Over the last few years, discretization technologies have appeared that support arbitrary approximation orders on general polytopal meshes. In this work, we focus on a particular instance of such technologies, the so-called Hybrid High-Order (HHO) methods originally introduced in [18, 17]; see also  for an introduction. So far, the literature on HHO methods has focused on the h-version of the method with uniform polynomial degree. Our goal is to provide a first example of variable-degree hp-HHO method and carry out a full hp-convergence analysis valid for fairly general meshes and arbitrary space dimension. Let , , denote a bounded connected polytopal domain with Lipschitz boundary. We consider the variable diffusion model problem: Find such that
where is a uniformly positive, symmetric, tensor-valued field on Ω, while denotes a volumetric source. For the sake of simplicity, we assume that is piecewise constant on a partition of Ω into polytopes. The weak formulation of problem (1.1) reads: Find such that
where we have used the notation for the usual inner products of both and . Here, the scalar-valued field u represents a potential, and the vector-valued field the corresponding flux.
For a given polytopal mesh of Ω, the hp-HHO discretization of problem (1.2) is based on two sets of degrees of freedom (DOFs):
skeletal DOFs, consisting in -variate polynomials of total degree on each mesh face F, and
elemental DOFs, consisting in d-variate polynomials of degree on each mesh element T, where denotes the lowest degree of skeletal DOFs on the boundary of T.
Skeletal DOFs are globally coupled, and can be alternatively interpreted as traces of the potential on the mesh faces or as Lagrange multipliers enforcing the continuity of the normal flux at the discrete level; cf. [1, 12] for further insight. Elemental DOFs, on the other hand, are auxiliary DOFs that can be locally eliminated by static condensation, as detailed in [12, Section 2.4] for the case where for all mesh faces F.
Two key ingredients are devised locally from skeletal and elemental DOFs attached to each mesh element T:
a reconstruction of the potential of degree (i.e., one degree higher than elemental DOFs in T) obtained solving a small Neumann problem, and
a stabilization term penalizing face-based residuals and polynomially consistent up to degree .
The local contributions obtained from these two ingredients are then assembled following a standard, finite element-like procedure. The resulting discretization has several appealing features, the most prominent of which are summarized hereafter:
It is valid for fairly general polytopal meshes.
The construction is dimension-independent, which can significantly ease the practical implementation.
It enables the local adaptation of the approximation order, a highly desirable feature when combined with a regularity estimator (whose development will be addressed in a separate work).
It exhibits only a moderate dependence on the diffusion coefficient .
It has a moderate computational cost thanks to the possibility of eliminating elemental DOFs locally via static condensation.
Parallel implementations can be simplified by the fact that processes communicate via skeletal unknowns only.
The seminal works on the p- and hp-conforming finite element method on standard meshes date back to the early 80s; cf. [7, 6, 5]. Starting from the late 90s, nonconforming methods on standard meshes supporting arbitrary-order have received a fair amount of attention; a (by far) nonexhaustive list of contributions focusing on scalar diffusive problems similar to the one considered here includes [29, 10, 28, 24, 31]. The possibility of refining both in h and in p on general meshes is, on the other hand, a much more recent research topic. We cite, in particular, hp-composite [3, 25] and polyhedral  discontinuous Galerkin methods, and the two-dimensional virtual element method proposed in .
The main novelty of the paper are hp-energy- and -estimates for HHO methods summarized in Section 3.2. These are the first results of this kind for HHO methods, and among the first for discontinuous skeletal methods in general (a prominent example of discontinuous skeletal methods are the hybridizable discontinuous Galerkin methods of ; cf.  for a precise study of their relation with HHO methods).
The cornerstone of the analysis is a novel extension of the classical Babuška–Suri hp-approximation results to regular mesh sequences in the sense of [16, Chapter 1] and arbitrary space dimension ; see Lemma 2.3 below. Similar results had been derived in  for and, under assumptions on the mesh different from the ones proposed here, in  for . A relevant difference with respect to classical approximation results based on the Bramble–Hilbert lemma (see, e.g., [23, Theorem 1.103]) is that we cannot rely here on the notion of reference element, since general polyhedral shapes are allowed. As a result, it turns out that a key point consists in showing that the regularity assumptions adapted to polyhedral meshes imply uniform bounds for the Lipschitz constant of mesh elements.
The resulting energy-norm estimate confirms the characteristic h-superconvergence behavior of HHO methods, whereas we have a more standard scaling as with respect to the polynomial degree of elemental DOFs. This scaling is analogous to the best available results for discontinuous Galerkin (dG) methods on rectangular meshes based on polynomials of degree , cf.  (on more general meshes, the scaling for the symmetric interior penalty dG method is , half a power less than for the hp-HHO method studied here). Classically, when elliptic regularity holds, the h-convergence order can be increased by 1 for the -norm. In our error estimates, the dependence on the diffusion coefficient is carefully tracked, showing full robustness with respect to its heterogeneity and only a moderate dependence with a power of on its local anisotropy when the error in the energy-norm is considered. These results improve the ones in  where, considering a different norm for the error, a power of 1 was found. Numerical experiments confirm the expected exponentially convergent behavior for both isotropic and strongly anisotropic diffusion coefficients on a variety of two-dimensional meshes.
The rest of the paper is organized as follows. In Section 2 we introduce the notation and state the basic results required in the analysis including, in particular, Lemma 2.3 (whose proof is detailed in Appendix A). In Section 3 we formulate the hp-HHO method, state our main results, and provide some numerical examples. The proofs of the main results, preceded by the required preparatory material, are collected in Section 4.
In this section we introduce the notation and state the basic results required in the analysis.
2.1 Mesh and Notation
Let denote a countable set of meshsizes having 0 as its unique accumulation point. We consider mesh sequences where, for all , is a finite collection of nonempty disjoint open polytopal elements such that and ( stands for the diameter of T). A hyperplanar closed connected subset F of is called a face if it has positive -dimensional measure and
either there exist distinct such that (and F is an interface)
or there exists such that (and F is a boundary face).
Interfaces are collected in , boundary faces in , and we let . It is assumed that the set of faces partitions the mesh skeleton in the sense that distinct faces have disjoint interiors and that . For all , the set collects the faces lying on the boundary of T and, for all , we denote by the normal vector to F pointing out of T.
The following assumptions on the mesh will be kept throughout the exposition.
Assumption 2.1 (Admissible Mesh Sequence).
We assume that is admissible in the sense of [16, Chapter 1], i.e., for all , admits a matching simplicial submesh and there exists a real number (the mesh regularity parameter) independent of h such that the following conditions hold:
for all and all simplices of diameter and inradius .
for all , all , and all such that .
Every mesh element is star-shaped with respect to every point of a ball of radius .
Assumption 2.2 (Compliant Mesh Sequence).
We assume that the mesh sequence is compliant with the partition on which the diffusion tensor is piecewise constant, so that jumps only occur at interfaces and, for all , we set
In what follows, for all , and denote the largest and smallest eigenvalue of , respectively, and the local anisotropy ratio.
2.2 Basic Results
For a subset X of Ω and an index q, denotes the Hilbert space of functions which are in together with their weak derivatives of order , equipped with the usual inner product and associated norm . When , we recover the Lebesgue space , and the subscript 0 is omitted from both the inner product and the norm. The subscript X is also omitted when . For a given integer , we denote by the space spanned by the restriction to X of d-variate polynomials of degree . For further use, we also introduce the -projector such that, for all ,
We recall hereafter a few known results on admissible mesh sequences and refer to [16, Chapter 1] and [14, 15] for a more comprehensive collection. By [16, Lemma 1.41], there exists an integer (possibly depending on d and ϱ) such that the maximum number of faces of one mesh element is bounded,
The following multiplicative trace inequality, valid for all , all , and all , is proved in [16, Lemma 1.49]:
where C only depends on d and ϱ. We also note the following local Poincaré’s inequality valid for all and all such that :
where when T is convex, while it can be estimated in terms of ϱ for nonconvex elements (cf., e.g., ).
The following functional analysis results lie at the heart of the hp-analysis carried out in Section 4.
Lemma 2.3 (Approximation).
There is a real number (possibly depending on d and ϱ) such that, for all , all , all integers , all , and all , there exists a polynomial satisfying, for all ,
See Appendix A. ∎
Lemma 2.4 (Discrete Trace Inequality).
There is a real number (possibly depending on d and ϱ) such that, for all , all , all integers , and all , it holds
In this section, we formulate the hp-HHO method, state our main results, and provide some numerical examples.
3.1 The hp-HHO Method
We present in this section an extension of the classical HHO method of  accounting for variable polynomial degrees. Let a vector of skeletal polynomial degrees be given. For all , we denote by the restriction of to , and we introduce the following local space of DOFs:
The standard HHO notation is used for a generic element of . We define the local potential reconstruction operator
such that, for all and all ,
Note that computing according to (3.2) requires to invert the -weighted stiffness matrix of , which can be efficiently accomplished by a Cholesky solver.
We define on the local bilinear form such that
where, for all , we let and the face-based residual operator
is such that, for all ,
The first contribution in is in charge of consistency, whereas the second ensures stability by a least-square penalty of the face-based residuals . This subtle form for ensures that the residual vanishes when its argument is the interpolate of a function in , and is required for high-order h-convergence (a detailed motivation is provided in [18, Remark 6]).
The global space of DOFs and its subspace with strongly enforced boundary conditions are defined, respectively, as
Note that interface DOFs in are single-valued, meaning that they match from one element to the adjacent one. We use the notation
for a generic DOF vector in and, for all , we denote by its restriction to T. For further use, we also introduce the global interpolator
such that, for all ,
and denote by its restriction to .
The hp-HHO discretization of problem (1.2) consists in seeking such that
where the global bilinear form on and the linear form on are assembled element-wise setting
Remark 3.1 (Static Condensation).
Using a standard static condensation procedure, it is possible to eliminate element-based DOFs locally and solve (3.7) by inverting a system in the skeletal unknowns only. For the sake of conciseness, we do not repeat the details here and refer instead to [12, Section 2.4]. Accounting for the strong enforcement of boundary conditions, the size of the system after static condensation is
Remark 3.2 (Finite Element Interpretation).
A finite element interpretation of the scheme (3.7) is possible following the extension proposed in [12, Remark 3] of the ideas originally developed in  in the context of nonconforming virtual element methods. For all , we denote by the usual jump operator (the sign is irrelevant), which we extend to boundary faces setting . Let
where, for all , we have introduced the local space
It can be proved that, for all ,
and it can be proved that is the unique element of such that with unique solution to (3.7).
3.2 Main Results
We next state our main results. The proofs are postponed to Section 4. For all , we denote by and the seminorms defined on by the bilinear forms and , respectively, and by the seminorm defined by the bilinear form on . We also introduce the penalty seminorm such that, for all ,
Note that is a norm on the subspace with strongly enforced boundary conditions (the arguments are analogous to those of [17, Proposition 5]). We will also need the global reconstruction operator
such that, for all ,
Finally, for the sake of conciseness, throughout the rest of the paper we denote by the inequality with real number independent of h, , and, for local inequalities on a mesh element , also of T.
Our first estimate concerns the error measured in energy-like norms.
Theorem 3.3 (Energy Error Estimate).
Assuming that for all , it holds
Consequently, denoting by the broken gradient on , we have that
See Section 4.3. ∎
In (3.10) and (3.11), we observe the characteristic improved h-convergence of HHO methods (cf. ), whereas, in terms of p-convergence, we have a more standard scaling as (i.e., half a power more than discontinuous Galerkin methods based on polynomials of degree , cf., e.g., ). In (3.11), we observe that the left-hand side has the same convergence rate (both in h and in p) as the interpolation error
as can be verified combining (4.2) and (4.4) below. Note that, in this case, the p-convergence is limited by the second term, which measures the discontinuity of the potential reconstruction at interfaces. An inspection of formulas (3.10) and (3.11) also shows that the method is fully robust with respect to the heterogeneity of the diffusion coefficient, while only a moderate dependence (with a power of ) is observed with respect to its local anisotropy ratio.
For the sake of completeness, we also provide an estimate of the -error between the piecewise polynomial fields and such that
To this end, we need elliptic regularity in the following form: For all , the unique element such that
satisfies the a priori estimate
The following result is proved in Section 4.4.
Convergence with p-refinement (regular test case).
Convergence with p-refinement (Le Potier’s test case).
3.3 Numerical Examples
We close this section with some numerical examples. The h-convergence properties of the method (3.7) have been numerically investigated in [18, Section 4]. To illustrate its p-convergence properties, we solve on the unit square domain the homogeneous Dirichlet problem with exact solution and right-hand side f chosen accordingly. We consider two values for the diffusion coefficients:
where denotes the identity matrix of dimension 2, , and . The choice (“regular” test case) yields a homogeneous isotropic problem, while the choice (“Le Potier’s” test case ) corresponds to a highly anisotropic problem where the principal axes of the diffusion tensor vary at each point of the domain. Figures 1 and 2 depict the energy- and -errors as a function of the number of skeletal DOFs (cf. (3.8)) when for all and for the proposed choices for on the meshes of Figure 3. In all the cases, the expected exponentially convergent behavior is observed. Interestingly, the best performance in terms of error vs. is obtained for the Cartesian and Voronoi meshes. A comparison of the results for the two values of the diffusion coefficients allows to appreciate the robustness of the method with respect to anisotropy.
Meshes considered in the p-convergence test of Section 3.3. The triangular, Cartesian, refined, and staggered meshes originate from the FVCA5 benchmark ; the hexagonal mesh was originally introduced in ; the Voronoi mesh was obtained using the PolyMesher algorithm of .
4 Convergence Analysis
In this section we prove the results stated in Section 3.2.
4.1 Consistency of the Potential Reconstruction
Preliminary to the convergence analysis is the study of the approximation properties of the potential reconstruction defined by (3.2) when its argument is the interpolate of a regular function. Let a mesh element be fixed. For any integer , we define the oblique elliptic projector such that, for all , and it holds
Proposition 4.1 (Characterization of ).
It holds, for all ,
For a generic , letting in (3.2), we infer, for all ,
where we have used the fact that and (cf. definition (3.1) of ) to cancel the projectors in the second line, and an integration by parts to conclude. ∎
We next study the approximation properties of , from which those of follow in the light of Proposition 4.1.
Lemma 4.2 (Approximation Properties of ).
For all integers , all mesh elements , all , and all , it holds
By definition (4.1) of , it holds
Hence, using (2.5) with , it is readily inferred that
To prove the second bound in (4.2), use the triangle inequality to infer
For the second term, we have
where we have used the discrete trace inequality (2.6) in the first line, the triangle inequality in the second line, the estimate (4.3) in the third, and the approximation result (2.5) with to conclude. To obtain the third bound in (4.2), after recalling that , we apply the local Poincaré inequality (2.4) to infer
and use the first and third bound in (4.2) to estimate the various terms. ∎
4.2 Consistency of the Stabilization Term
The consistency properties of the stabilization bilinear form defined by (3.4) are summarized in the following lemma.
Lemma 4.3 (Consistency of the Stabilization Term).
For all , all , and all , it holds
Let and and set, for the sake of brevity,
Using the triangle inequality and the -stability of , we infer,
For the first term, the approximation properties (4.2) of (with and ) readily yield
4.3 Energy Error Estimate
Proof of Theorem 3.3.
We start by noting the following abstract error estimate:
with nonconformity error
To prove (4.8), it suffices to observe that
where we have used the definition of the -norm in the first line, the linearity of in its first argument in the second line, and the discrete problem (3.7) in the third. The conclusion follows dividing both sides by , using linearity, and passing to the supremum.
We next bound the nonconformity error for a generic vector of DOFs . A preliminary step consists in finding a more appropriate rewriting for . Observing that a.e. in Ω, integrating by parts element-by-element, and using the continuity of the normal component of across interfaces together with the strongly enforced boundary conditions in to insert into the second term in parentheses, we infer
Setting, for the sake of conciseness (cf. Proposition 4.1),
and using the definition (3.2) of with , we have
For the second term, the Cauchy–Schwarz inequality followed by (4.4) (with ) readily yields
Using (4.14)–(4.15) to estimate the right-hand side of (4.13), applying the Cauchy–Schwarz inequality, and passing to the supremum yields (3.10). To prove (3.11), it suffices to observe that, inserting and using the triangle inequality, we obtain
Remark 4.4 (Nonconformity Error).
The name nonconformity error for the quantity (4.9) is reminiscent of the Finite Element literature. As a matter of fact corresponds to the contribution that appears in the estimate provided by the Second Strang Lemma when nonconforming methods are considered; see, e.g., [23, Lemma 2.25] and, in particular, the third term in the right-hand side of [23, (2.20)].
Proposition 4.5 (Estimate of Boundary Difference Seminorm).
It holds, for all ,
Let , , and set, for the sake of brevity, . We have, for all ,
where we have used the fact that (cf. (3.1)) to infer that and thus insert in the first line, added and subtracted in the second line, used the triangle inequality together with the definition (3.5) of the face-based residual and the -stability of in the third. To conclude, we observe that, if , then the discrete trace inequality (2.6) followed by Poincaré’s inequality yield
while, if , then
where we have inserted in the first line, used the discrete trace inequality (2.6) in the second line, the -optimality of together with the approximation properties (2.5) (with and ) in the third line, and we have concluded observing that and using the local Poincaré inequality (2.4) to infer
4.4 -Error Estimate
Proof of Theorem 3.4.
For the sake of brevity, we also let (recall the definition (3.9) of ), so that for all . We start by observing that
where we have used the fact that a.e. in Ω followed by element-by-element partial integration together with the continuity of the normal component of across interfaces and the strongly enforced boundary conditions in to insert into the last term.
In view of adding and subtracting to the right-hand side of (4.19), we next provide two useful reformulations of this quantity. First, we have
Using the Cauchy–Schwarz inequality, the approximation properties (4.2) of (with and ) together with the consistency properties (4.4) of (with ) for the first factor, and the bound (4.16) for the second factor, we get
For the second term, the Cauchy–Schwarz inequality followed by the approximation properties (4.2) of (with ) and (with ), and the consistency properties (4.4) of (with and for the first and second factor, respectively) yield
Finally, for the third term we have, when ,
while, when ,
where we have used the optimality of in the -norm to pass to the second line and the approximation properties (2.5) of to conclude. Using (4.23)–(4.26) to bound the right-hand side of (4.22), and recalling the energy error estimate (3.10) and elliptic regularity (3.13) concludes the proof. ∎
This is some additional text, but it also contains. and some more information.
A Proof of Lemma 2.3
Let be a L-Lipschitz set (that is, such that its boundary can be locally parametrized by means of L-Lipschitz functions) with , and fix and a d-cube containing . In [6, Lemma 3.1], the following is shown: Suppose that is a cube, then given a function , there exists a polynomial that satisfies
for . Actually, by analyzing the proof of that result, we can reinforce that statement not only for cubes, but for sets which are regular enough, in the sense that there has to exist an extension operator such that
So the problem is to show the existence of such an extension (in any dimension ). The minimal assumptions such that this is doable are studied in [32, Theorems 5 and 5, p. 181]. Namely, by means of a careful inspection of [32, Theorems 5 and 5] and their proofs, and in particular formulas (25), (30) and the end of the proof of Theorem 5 at p. 192, we get that the constant C in (A.2) depends on the Lipschitz constant L and on the (minimal) number of L -Lipschitz coverings of , that is, the number of open sets which cover and in each of whom can be parametrized by means of an L-Lipschitz function.1 Thus, to get the hp-estimate (2.5), we have to show two facts:
second, that (A.1) actually holds with a uniform bounding constant for the elements of the mesh.
(i) Proof of (2.5) for regular elements. Assume for the moment that any satisfies the regularity assumptions needed to have the extension (A.2), and thus (A.1), with in place of . We additionally assume, without loss of generality, that the barycenter of T (and thus of ) is . Then, by homogeneity, we get that, for every , letting ,
Thus, setting , , and with a generic polynomial of degree l, we get by (A.1) (applied to and in place of v and , respectively),
Now, [22, Theorem 3.2] states that
for any . Thus by the definition of the -norm we conclude that
Notice that we exploited again (A.3) to scale back on T, and the fact that .
(ii) Proof of regularity under Assumption 2.1. To conclude the proof, we are left to show that Assumption 2.1 entails a uniform bound only in terms of ϱ for the Lipschitz constant of every element . To this aim, consider . Then, for some (convex) element of the submesh contained in T. Since the sets S form a submesh of , a uniform bound on the Lipschitz regularity of would imply a bound on the Lipschitz regularity of . Thus, we focus on the regularity of S. Since S is convex, we can cover by means of open sets , such that admits a local convex (and thus Lipschitz) parametrization , i.e., there exists an orthogonal coordinate system such that is the graph of a Lipschitz function . This bound on the number of open sets is crucial to get [32, Theorem 5] to work (clearly, thanks to (2.2), the bound on the number of Lipschitz coverings of T is bounded by a constant ). We claim that each is -Lipschitz.
Suppose that and set . Up to a rotation and a rescaling, we can suppose that and . Let now be the inradius of S and be its diameter. By Assumption 2.1, we know that . Let be a ball contained in S of radius . Up to a further rotation of center of the coordinate system, we can suppose that is centered on the -axis. In place of , it is useful to consider its Lipschitz extension defined by, denoting by the usual Euclidian norm,
We know that is Lipschitz on and that (see for instance [2, Proposition 2.12]). Moreover, it is clear that is convex on . The fact that and it is centered on the -axis (without loss of generality, we can suppose that its center is with ) translates into the fact that is contained in the epigraph of and its center has distance from at most . Let now , where is the subdifferential of . Then, for every we have
By choosing , with , we get the inequality
Since the epigraph of contains , which is centered at a height less than on the -axis, and by the convexity of , we have that the truncated cone
and so, by Assumption 2.1,
Let now . Then
Since is arbitrary, this shows that , and so that is -Lipschitz.
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About the article
Published Online: 2017-06-09
Funding Source: Tel Aviv University
Award identifier / Grant number: project HHOMM (ANR-15-CE40-0005)
We acknowledge the support of Tel Aviv University through project HHOMM (ANR-15-CE40-0005).
Citation Information: Computational Methods in Applied Mathematics, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2017-0009.
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