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

07/08/2021
by   F. Dassi, et al.
University of Bonn
Universität Wien
0

We present two a posteriori error estimators for the virtual element method (VEM) based on global and local flux reconstruction in the spirit of [5]. The proposed error estimators are reliable and efficient for the h-, p-, and hp-versions of the VEM. This solves a partial limitation of our former approach in [6], which was based on solving local nonhybridized mixed problems. Differently from the finite element setting, the proof of the efficiency turns out to be simpler, as the flux reconstruction in the VEM does not require the existence of polynomial, stable, divergence right-inverse operators. Rather, we only need to construct right-inverse operators in virtual element spaces, exploiting only the implicit definition of virtual element functions. The theoretical results are validated by some numerical experiments on a benchmark problem.

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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 -, -, and 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  and Conjecture ], 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 , denotes the standard Sobolev space of order , which we endow with the standard inner product , norm , and seminorm 

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

and ,

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

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

The model problem.

Given a polygonal domain  and , we consider the Poisson problem: find  such that

(1)

where  denotes the boundary of .

The weak formulation of (1) reads

(2)

where we have set

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 of polygonal meshes with straight edges. Hanging nodes are naturally incorporated in such meshes. Given a mesh , we denote its set of edges and vertices by  and . We split  into the set of boundary and internal edges as follows:

With each element , we associate its diameter 

, the outward pointing unit normal vector 

, its centroid , and its set of edges  and vertices . Moreover, we associate with each edge its length  and any of the two unit normal vectors .

We consider standard properties for the sequence of meshes [3]: there exist two positive constants  and , such that, for all ,

  • every  is star-shaped with respect to a ball  with radius ;

  • given , for every of its edges , .

Associated with each mesh  and , we define standard broken polynomial and Sobolev spaces:

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

Henceforth, for ease of presentation, we assume that 

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

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 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  be a decomposition of  into triangles. We consider the standard finite element approximation of (2):

(3)

where we have set

We define a continuous residual  as

(4)

An integration by parts yields

where we have introduced the bulk and internal edge residuals

(5)

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

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 , whereas we let , where we have set .

Define a flux  in a distributional sense as follows:

(6)

We can interpret  as follows:

Using (4) and (6) implies that  satisfies

(7)

The information about the homogeneous Dirichlet boundary conditions is implicitly contained in the two residuals through . We rewrite (7) as a hybridized mixed method with flux and primal variable given by  and , respectively. More precisely, we write

(8)

where we picked the spaces

(9)

being  the local Raviart-Thomas space of order  on element , and the bilinear forms

(10)

In the literature, the space  is known as the broken, hybridized Raviart-Thomas space of order 

. 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 

.

Next, we discuss the localized version of the above problem. With each vertex , we associate the partition of unity function , i.e., the standard hat function with value  at  and  at all other vertices, and localize the bulk and edge residuals in (5) as follows: for all ,

(11)

Proceeding as above, on each vertex patch

(12)

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

whereas, if  is a boundary vertex,

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

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

is reliable and efficient with constants independent of  and .

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 , for all , we introduce the primal local space

We have that . Next, consider  any basis of , whose elements are invariant with respect to translations and dilations. The following set of linear functionals of  is a set of unisolvent degrees of freedom [3]: given ,

  • the point values of  at the vertices of ;

  • for each , the point values of  at the  internal Gauß-Lobatto nodes of ;

  • the scaled moments

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

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 defined as follows: given  in ,

The second one is the projector defined as follows: for all  in and  in ,

3.1.3. The primal virtual element method.

Following [3], we split the bilinear form  into local contributions:

We approximate each local bilinear form  with the following discrete counterpart: for all  and  in ,

where the stabilizing term  is any bilinear form computable via the degrees of freedom and satisfying the following bounds: there exist positive stability constants  and  such that

(13)

Explicit choices of  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  and  depend on  algebraically only.

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

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

The primal virtual element method reads as follows:

(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 , for all , we introduce the local mixed space [4]

We have that . Next, consider the splitting of  into

where  is any completion of  in . Denote the dimension of  and  by  and , respectively. Moreover, consider and any bases of  and , 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  is a set of unisolvent degrees of freedom [4]: given ,

  • for each , the point values of  at the Gauß quadrature nodes of ;

  • the gradient-type moments up to order 

  • the “completion”-type moments up to order 

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

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

3.2.2. Projectors

Through the degrees of freedom detailed in Section 3.2.1, it is possible to compute [4] a vector, piecewise orthogonal projector , which is defined as follows: for all  and ,

3.2.3. Mixed virtual element bilinear forms.

Following [4], we split the bilinear form into local contributions:

We approximate each local bilinear form  with the following discrete counterpart: for all  and  in ,

(15)

where the stabilizing term  is any bilinear form computable via the degrees of freedom and satisfying the following bounds: there exist positive stability constants  and  such that

(16)

An explicit choice of  will be derived and analyzed in Appendix A below. Importantly, we shall show that the stability constants depend on  algebraically only.

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

The choice of the degrees of freedom in Section 3.2.1 allows for the explicit computation of the local divergence of any function  on each element ; see [4]. Therefore, we also define the exactly computable bilinear form

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 , we introduce the hybridization space

The following set of linear functionals is a global set of degrees of freedom for : given ,

  • for each , the point values of  at the Gauß nodes of .

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 : for any ,

We define the bilinear form as

Introduce the weighted, broken divergence norm:

To analyze hybridized virtual elements, it is convenient to prove the existence of a stable, broken lifting operator such that

(17)

where  is a positive constant, independent of the discretization parameters.

Proposition 3.1.

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

Proof.

For all , we have to construct a flux  satisfying (17). To the aim, we define in each element  as , where  is the solution to the following local elliptic problems on each :

where  is fixed so that the following compatibility condition is valid:

(18)

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

The flux  satisfies the following properties:

(19)

Additionally, for all , we observe that

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