Non-conforming harmonic virtual element method: h- and p-versions

Non-conforming harmonic virtual element method: $h$- and $p$-versions

Abstract

We study the - and -versions of non-conforming harmonic virtual element methods (VEM) for the approximation of the Dirichlet-Laplace problem on a 2D polygonal domain, providing quasi-optimal error bounds. Harmonic VEM do not make use of internal degrees of freedom. This leads to a faster convergence, in terms of the number of degrees of freedom, as compared to standard VEM. Importantly, the technical tools used in our -analysis can be employed as well in the analysis of more general non-conforming finite element methods and VEM. The theoretical results are validated in a series of numerical experiments. The -version of the method is numerically tested, demonstrating exponential convergence with rate given by the square root of the number of degrees of freedom.

AMS subject classification: 65N30, 65N12, 65N15, 35J05, 31A05

Keywords: Virtual element methods, non-conforming methods, Laplace problem, approximation by harmonic functions, error bounds, polytopal meshes

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Acknowledgement

The authors have been funded by the Austrian Science Fund (FWF) through the projects P 29197-N32 and F 65.

1 Introduction

In recent years, Galerkin methods based on polygonal/polyhedral meshes have attracted a lot of attention, owing to their flexibility in dealing with complex geometries. The virtual element method (VEM) introduced in [17, 21] is one of them. VEM can be seen as a generalization of standard finite element methods (FEM) to polytopal meshes. The main feature of VEM, in addition to the fact that they allow for general meshes, is that they are based on trial and test spaces that consist of solutions to local problems mimicking the target one. These functions are not known in a closed form, which is at the origin of the name “virtual”. Importantly, the construction of the method does not rely on an explicit representation of the basis functions, but rather on the explicit knowledge of degrees of freedom. This allows to compute certain projection operators from local VE spaces into polynomial ones, which are instrumental in the definition of proper bilinear forms.

Owing to its flexibility and simplicitity of the implementation, despite its novelty, the basic VEM paradigm has already been extended to highly-regular [25], non-conforming [9, 36], and serendipity [18] approximating spaces, combined with domain decomposition techniques [30], adaptive mesh refinement [34], adapted to curved domains [26], and applied to a wide variety of problems; among them, we recall general second-order elliptic problems [19], eigenvalue problems [50, 42], Stokes problem [5], elasticity problems [20, 16], Helmholtz problem [51], Cahn-Hilliard equation [6], discrete fracture network simulations [27], and topology optimization [41].

In this paper, we introduce and analyze non-conforming harmonic VEM for the approximation of the Dirichlet-Laplace problem on polygonal domains. These methods can be seen as the intermediate conformity level between the continuous harmonic VEM developed in [38], and the harmonic discontinuous Galerkin FEM (DG-FEM) of [44, 46, 45]. As typically done in non-conforming methods, instead of requiring -continuity of test and trial functions over the entire physical domain, one only imposes that the moments, up to a certain order, of their jumps across two adjacent elements are zero. We highlight that non-conforming VEM were introduced in [9] for the approximation of the Poisson problem and were subsequently extended to the approximation of general elliptic and Stokes problems in [36, 35], respectively. Our method inherits the structure of that of [9], but makes use of harmonic basis functions, which yield to faster convergence, when approximating harmonic solutions, compared to standard basis functions.

We are particularly interested in the investigation of the - and -versions of these methods. In the former version, convergence is achieved by fixing the dimension of local spaces and refining the mesh, whereas, in the latter, by keeping fix a single mesh and increasing the dimension of local spaces. A combination of the two goes by the name of -version. The literature regarding the - and -versions of VEM is restricted to [22, 23, 47, 40, 7], in addition to the above-mentioned work [38]; the literature regarding other polygonal methods restricts, to the best of our knowledge, to hybrid-high order methods, see [3], and to DG-FEM on polytopal grids, see [33] and the references therein. We derive quasi-optimal error bounds in the broken norm and in the norm, which are explicit in terms of the mesh size and of the degree of accuracy of the method. Although not covered by our theoretical analysis, we provide numerical evidence that, similarly as for the harmonic VEM and harmonic DG-FEM [38, 44], the convergence of the -version of the non-conforming harmonic VEM is faster than the one of standard FEM [10, 52] and VEM [22, 23]. Indeed, one gets , in the former case, and , in the latter, where denotes the number of degrees of freedom and is a positive constant.

We stress that, in the high-order case, the construction of an explicit basis for non-conforming harmonic VEM, as well as for non-conforming standard VEM, is much simpler than for non-conforming FEM, see for instance [39].

The tools that we employ in the forthcoming -analysis of non-conforming harmonic VEM can actually be employed as well in the -analysis of non-conforming FEM and of non-conforming VEM. For instance, our argument to trace back best approximation estimates by means of non-conforming harmonic VE functions to best approximation estimates by means of discontinuous harmonic polynomials (Proposition 3.1) extends to the non-harmonic case (Proposition 3.8). This provides a useful tool in order to develop a -analysis of the non-conforming VEM of [9].

The design and analysis of non-conforming harmonic VEM developed in this paper pave the way for the study of VEM for the Helmholtz problem in a truly Trefftz setting, alternative to the plane wave VEM of [51], which was based on a partition of unity approach. In fact, the non-conforming framework seems to be the most appropriate one in order to design virtual Helmholtz-Trefftz approximation spaces.

The outline of this paper is as follows. In Section 2, the model problem is formulated and the concept of regular polygonal decompositions needed for the analysis is introduced; besides, we recall the definition of non-conforming Sobolev spaces subordinated to polygonal decompositions of the physical domain. Section 3 is dedicated to the construction of the 2D non-conforming harmonic VEM and to the analysis of its - and -versions; further, a hint for the extension to the 3D case is given. Next, in Section 4, numerical results validating the theoretical convergence estimates are presented; a numerical investigation of the full -version of the method is also provided. Finally, details on the implementation of the method are given in Appendix 6.

Notation We fix here once and for all the notation employed throughout the paper. Given any domain , , and , we denote by and the spaces of polynomials and harmonic polynomials up to order over , respectively; moreover, we set .

We use the standard notation for Sobolev spaces, norms, seminorms and inner products, see [2]. More precisely, we denote the Sobolev space of functions with square integrable weak derivatives up to order over by , and the corresponding seminorms and norms by and , respectively. Sobolev spaces of non-integer order can be defined, for instance, by interpolation theory [53]. In addition, for bounded , denotes the space of traces of functions; and are the Sobolev spaces of functions with traces equal to zero and equal to a given function , respectively. Further, is the usual inner product over .

We employ the following multi-index notation: for ,

with , and where denotes the -th partial derivative along direction .

In the sequel, we also use the notation meaning that there exists a constant , independent of and , but possibly dependent on the shape of the domain/mesh elements, such that . Finally, we use the notation in lieu of and simultaneously.

2 Continuous problem, polygonal decompositions and functional setting

Here, we want to set the target problem and some basic notation we need for the construction of the non-conforming harmonic virtual element method (VEM). More precisely, the outline of the section is as follows. In Section 2.1, we introduce the model problem, that is a Laplace problem on a polygonal domain. Then, in Section 2.2, we define the concept of regular decompositions into polygons of the physical domain of the problem. Finally, in Section 2.3, we describe non-conforming Sobolev spaces over such polygonal decompositions.

2.1 The continuous problem

The target problem we aim to approximate is a Laplace problem over a polygonal domain with boundary . More precisely, given , we look for a function solving

(1)

The weak formulation of (1) reads

(2)

where

(3)

Well-posedness of problem (2) follows from a lifting argument and the Lax-Milgram lemma.

2.2 Regular polygonal decompositions

In this section, we introduce the concept of regular sequences of polygonal decompositions of the domain , which will be needed in the forthcoming analysis of the method.

Let be a sequence of conforming polygonal decompositions of ; by conforming, we mean that, for each , every internal edge of is contained in the boundary of precisely two elements in the decomposition. This automatically includes the possibility of dealing with hanging nodes.

For all , with each , we associate , and , which denote its set of edges, internal edges and boundary edges, respectively. Moreover, with each element of , we associate , the set of its edges. Finally, we set for all and for all ,

and we denote by the centroid of .

We say that is a regular sequence of polygonal decompositions if the following assumptions are satisfied:

  • there exists a positive constant such that, for all and for all , for all edges of ;

  • there exists a positive constant such that, for all and for all , is star-shaped with respect to a ball of radius greater than or equal to ;

  • there exists a constant such that, for all and for all , card(), that is, the number of edges of each element is uniformly bounded.

We point out that, in this definition, we are not requiring any quasi-uniformity on the size of the elements. A discussion of VEM under more general mesh assumptions is the topic of [24, 32].

For future use, we also define local bilinear forms on polygons as

(4)

2.3 Non-conforming Sobolev spaces

Having introduced the concept of regular sequences of meshes, we pinpoint the concept of sequences of broken and non-conforming Sobolev spaces, along with their norms. For all and , we define the broken Sobolev spaces on as

and the corresponding broken seminorms and norms

(5)

Particular emphasis is stressed on the broken bilinear form

In order to define non-conforming Sobolev spaces associated with polygonal decompositions, we need to fix some additional notation. In particular, given any internal edge shared by the polygons and in , we denote by the two outer normal unit vectors with respect to . For simplicity, we will later only write instead of . Moreover, for boundary edges , we introduce the normal unit vector pointing outside . Having this, for any , we set the jump operator across an edge to

(6)

Finally, we introduce the global non-conforming Sobolev space of order with respect to the decomposition incorporating boundary conditions in a non-conforming sense: Given and , we define

(7)

where is either of the two normal unit vectors to , but fixed, if , and , if . In the homogeneous case, definition (7) becomes

(8)

Importantly, the seminorm is actually a norm for functions in . In [31], the validity of the following Poincaré  inequality was proven: there exists a positive constant only depending on such that, for all ,

3 Non-conforming harmonic virtual element methods

In this section, we introduce a non-conforming harmonic virtual element method for the approximation of problem (2) and investigate its - and -versions. To this purpose, in addition to the geometric assumptions (A1)-(A3) on the sequence of meshes , we will also require the following quasi-uniformity assumption:

  • there exists a constant such that, for all and for all and in with , it holds .

We want to approximate problem (2) with a method of the following type:

(9)

where the space of trial functions and the space of test functions are finite dimensional (non-conforming) spaces on a mesh , “mimicking” the infinite dimensional spaces and , defined in (3), respectively. Moreover, is a computable discrete bilinear form mimicking its continuous counterpart defined again in (3). Such approximation spaces and discrete bilinear forms have to be tailored so that method (9) is well-posed and provides “good” - and -approximation estimates.

The outline of this section is as follows. We first introduce suitable global approximation spaces and in Section 3.1, highlighting their approximation properties in Section 3.2. Next, in Section 3.3, we define and provide an explicit discrete bilinear form and, moreover, we discuss its properties. An abstract error analysis is carried out in Section 3.4; such analysis is instrumental for the - and -error estimates proved in Section 3.5. error bounds are provided in Section 3.6. Finally, in Section 3.7, we give a hint concerning the extension to the 3D case and we stress the main differences between the 2D and 3D cases. Some details on the implementation of the method are presented in Appendix 6.

3.1 Non-conforming harmonic virtual element spaces

The aim of the present section is to introduce non-conforming harmonic virtual element spaces with uniform degree of accuracy. To this purpose, we begin with the description of local harmonic VE spaces, modifying those in [38] into a new setting suited for building global non-conforming spaces.

Let be a given parameter. For all and for all , we set

(10)

In words, consists of harmonic functions with piecewise (discontinuous) polynomial normal traces on the boundary of .

The space has dimension , being the number of edges of . A set of degrees of freedom for is given by

(11)

where is any basis of . These degrees of freedom are in fact unisolvent since, if has all the degrees of freedom equal to , then

which implies that is constant. This, in addition to

for some edge , implies , providing unisolvence.

We denote by the local canonical basis associated with the set of degrees of freedom (11), namely

(12)

We underline that the indices and refer to the edge, whereas the indices and refer to the polynomial employed in the definition of the local degrees of freedom (11).

It is worth to note that the local canonical basis consists of functions that are not explicitly known inside the element and even their polynomial normal traces over the boundary are unknown.

By employing the degrees of freedom defined in (11), it is possible to compute the following two projectors. The first one is the edge projector onto the space of polynomials of degree

(13)

The second one is the bulk projector onto the space of harmonic polynomials of degree

(14)

where the last condition is imposed in order to define the projector in a unique way.

We are ready to define global non-conforming harmonic VE spaces, which incorporate Dirichlet boundary conditions in a “non-conforming sense”. Let be a given parameter. Then, for any , we set

(15)

We observe the following facts:

  • Definition (15) includes the space of test functions , by selecting .

  • The parameter in (15) indicates the level of non-conformity of the method. The fact that the non-conformity is defined with respect to Dirichlet traces allows us to easily couple the local degrees of freedom into a global set, provided that we choose the same value for the non-conformity parameter and for the polynomial degree entering definition (10) of the local spaces. The resulting global set of degrees of freedom is of dimension .

  • Dirichlet boundary conditions on are imposed weakly via the definition of the non-conforming spaces (7) and (8). For instance, given a Dirichlet datum , on all boundary edges , we set

Remark 1.

We highlight that, at the discrete level, one should also take into account the approximation of the Dirichlet boundary condition . In practice, assuming , for any arbitrarily small, and denoting by the approximation of obtained by interpolating at the Gauß-Lobatto nodes on each edge in , one should define the trial space as

With this definition, in the forthcoming analysis (see Proposition 3.1, Theorem 3.3, Theorem 3.6, Proposition 3.8, and Theorem 3.9 below), an additional term related to the approximation of the Dirichlet datum via Gauß-Lobatto interpolants should be taken into account. However, following [29, Theorem 4.2, Theorem 4.5], it is possible to show that the - and -rates of convergence of the method are not spoilt by this term. For this reason and for the sake of simplicity, we will neglect in the following the presence of this term and assume that the approximation space is the one defined in (15).

3.2 Approximation properties of functions in non-conforming harmonic virtual element spaces

In this section, we deal with approximation properties of functions in the non-conforming harmonic VE spaces and .

Since - and -approximation properties of harmonic functions via harmonic polynomials are known, see e.g. [14, 44], we want to relate best approximation estimates in the non-conforming harmonic VE spaces to the corresponding ones in discontinuous harmonic polynomial spaces. In particular, we prove the following result.

Proposition 3.1.

Given , let , where is defined in (3). For any polygonal partition of , there exists , with introduced in (15), such that

where is the space of discontinuous piecewise harmonic polynomials of degree at most , that is,

(16)
Proof.

Define by

(17)

that is, we fix the degrees of freedom (11) of to be equal to the values of the same functionals applied to the solution . Having this, it holds

(18)

where is defined in (16). We focus on the second term on the right-hand side of (18). By integrating by parts and using (17), together with the definition of the space (15), we get

(19)

By expanding the right-hand side of (19) and using the Cauchy-Schwarz inequality, we obtain

Inserting this into (18) gives the result. ∎

We remark that, with a similar proof of that of Proposition 3.1, one can prove an equivalent result for the non-conforming (non-harmonic) VE spaces of [9]; see Proposition 3.8 below.

3.3 Discrete bilinear forms

In this section, we complete the definition of the method (9) by introducing a suitable bilinear form , which is explicitly computable. We follow here the typical VEM gospel [17, 23, 38]. It is important to highlight that the local bilinear forms defined in (4) are not explicitly computable on the whole discrete spaces since an explicit representation of functions in the harmonic VE spaces is not available in closed form.

Therefore, we aim at introducing explicit computable discrete bilinear forms which mimic their continuous counterparts . To this purpose, we observe that the Pythagorean theorem yields

(20)

where we recall that is defined in (14). The first term on the right-hand side of (20) is computable, whereas the second is not. Thus, following [38] and the references therein, we replace this term by a computable symmetric bilinear form , such that

(21)

where and are two positive constants which may depend on , but are independent of and, in particular, of .

Hence, depending on the choice of the stabilization, a class of candidates for the local discrete symmetric bilinear forms is

(22)

The forms satisfy the two following properties:

  • -harmonic consistency: for all and for all ,

    (23)
  • stability: for all and for all ,

    (24)

    where and .

Owing to property (P1), can be addressed to as degree of accuracy of the method, since whenever either of its two entries is a harmonic polynomial of degree , the local discrete bilinear form can be computed exactly, up to machine precision. Moreover, since is assumed to be symmetric, (P2) implies continuity

(25)

The global discrete bilinear form is defined as

(26)

for all . The remainder of this section is devoted to introduce an explicit stabilization with explicit bounds of the constants and .

For all , we define

(27)

For this choice of stabilization forms, the following result holds true.

Theorem 3.2.

Assume that (A1) and (A2) hold true. Then, for any , the stabilization defined in (27) satisfies (21) with the bounds

(28)

where the hidden constants in (28) are independent of and .

Proof.

We assume, without loss of generality, that ; the general result follows from a scaling argument.

For any function in , we have

(29)

where we have set, with an abuse of notation, . We prove that

(30)

To this end, we set, for the sake of simplicity, , and consider the case . One has with for all . In general, . Further, we introduce the piecewise bubble function defined edgewise as

where is the linear transformation mapping the interval to the edge , and is the 1D quadratic bubble function .

From the definition of the norm, the fact that , and , we have

(31)

We have the two following polynomial -inverse inequalities:

(32)

The first one is a direct consequence of the fact that the range of is , and the second one follows from [15, Lemma 2]. Using (32) and interpolation theory, and summing over all edges lead to

which, together with (31), gives

where, in the last inequality, [28, Lemma 4] was used. The bound (30) follows immediately.

From (29) and (30), taking also into account that in , we get

where in the last step we have used a Neumann trace inequality, see e.g. [52, Theorem A.33]. This proves the first inequality of (21) with . The second one follows instead from

where we have used the stability of the projection, the trace inequality, and the Poincaré  inequality, see [31], which is valid since and thus has zero mean value on . This gives . ∎

Owing to (24) and (28) one also has

3.4 Abstract error analysis

Along the lines of [17, 22, 38], we provide in this section an abstract error analysis of the method (9), taking into account the non-conformity of the approximation. To this purpose, we introduce the auxiliary bilinear form

(33)

The following convergence result holds true.

Theorem 3.3.

Assume that (A1) and (A2) hold true and consider the non-conforming harmonic VEM (9) defined by choosing the harmonic VE spaces as in (15) and (10), with level of non-conformity, as well as degree of accuracy, equal to , and by choosing the discrete bilinear form as in (26) and (22), with stabilization form satisfying (21). Then, the method is well-posed and the following bound holds true:

(34)

where we recall that is defined in (16), is given in (33), and the stability constants and are introduced in (24).

Proof.

The well-posedness of the method follows directly from (24) and the Lax-Milgram lemma.

For the bound (34), we observe that

We estimate the second term on the right-hand side. Set