Adaptive virtual elements based on hybridized, reliable, and efficient flux reconstructions

1. Introduction

In [7], we analyzed certain error estimators for the virtual element method (VEM) [3], based on global and local flux reconstruction via mixed problems. To the aim, we mainly followed the finite element (FE) and discontinuous Galerkin (dG) approach of [9] and realized that the VEM contains too many variational crimes to allow for a localized, efficient error estimator with such an approach. In this paper, we undertake a slightly different path, i.e., we design error estimators by means of flux reconstruction via hybridized mixed methods. Notably, we adapt the finite element framework [6] to the virtual element one, which is based on polygonal meshes that are more suited for mesh adaptation as they naturally handle hanging nodes.

We shall design two error estimators based on global and local flux reconstructions, and prove their reliability and efficiency for the $h$ -, $p$ -, and $h p$ versions of the VEM. In both cases, the proof of the efficiency turns out to be more straightforward than the finite element one. Indeed, when using virtual elements, the existence of a polynomial, divergence right-inverse operator, see [6, Theorem $5$ and Conjecture $6$ ], is not required.

As a pivot result, we also prove the existence of a stabilization term for mixed virtual elements, with stability constants having explicit dependence on the degree of accuracy of the method. Furthermore, we discuss in details the flux reconstruction, well posedness of the hybridized mixed VEM; see also [8] for the hybridization of the mixed VEM in linear elasticity.

Notation.

We consider standard notation for Sobolev spaces. Given a domain $D$ , $H^{s} (D)$ denotes the standard Sobolev space of order $s$ , which we endow with the standard inner product $(\cdot, \cdot)_{s, D}$ , norm $∥ \cdot ∥_{s, D}$ , and seminorm $| \cdot |_{s, D}$

. Fractional and negative Sobolev spaces are defined via interpolation and duality, respectively. Recall the definition of standard differential equations in two dimensions: given

v : R^{2} \to R

and

τ = (τ_{1}, τ_{2}) : R^{2} \to R^{2}

\begin{matrix} div τ := \partial_{x} τ_{1} + \partial_{y} τ_{2}, r o t τ := \partial_{y} τ_{1} - \partial_{x} τ_{2}, c u r l v := (\partial_{y} v; - \partial_{x} v) . \end{matrix}

In light of the above definitions, we also introduce the spaces

H (div, D) := {τ \in [L^{2} (D)]^{2} ∣ d i v τ \in L^{2} (D)}, H (rot, D) := {τ \in [L^{2} (D)]^{2} ∣ rot τ \in L^{2} (D)} .

Given two positive quantities $a$ and $b$ , we shall write “ $a ≲ b$ ” instead of “there exists a positive constant $C$ , independent of the discretization parameters, such that $a \leq C b$ ”.

The model problem.

Given a polygonal domain $Ω \subset R^{2}$ and $f \in L^{2} (Ω)$ , we consider the Poisson problem: find $˜ u$ such that

{\begin{matrix} - Δ ˜ u = f & in Ω ˜ u = 0 & on \partial Ω, \end{matrix}

(1)

where $\partial Ω$ denotes the boundary of $Ω$ .

The weak formulation of (1) reads

{\begin{matrix} find ˜ u \in ˜ V such that ˜ a (˜ u, ˜ v) = (f, ˜ v)_{0, Ω} \forall ˜ v \in ˜ V, \end{matrix}

(2)

where we have set

\begin{matrix} ˜ V := H_{0}^{1} (Ω), ˜ a (˜ u, ˜ v) := (\nabla ˜ u, \nabla ˜ v)_{0, Ω} \forall ˜ u, ˜ v \in ˜ V . \end{matrix}

Problem (2) is well posed thanks to the Lax-Milgram lemma.

Mesh assumptions.

We employ standard notation for polygonal meshes. More precisely, we consider sequences ${T_{n}}_{n \in N}$ of polygonal meshes with straight edges. Hanging nodes are naturally incorporated in such meshes. Given a mesh $T_{n}$ , we denote its set of edges and vertices by $E_{n}$ and $V_{n}$ . We split $E_{n}$ into the set of boundary and internal edges as follows:

E_{n}^{D} := {e \in E_{n} ∣ e \subset \partial Ω}, E_{n}^{I} := E_{n} ∖ E_{n}^{D} .

With each element $K \in T_{n}$ , we associate its diameter $h_{K}$

, the outward pointing unit normal vector

n_{K}

, its centroid

x_{K}

, and its set of edges

E^{K}

and vertices

V^{K}

. Moreover, we associate with each edge

e \in E_{n}

its length

h_{e}

and any of the two unit normal vectors

n_{e}

We consider standard properties for the sequence of meshes [3]: there exist two positive constants $γ$ and $˜ γ$ , such that, for all $n \in N$ ,

every $K \in T_{n}$ is star-shaped with respect to a ball $B$ with radius $r_{B} \geq γ h_{K}$ ;
given $K \in T_{n}$ , for every of its edges $e \in E^{K}$ , $h_{K} \leq ˜ γ h_{e}$ .

Associated with each mesh $T_{n}$ and $p \in N$ , we define standard broken polynomial and Sobolev spaces:

\begin{matrix} P_{p} (T_{n}) := {q_{p} \in L^{2} (Ω) ∣ q_{p}_{| K} \in P_{p} (K) \forall K \in T_{n}}; H^{1} (T_{n}) := {v \in L^{2} (Ω) ∣ v_{| K} \in H^{1} (K) \forall K \in T_{n}}; H (div, T_{n}) := {τ \in [L^{2} (Ω)]^{2} ∣ τ_{| K} \in H (div, Ω)} . \end{matrix}

The first and second spaces can be generalized to the corresponding vector spaces.

Henceforth, for ease of presentation, we assume that $f \in P_{p - 1} (T_{n})$

. Moreover, throughout the paper, we assume a uniform distribution of the polynomial degree

p

for the sake of presentation. The general case involving nonuniform distributions of polynomial degrees is a trivial generalization.

Structure of the paper.

In Section 2, we review the hybridized global and local flux reconstruction in the finite element setting. After describing virtual elements for the primal and the hybridized mixed formulations in Section 3, we design and analyze the hybridized global and local flux reconstructions in the virtual element setting in Section 4. Notably, we introduce a reliable and efficient error estimator. Section 5 devotes to presenting numerical experiments validating the theoretical predictions. Finally, we state conclusions in Section 6

2. Hybridized global/local flux reconstruction in the finite element setting

Here, we briefly recall the approach of [6] providing $p$ robust, reliable and efficient error estimators on simplicial and Cartesian meshes for the finite element method. This will pave the way to design reliable and efficient error estimators for the virtual element method.

Let $T_{n}$ be a decomposition of $Ω$ into triangles. We consider the standard finite element approximation of (2):

{\begin{matrix} find {˜ u}_{p} \in {˜ V}_{p} such that% ˜ a ({˜ u}_{p}, {˜ v}_{p}) = (f, {˜ v}_{p})_{0, Ω} \forall {˜ v}_{p} \in {˜ V}_{p}, \end{matrix}

(3)

where we have set

{˜ V}_{p} := P_{p} (T_{n}) \cap H_{0}^{1} (Ω) .

We define a continuous residual $r \in [H_{0}^{1} (Ω)]^{*}$ as

⟨ r, ˜ v ⟩ := (f, ˜ v)_{0, Ω} - ˜ a ({˜ u}_{p}, ˜ v) \forall ˜ v \in H_{0}^{1} (Ω) .

(4)

An integration by parts yields

⟨ r, ˜ v ⟩ := \sum K \in T_{n} (r^{K}, ˜ v)_{0, K} + \sum e \in E_{n} (r^{e}, ˜ v)_{0, e} \forall ˜ v \in ˜ V,

where we have introduced the bulk and internal edge residuals

r^{K} := f + Δ {˜ u}_{p} \in P_{p - 1} (K) \forall K \in T_{n}, r^{e} := ⟦ \nabla {˜ u}_{p} ⟧ \in P_{p - 1} (e) \forall e \in E_{n}^{I},

(5)

for the standard definition of the jump operator on an internal edge $e$ :

⟦ τ ⟧_{e} := ⟦ τ ⟧ := n_{K_{1}} \cdot τ_{K_{1}} + n_{K_{2}} \cdot τ_{K_{2}}, e \subset_{1} \cap_{2}, \forall τ_{p} \in H (div, T_{n}) .

The initial step towards the design of the hybridized flux reconstruction consists in letting the bulk and edge residual belong to different polynomial spaces. Notably, we assume that $r^{K} \in {˜ V}_{p}$ , whereas we let $r^{e} \in Λ_{p}^{I}$ , where we have set $Λ_{p}^{I} := P_{p} (E_{n}^{I})$ .

Define a flux $σ_{p - 1}^{Δ}$ in a distributional sense as follows:

div σ_{p - 1}^{Δ} = r \in (H_{0}^{1} (Ω))^{*} .

(6)

We can interpret $σ_{p - 1}^{Δ}$ as follows:

σ_{p}^{Δ} \approx \nabla {˜ u}_{p} - \nabla ˜ u \in (H_{0}^{1} (Ω))^{*} .

Using (4) and (6) implies that $σ_{p - 1}^{Δ}$ satisfies

{\begin{matrix} div σ_{p - 1}^{Δ} = r^{K} \in P_{p - 1} (K) & in all K \in T_{n} ⟦ σ_{p - 1}^{Δ} ⟧ = r^{e} \in P_{p - 1} (e) & on all e \in E_{n}^{I} . \end{matrix}

(7)

The information about the homogeneous Dirichlet boundary conditions is implicitly contained in the two residuals through ${˜ u}_{p}$ . We rewrite (7) as a hybridized mixed method with flux and primal variable given by $σ_{p}^{Δ}$ and $u_{p - 1}$ , respectively. More precisely, we write

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} find (σ_{p - 1}^{Δ}, u_{p - 1}, λ_{p - 1}^{I}) \in Σ_{p - 1} \times V_{p - 1} \times Λ_{p - 1}^{I} such that a (σ_{p - 1}^{Δ}, τ_{p - 1}) + b (τ_{p - 1}, u_{p - 1}) + c (τ_{p - 1}, λ_{p - 1}^{I}) = 0 & \forall τ_{p - 1} \in Σ_{p - 1} b (σ_{p - 1}^{Δ}, v_{p - 1}) = (r^{K}, v_{p - 1})_{0, Ω} & \forall v_{p - 1} \in V_{p - 1} c (σ_{p - 1}^{Δ}, μ_{p - 1}^{I}) = (r^{e}, μ_{p - 1}^{I})_{0, E_{n}^{I}} & \forall μ_{p - 1}^{I} \in Λ_{p - 1}^{I}, \end{matrix}

(8)

where we picked the spaces

\begin{matrix} Σ_{p - 1} & := {τ_{p - 1} \in [L^{2} (Ω)]^{2} ∣ τ_{p - 1}_{| K} \in {R T}_{p - 1} (K) \forall K \in T_{n}}, V_{p - 1} & := P_{p - 1} (T_{n}), Λ_{p - 1} & := P_{p - 1} (E_{n}^{I}), \end{matrix}

(9)

being ${R T}_{p} (K)$ the local Raviart-Thomas space of order $p \in N_{0}$ on element $K$ , and the bilinear forms

\begin{matrix} a (σ_{p - 1}, τ_{p - 1}) & := (σ_{p - 1}, τ_{p - 1})_{0, Ω}, b (τ_{p - 1}, u_{p - 1}) & := - (div τ_{p - 1}, u_{p - 1})_{0, Ω}, c (τ_{p - 1}, μ_{p - 1}^{I}) & := (⟦ τ_{p - 1} ⟧, μ_{p - 1}^{I})_{0, E_{n}^{I}} . \end{matrix}

(10)

In the literature, the space $Σ_{p - 1}$ is known as the broken, hybridized Raviart-Thomas space of order $p - 1$

. No degrees of freedom coupling takes place at the interface between two elements. Rather, the communication between local Raviart-Thomas spaces takes place in a mortar fashion through the Lagrange multipliers in the space

Λ_{p - 1}

Next, we discuss the localized version of the above problem. With each vertex $ν \in V_{n}$ , we associate the partition of unity function ${˜ φ}_{ν} \in {˜ V}_{p}$ , i.e., the standard hat function with value $1$ at $ν$ and $0$ at all other vertices, and localize the bulk and edge residuals in (5) as follows: for all $ν \in V_{n}$ ,

r_{ν}^{K} := {˜ φ}_{ν} (f + Δ {˜ u}_{p}) \in P_{p} (K) \forall K \in T_{n}, r_{ν}^{e} := {˜ φ}_{ν} ⟦ \nabla {˜ u}_{p} ⟧ \in P_{p} (e) \forall e \in E_{n}^{I} .

(11)

Proceeding as above, on each vertex patch

ω_{ν} := ⋃ {K \in T_{n} ∣ ν is a vertex of K},

(12)

we introduce the following local problems: if $ν$ is an interior vertex,

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} find (σ_{p}^{ν}, u_{p}^{ν}, λ_{p}^{ν, I}) \in Σ_{p}^{0} (ω_{ν}) \times V_{p}^{*} (ω_{ν}) \times Λ_{p}^{I} (ω_{ν}) such that a^{ω_{ν}} (σ_{p}^{ν}, τ_{p}^{ν}) + b^{ω_{ν}} (τ_{p}^{ν}, u_{p}^{ν}) + c^{ω_{ν}} (τ_{p}^{ν}, λ_{p}^{ν, I}) = 0 & \forall τ_{p}^{ν} \in Σ_{p}^{0} (ω_{ν}) b^{ω_{ν}} (σ_{p}^{ν}, v_{p}^{ν}) = (r_{ν}^{K}, v_{p}^{ν})_{0, ω_{ν}} & \forall v_{p}^{ν} \in V_{p}^{*} (ω_{ν}) c^{ω_{ν}} (σ_{p}^{ν}, μ_{p}^{ν, I}) = (r_{ν}^{e}, μ_{p}^{ν, I})_{0, E_{n}^{I} (ω_{ν})} & \forall μ_{p}^{ν, I} \in Λ_{p}^{I} (E_{n}^{I} (ω_{ν})), \end{matrix}

whereas, if $ν$ is a boundary vertex,

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} find (σ_{p}^{ν}, u_{p}^{ν}, λ_{p}^{ν, I}) \in Σ_{p}^{0, \partial Ω} (ω_{ν}) \times V_{p} (ω_{ν}) \times Λ_{p}^{I} (ω_{ν}) such that a^{ω_{ν}} (σ_{p}^{ν}, τ_{p}^{ν}) + b^{ω_{ν}} (τ_{p}^{ν}, u_{p}^{ν}) + c^{ω_{ν}} (τ_{p}^{ν}, λ_{p}^{ν, I}) = 0 & \forall τ_{p}^{ν} \in Σ_{p}^{0, \partial Ω} (ω_{ν}) b^{ω_{ν}} (σ_{p}^{ν}, v_{p}) = (r_{ν}^{K}, v_{p}^{ν})_{0, ω_{ν}} & \forall v_{p}^{ν} \in V_{p} (ω_{ν}) c^{ω_{ν}} (σ_{p}^{ν}, μ_{p}^{ν, I}) = (r_{ν}^{e}, μ_{p}^{ν, I})_{0, E_{n}^{I} (ω_{ν})} & \forall μ_{p}^{ν, I} \in Λ_{p}^{I} (E_{n}^{I} (ω_{ν})), \end{matrix}

where the involved spaces and bilinear forms are the obvious counterparts of those in (9) and (10), and

\begin{matrix} Σ_{p}^{0} (ω_{ν}) & := {τ_{p} \in Σ_{p} (ω_{ν}) ∣ n_{e} \cdot τ_{p}_{| e} = 0 \forall e \subset \partial ω_{ν}}, Σ_{p}^{0, \partial Ω} (ω_{ν}) & := {τ_{p} \in Σ_{p} (ω_{ν}) ∣ n_{e} \cdot τ_{p}_{| e} = 0 \forall e \subset \partial ω_{ν}, ν ⊈ e}, V_{p}^{*} (ω_{ν}) & := {v_{p} \in V_{p} (ω_{ν}) ∣ ∣ \int_{ω_{ν}} v_{p} = 0} . \end{matrix}

In [6], it was proven that the error estimator

σ_{p} := \sum ν \in V_{n} σ_{p}^{ν}

is reliable and efficient with constants independent of $h$ and $p$ .

In Sections 3 and 4 below, we shall generalize the finite element setting to the virtual element one.

3. The virtual element setting

Here, we introduce several virtual element spaces, which we shall use in the design of the hybridized local flux reconstruction in Section 4 below. Primal virtual elements are detailed in Section 3.1, whereas we design couples of broken, mixed virtual elements in Section 3.2. We also introduce skeletal, broken polynomial spaces in Section 3.3, that will play the role of the hybridization space. As a matter of notation, we shall denote the primal, scalar variable with a tilde on top when considered as solution to the primal formulation, and without tilde when appearing in the mixed formulation.

3.1. Primal virtual elements

Here, we follow [3] and briefly recall the primal virtual element method for the Poisson problem (2).

3.1.1. Primal virtual element spaces.

Given $p \in N$ , for all $K \in T_{n}$ , we introduce the primal local space

{˜ V}_{p} (K) := {{˜ v}_{p} \in C^{0} (¯ ¯¯¯ ¯ K) ∣ {˜ v}_{p}_{| e} \in P_{p} (e) \forall e \in E_{n}, Δ {˜ v}_{p} \in P_{p - 2} (K)} .

We have that $P_{p} (K) \subseteq {˜ V}_{p} (K)$ . Next, consider ${m_{α}^{K}}_{| α | = 0}^{p - 2}$ any basis of $P_{p - 2} (K)$ , whose elements are invariant with respect to translations and dilations. The following set of linear functionals of ${˜ V}_{p} (K)$ is a set of unisolvent degrees of freedom [3]: given ${˜ v}_{p} \in {˜ V}_{p} (K)$ ,

the point values of ${˜ v}_{p}$ at the vertices of $K$ ;
for each $e \in E^{K}$ , the point values of ${˜ v}_{p}$ at the $p$ internal Gauß-Lobatto nodes of $e$ ;
the scaled moments

$\frac{1}{| K |} \int_{K} {˜ v}_{p} m_{α}^{K} \forall | α | = 0, \dots, p - 2.$

The global virtual element space together with its set of degrees of freedom is constructed via a standard $H^{1}$ -conforming coupling of their local counterparts and strongly imposing the Dirichlet boundary conditions:

{˜ V}_{p} := {{˜ v}_{p} \in C^{0} (¯ ¯¯ ¯ Ω) \cap H_{0}^{1} (Ω) ∣ {˜ v}_{p}_{| K} \in {˜ V}_{p} (K) \forall K \in T_{n}} .

3.1.2. Projectors

Through the degrees of freedom detailed in Section 3.1.1, it is possible to compute two local orthogonal projectors onto polynomial space. The first one is the energy projector ${˜ Π}_{p}^{\nabla} : {˜ V}_{p} (K) \to P_{p} (K)$ defined as follows: given ${˜ v}_{p}$ in ${˜ V}_{p} (K)$ ,

\int_{K} \nabla q_{p}^{K} \cdot \nabla ({˜ v}_{p} - {˜ Π}_{p}^{\nabla} {˜ v}_{p}) = 0 \forall q_{p}^{K} \in P_{p} (K), \int_{\partial K} ({˜ v}_{p} - {˜ Π}_{p}^{\nabla} {˜ v}_{p}) = 0.

The second one is the $L^{2}$ projector ${˜ Π}_{p - 2}^{0} : {˜ V}_{p} (K) \to P_{p - 2} (K)$ defined as follows: for all $q_{p - 2}^{K}$ in $P_{p} (K)$ and ${˜ v}_{p}$ in ${˜ V}_{p} (K)$ ,

\int_{K} q_{p - 2}^{K} ({˜ v}_{p} - {˜ Π}_{p - 2}^{0} {˜ v}_{p}) = 0.

3.1.3. The primal virtual element method.

Following [3], we split the bilinear form $˜ a (\cdot, \cdot)$ into local contributions:

˜ a (u, v) = \sum K \in T_{n} {˜ a}^{K} (u_{| K}, v_{| K}) := \sum K \in T_{n} (\nabla u, \nabla v)_{0, K} \forall u, v \in H^{1} (Ω) .

We approximate each local bilinear form ${˜ a}^{K} (\cdot, \cdot)$ with the following discrete counterpart: for all ${˜ u}_{p}$ and ${˜ v}_{p}$ in ${˜ V}_{p} (K)$ ,

{˜ a}_{p}^{K} ({˜ u}_{p}, {˜ v}_{p}) := {˜ a}^{K} ({˜ Π}_{p}^{\nabla} {˜ u}_{p}, {˜ Π}_{p}^{\nabla} {˜ v}_{p}) + {˜ S}^{K} ((I - {˜ Π}_{p}^{\nabla}) {˜ u}_{p}, (I - {˜ Π}_{p}^{\nabla}) {˜ v}_{p}),

where the stabilizing term ${˜ S}^{K} (\cdot, \cdot)$ is any bilinear form computable via the degrees of freedom and satisfying the following bounds: there exist positive stability constants ${˜ α}_{*}$ and ${˜ α}^{*}$ such that

{˜ α}_{*} | {˜ v}_{p} |_{1, K}^{2} \leq {˜ S}^{K} ({˜ v}_{p}, {˜ v}_{p}) \leq {˜ α}^{*} | {˜ v}_{p} |_{1, K}^{2} \forall {˜ v}_{p} \in ker ({˜ Π}_{p}^{\nabla}) .

(13)

Explicit choices of ${˜ S}^{K} (\cdot, \cdot)$ will be discussed in Section 5 below. In the literature [5], it is possible to find explicit choices of the stabilizing term such that the stability constants ${˜ α}_{*}$ and ${˜ α}^{*}$ depend on $p$ algebraically only.

The global discrete bilinear form for the primal formulation is given by the sum of the above local contributions:

{˜ a}_{p} ({˜ u}_{p}, {˜ v}_{p}) := \sum K \in T_{n} {˜ a}_{p}^{K} ({˜ u}_{p}_{| K}, {˜ v}_{p}_{| K}) .

As for the discretization of the right-hand side, we pick

(f_{p}, {˜ v}_{p})_{0, Ω} := \sum K \in T_{n} (f, {˜ Π}_{p - 2}^{0} {˜ v}_{p})_{0, K} .

The primal virtual element method reads as follows:

{\begin{matrix} find {˜ u}_{p} \in {˜ V}_{p} such that% {˜ a}_{p} ({˜ u}_{p}, {˜ v}_{p}) = (f_{p}, {˜ v}_{p})_{0, Ω} \forall {˜ v}_{p} \in {˜ V}_{p} . \end{matrix}

(14)

Method (14) is well posed; see [3].

3.2. Mixed virtual elements

Here, we introduce stable couples of broken, mixed virtual element spaces. As for the flux space, the local spaces are those introduced in [4], yet the global space consists of piecewise discontinuous functions as in [8] for the elasticity case. Furthermore, we design suitable bilinear forms appearing in the hybridized mixed formulations of Section 4 below.

3.2.1. Mixed virtual element spaces.

Given $p \in N_{0}$ , for all $K \in T_{n}$ , we introduce the local mixed space [4]

\begin{matrix} Σ_{p} (K) := {τ_{p} \in H (div, K) \cap H (rot, Ω) & ∣ n_{K} \cdot τ_{p}_{| e} \in P_{p} (e) \forall e \in E^{K}, div τ_{p} \in P_{p} (K), rot τ_{p} \in P_{p - 1} (K)} . \end{matrix}

We have that $[P_{p} (K)]^{2} \subseteq Σ_{p} (K)$ . Next, consider the splitting of $[P_{p} (K)]^{2}$ into

[P_{p} (K)]^{2} = G_{p} (K) \oplus G_{p}^{⊥} (K), G_{p} (K) := \nabla P_{p + 1} (K),

where $G^{⊥} (K)$ is any completion of $G_{p} (K)$ in $[P_{p} (K)]^{2}$ . Denote the dimension of $G_{p} (K)$ and $G_{p}^{⊥} (K)$ by $g_{p}$ and $g_{p}^{⊥}$ , respectively. Moreover, consider ${g_{α}}_{α = 1}^{g_{p}}$ and ${g_{α}^{⊥}}_{α = 1}^{g_{p}^{⊥}}$ any bases of $G_{p} (K)$ and $G_{p}^{⊥} (K)$ , consisting of elements that are invariant with respect to translations and dilations; we refer to [bricksVEMmixed] for explicit choices of such bases. The following set of linear functionals of $Σ_{p} (K)$ is a set of unisolvent degrees of freedom [4]: given $τ_{p} \in Σ_{p} (K)$ ,

for each $e \in E^{K}$ , the point values of $τ_{p}$ at the $p + 1$ Gauß quadrature nodes of $e$ ;
the gradient-type moments up to order $p - 1$

$\frac{1}{| K |} \int_{K} τ_{p} \cdot g_{α} \forall α = 1, \dots, g_{p - 1};$
the “completion”-type moments up to order $p$

$\frac{1}{| K |} \int_{K} τ_{p} \cdot g_{α}^{⊥} \forall α = 1, \dots, g_{p}^{⊥} .$

Differently from the standard Raviart-Thomas approach of [4], we do not construct the corresponding global space by imposing $H (div)$ conformity. Rather, we hybridize the space, see [8, 1], and therefore consider the following global discontinuous mixed space:

Σ_{p} := {τ_{p} \in [L^{2} (Ω)]^{2} ∣ τ_{p}_{| K} \in Σ_{p} (K)} .

Since we do not match the edge degrees of freedom, we do not impose any boundary condition in the space. Rather, the boundary conditions will be imposed in a mortar way through the hybridized variable.

We further introduce the space taking care of the scalar variable. To the aim, we simply set

V_{p} := P_{p} (T_{n}) .

3.2.2. Projectors

Through the degrees of freedom detailed in Section 3.2.1, it is possible to compute [4] a vector, piecewise $L^{2}$ orthogonal projector $Π_{p}^{0} : Σ_{p} (K) \to [P_{p} (K)]^{2}$ , which is defined as follows: for all $q_{p}^{K}$ and $τ_{p} \in Σ_{p} (K)$ ,

(q_{p}^{K}, τ_{p} - Π_{p}^{0} τ_{p})_{0, K} = 0.

3.2.3. Mixed virtual element bilinear forms.

Following [4], we split the bilinear form $a (\cdot, \cdot) := (\cdot, \cdot)_{0, Ω}$ into local contributions:

a (σ, τ) = \sum K \in T_{n} a^{K} (σ_{| K}, τ_{| K}) := \sum K \in T_{n} (σ, τ)_{0, K} \forall σ, τ \in [L^{2} (Ω)]^{2} .

We approximate each local bilinear form $a_{p}^{K} (\cdot, \cdot)$ with the following discrete counterpart: for all $σ_{p}$ and $τ_{p}$ in $Σ_{p} (K)$ ,

a_{p}^{K} (σ_{p}, τ_{p}) = a^{K} (Π_{p}^{0} σ_{p}, Π_{p}^{0} τ_{p}) + S^{K} ((I - Π_{p}^{0}) σ_{p}, (I - Π_{p}^{0}) τ_{p}),

(15)

where the stabilizing term $S^{K} (\cdot, \cdot)$ is any bilinear form computable via the degrees of freedom and satisfying the following bounds: there exist positive stability constants $α_{*}$ and $α^{*}$ such that

α_{*} ∥ τ_{p} ∥_{0, K}^{2} \leq S^{K} (τ_{p}, τ_{p}) \leq α^{*} ∥ τ_{p} ∥_{0, K}^{2} \forall τ_{p} \in ker (Π_{p}^{0}) .

(16)

An explicit choice of $S^{K} (\cdot, \cdot)$ will be derived and analyzed in Appendix A below. Importantly, we shall show that the stability constants depend on $p$ algebraically only.

The global discrete bilinear form for mixed virtual elements is given by the sum of the above local contributions:

a_{p} (σ_{p}, τ_{p}) := \sum K \in T_{n} a_{p}^{K} (σ_{p}_{| K}, τ_{p}_{| K}) .

The choice of the degrees of freedom in Section 3.2.1 allows for the explicit computation of the local divergence of any function $τ_{p} \in Σ_{p} (K)$ on each element $K \in T_{n}$ ; see [4]. Therefore, we also define the exactly computable bilinear form

b (τ_{p}, v_{p}) := \sum K \in T_{n} (div τ_{p}, v_{p})_{0, K} \forall τ_{p} \in Σ_{p}, \forall v_{p} \in V_{p} .

3.3. Hybridized virtual elements

Here, we introduce piecewise discontinuous polynomial spaces, and the related bilinear forms and their properties, which will be instrumental in the design of the hybridized flux reconstruction of Section 4 below.

3.3.1. Hybrid virtual element spaces.

Given $p \in T_{n}$ , we introduce the hybridization space

Λ_{p}^{I} := P_{p} (E_{n}^{I}) .

The following set of linear functionals is a global set of degrees of freedom for $Λ_{p}^{I}$ : given $λ_{p}^{I} \in Λ_{p}^{I}$ ,

for each $e \in E_{n}^{I}$ , the point values of $λ_{p}^{I}$ at the $p + 1$ Gauß nodes of $e$ .

3.3.2. Hybrid virtual element bilinear forms.

We introduce a bilinear form that will be instrumental in the design of the hybridized virtual element method. To the aim, we first define the jump operator across an internal edge $e \in E_{n}^{I}$ : for any $τ \in H (div, T_{n})$ ,

⟦ τ ⟧_{e} := ⟦ τ ⟧ = τ_{| K^{+}} \cdot n_{K^{+}} + τ_{| K^{-}} \cdot n_{K^{-}} \forall e \in E_{n}^{I}, e \in E^{K^{+}} \cap E^{K^{-}} .

We define the bilinear form $c (\cdot, \cdot) : Σ_{p} \times Λ_{p}^{I} \to R$ as

c (τ_{p}, μ_{p}^{I}) := \sum e \in E_{n}^{I} (⟦ τ_{p} ⟧, μ_{p}^{I})_{0, e} \forall τ_{p} \in Σ_{p}, \forall μ_{p}^{I} \in Λ_{p}^{I} .

Introduce the weighted, broken divergence norm:

∥ τ ∥_{div, T_{n}}^{2} = \sum K \in T_{n} (h_{K}^{- 1} ∥ τ ∥_{0, K}^{2} + h_{K} ∥ div τ ∥_{0, K}^{2}) \forall τ \in H (div, T_{n}) .

To analyze hybridized virtual elements, it is convenient to prove the existence of a stable, broken $H (div)$ lifting operator $L : L^{2} (E_{n}^{I}) \to H (div, T_{n})$ such that

∥ L (μ_{p}^{I}) ∥_{div, T_{n}} \leq β_{c} ∥ μ_{p}^{I} ∥_{0, E_{n}^{I}}, ⟦ L (μ_{p}^{I}) ⟧_{e} = μ_{p}^{I} \forall e \in E_{n}^{I},

(17)

where $β_{c}$ is a positive constant, independent of the discretization parameters.

Proposition 3.1.

There exists a stable lifting operator $L$ as in (17).

Proof.

For all $μ_{p}^{I} \in Λ_{p}^{I}$ , we have to construct a flux $L (μ_{p}^{I}) = τ_{p} \in Σ_{p}$ satisfying (17). To the aim, we define $τ_{p}$ in each element $K$ as $τ_{p} = \nabla φ$ , where $φ$ is the solution to the following local elliptic problems on each $K \in T_{n}$ :

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} Δ φ = c_{K} & in K n_{K} \cdot \nabla φ = \frac{1}{2} μ_{p}^{I} & on each e \in E^{K} \cap E_{n}^{I} n_{K} \cdot \nabla φ = 0 & on each e \in E^{K} \cap E_{n}^{D} \int_{K} φ = 0, \end{matrix}

where $c_{K} \in R$ is fixed so that the following compatibility condition is valid:

c_{K} = \frac{1}{2} | K |^{- 1} \sum e \in E^{K} \cap E_{n}^{I} (μ_{p}^{I}, 1)_{0, e} .

(18)

The above Neumann problem is well posed since the compatibility condition (18) is valid.

The flux $τ_{p} = \nabla φ$ satisfies the following properties:

\begin{matrix} div τ_{p}_{| K} = c_{K}, ∥ τ_{p} ∥_{0, K} = | φ |_{1, K}, ∥ d i v τ_{p} ∥_{0, K} = | K |^{\frac{1}{2}} | c_{K} | \forall K \in T_{n}, ⟦ τ_{p} ⟧_{e} = μ_{p}^{I}_{| e} \forall e \in E_{n}^{I} . \end{matrix}

(19)

Additionally, for all $K \in T_{n}$ , we observe that

∥ τ_{p} ∥_{0, K}^{2} = \int_{K} \nabla φ \cdot \nabla φ = - \int_{K} (Δ φ) φ + \int_{\partial K} (n_{K} \cdot \nabla φ) φ .

The first term on the right-hand side is zero, since $Δ φ$ is constant and $φ$ has zero average over $K$ . Thus, the trace and the Poincaré inequalities yield

∥ τ_{p} ∥_{0, K}^{2} \leq ∥ n_{K} \cdot \nabla φ ∥_{0, \partial K ∖ \partial Ω} ∥ φ ∥_{0, \partial K} ≲ ∥ μ_{p}^{I} ∥_{0, \partial K ∖ \partial Ω} h_{K}^{\frac{1}{2}} | φ |_{1, K}

Adaptive virtual elements based on hybridized, reliable, and efficient flux reconstructions

Authors

Adaptive virtual element methods with equilibrated flux

A new paradigm for enriching virtual element methods

Interior estimates for the Virtual Element Method

Flux recovery for Cut finite element method and its application in a posteriori error estimation

A numerical study of the dispersion and dissipation properties of virtual element methods for the Helmholtz problem

A posteriori error analysis for a Lagrange multiplier method for a Stokes/Biot fluid-poroelastic structure interaction model

A virtual element-based flux recovery on quadtree

1. Introduction

Notation.

The model problem.

Mesh assumptions.

Structure of the paper.

2. Hybridized global/local flux reconstruction in the finite element setting

3. The virtual element setting

3.1. Primal virtual elements

3.1.1. Primal virtual element spaces.

3.1.2. Projectors

3.1.3. The primal virtual element method.

3.2. Mixed virtual elements

3.2.1. Mixed virtual element spaces.

3.2.2. Projectors

3.2.3. Mixed virtual element bilinear forms.

3.3. Hybridized virtual elements

3.3.1. Hybrid virtual element spaces.

3.3.2. Hybrid virtual element bilinear forms.

Proposition 3.1.

Proof.

Adaptive virtual elements based on hybridized, reliable, and efficient flux reconstructions

Authors

Related Research

Adaptive virtual element methods with equilibrated flux

A new paradigm for enriching virtual element methods

Interior estimates for the Virtual Element Method

Flux recovery for Cut finite element method and its application in a posteriori error estimation

A numerical study of the dispersion and dissipation properties of virtual element methods for the Helmholtz problem

A posteriori error analysis for a Lagrange multiplier method for a Stokes/Biot fluid-poroelastic structure interaction model

A virtual element-based flux recovery on quadtree

This week in AI

1. Introduction

Notation.

The model problem.

Mesh assumptions.

Structure of the paper.

2. Hybridized global/local flux reconstruction in the finite element setting

3. The virtual element setting

3.1. Primal virtual elements

3.1.1. Primal virtual element spaces.

3.1.2. Projectors

3.1.3. The primal virtual element method.

3.2. Mixed virtual elements

3.2.1. Mixed virtual element spaces.

3.2.2. Projectors

3.2.3. Mixed virtual element bilinear forms.

3.3. Hybridized virtual elements

3.3.1. Hybrid virtual element spaces.

3.3.2. Hybrid virtual element bilinear forms.

Proposition 3.1.

Proof.