h K ⁢ ( p , v h ) = a K ⁢ ( p , v h ) .

2. Stability: There exist two positive constants α * and α * , independent of h K and K, such that

   α * ⁢ a K ⁢ ( v h , v h ) ≤ a h K ⁢ ( v h , v h ) ≤ α * ⁢ a K ⁢ ( v h , v h )   for all  ⁢ v h ∈ U h ⁢ ( K ) ,

   where U h ⁢ ( K ) is a local virtual element space under discussion, a K ⁢ ( ⋅ , ⋅ ) is the local bilinear form of the continuous problem and a h K ⁢ ( ⋅ , ⋅ ) is a computable approximate bilinear form.

To this end, we first define a local H 1 -elliptic projection operator Π 1 ∇ as in (2.3). For simplicity, we also write Π 1 ∇ for the related elementwise defined global operator. In view of the consistency condition (5.3), a natural construction of the bilinear form is

for all v , w ∈ W 1 ⁢ ( K ) (see [1, 5]). The stabilization term in the implementation is realized as

In what follows, denote by W 1 , h a finite-dimensional subspace of V, produced by combining all W 1 ⁢ ( K ) for K ∈ 𝒯 h in a standard way [5]. The consistent VEM proposed in [5] is to find u h ∈ W 1 , h such that

where

and 〈 ⋅ , ⋅ 〉 is the duality pairing between W 1 , h ′ and W 1 , h . A computable approximation of the right-hand side is given by

On the other hand, following the ideas in [13] for solving two-dimensional reaction-diffusion problems, we suggest a non-consistent VEM for solving (5.2), which is related to the approximate bilinear form

for all v , w ∈ V 1 ⁢ ( K ) . It is called the non-consistent bilinear form since the condition (5.3) is no longer valid. The corresponding VEM is: Find u h ∈ V 1 , h such that

where V 1 , h is the global version of V 1 ⁢ ( K ) , and

It is remarked that the notation 〈 ⋅ , ⋅ 〉 also applies to V 1 , h . The stabilization term and the approximation of the right-hand side are the same as in (5.4) and (5.6) for the consistent VEM.

We denote by u h W and u h V the VEM solutions to the methods (5.5) and (5.7), respectively. An interesting fact is that these two methods lead to the same linear system with the degrees of freedom as an unknown vector, due to the relation that Π 1 ∇ ⁢ w = Π 1 0 ⁢ w for a function w ∈ W 1 ⁢ ( K ) . In fact, for k = 1 the constraint in (2.2) gives

which implies Π 1 ∇ ⁢ w - Π 1 0 ⁢ w = 0 by noting Π 1 ∇ ⁢ w - Π 1 0 ⁢ w ∈ ℙ 1 ⁢ ( K ) . In addition, if w belongs to V 1 ⁢ ( K ) ∪ W 1 ⁢ ( K ) , we can also show that Π 1 ∇ ⁢ w is computable with respect to the DoFs of w. In fact, by integration by parts and using the properties that w | F ∈ W 1 ( F ) and Π 1 , F ∇ ⁢ w = Π 1 , F 0 ⁢ w for all w ∈ W 1 ⁢ ( F ) , the right-hand sides of the first equation and the constraint in (2.3) can be computed respectively as

and

where 𝐧 F is the unit outward normal to F ⊂ ∂ ⁡ K .

With the help of the estimates established beforehand under the mesh assumptions (A1) and (A2), we may apply the tricky but similar arguments in [13] to derive the optimal error estimates for both methods in H 1 and L 2 norms: