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A Hybrid High-Order method for Leray-Lions elliptic equations on general meshes

Daniele A. Di Pietro, Jérôme Droniou

arXiv:1508.01918v3math.NAcs.NA

TL;DR

Steady Leray–Lions problems call for an HHO discretization that supports arbitrary orders and general polytopal meshes. The paper builds it from local gradient reconstruction and high-order stabilization, and proves strong convergence of discrete solutions in L^q(Ω) for q < p*.

  • Problem

    The paper addresses numerical approximation of steady Leray–Lions equations while supporting arbitrary polynomial orders and general polytopal meshes.

    Source: §1 · p. 1

  • Method

    The HHO method combines element- and face-based polynomial degrees of freedom with a local gradient reconstruction and a generalized high-order stabilization.

    Source: §1 · p. 2

  • Results

    The discrete solutions converge strongly in L^q(Ω) for every q < p*.

    Source: §4.3 · p. 9

  • Takeaways & Limitations

    For k = 0, the method belongs to the mixed-hybrid Mimetic Finite Differences, Hybrid Finite Volumes, and Mixed Finite Volumes family, and tests show good approximation on varied meshes.

    Source: §8 · p. 21

    A simplification increases element-based degrees of freedom, making local reconstruction and static-condensation problems more onerous.

    Source: §4.3 · p. 9

Abstract

from arXiv · show

In this work, we develop and analyze a Hybrid High-Order (HHO) method for steady non-linear Leray-Lions problems. The proposed method has several assets, including the support for arbitrary approximation orders and general polytopal meshes. This is achieved by combining two key ingredients devised at the local level: a gradient reconstruction and a high-order stabilization term that generalizes the one originally introduced in the linear case. The convergence analysis is carried out using a compactness technique. Extending this technique to HHO methods has prompted us to develop a set of discrete functional analysis tools whose interest goes beyond the specific problem and method addressed in this work: (direct and) reverse Lebesgue and Sobolev embeddings for local polynomial spaces, $L^{p}$-stability and $W^{s,p}$-approximation properties for $L^{2}$-projectors on such spaces, and Sobolev embeddings for hybrid polynomial spaces. Numerical tests are presented to validate the theoretical results for the original method and variants thereof.

1 Introduction

The paper develops an HHO method for steady nonlinear Leray–Lions equations that supports arbitrary polynomial orders and general polytopal meshes. Its convergence analysis uses compactness and is accompanied by discrete functional-analysis results for local and hybrid polynomial spaces.

1 Introduction

  • The method supports arbitrary polynomial orders, including k = 0, on general polytopal meshes, with element-local coupling and static condensation of element unknowns.The construction is dimension-independent and also supports nonmatching interfaces.Source: §1 · p. 1
  • HHO uses discontinuous polynomial degrees of freedom on elements and faces, combining a discrete gradient with a high-order penalty that vanishes for degree-(k+1) polynomials.Only face-based degrees of freedom are globally coupled.Source: §1 · p. 2
  • For the nonlinear problem, the gradient reconstruction uses all vector-valued polynomials of degree at most k, while the generalized stabilization preserves the degree-(k+1) vanishing property.For interpolants of smooth functions, the discrete gradient commutes with the L2-projector and thereby has Lp-stability properties.Source: §1 · p. 2
  • The convergence proof uses a compactness argument that needs only the solution regularity inherent to the weak formulation, but does not provide a convergence-rate estimate.Rate analysis for smooth solutions is deferred to future work.Source: §1 · p. 2
  • Local polynomial spaces admit direct and reverse Lebesgue and Sobolev embeddings; the reverse inequalities require only non-degeneracy of the element geometry and support HHO stability.The reverse inequalities bound the largest-exponent (semi-)norm by the lowest-exponent (semi-)norm.Source: §1 · pp. 2–3
  • The local L2-projector is Lp-stable for every p from 1 to infinity and has optimal approximation properties, including in the W^{s,p} setting.The results hold under very general geometric assumptions.Source: §1 · p. 3
  • For hybrid polynomial spaces with element and face degrees of freedom, the paper proves discrete Sobolev embeddings and compactness for the discrete gradient reconstruction.The analysis introduces a suitable discrete W^{1,p}-like norm.Source: §1 · p. 3

2 Continuous setting

This section specifies assumptions on the nonlinear operator and gives the weak formulation for the steady Leray–Lions problem. It also discusses model examples and conditions under which uniqueness can be obtained.

2 Continuous setting

  • The operator is assumed to be Carathéodory, monotone in its gradient argument, and coercive with a p-growth lower bound.The section sets p and its dual and Sobolev exponents before presenting the classical weak formulation.Source: §2 · p. 3
  • The p-Laplace equation is presented as the simplest type of Leray–Lions operator.The discussion also identifies glacier-motion and turbulent-flow models as applications.Source: §2 · p. 4
  • When the operator is independent of the solution value and strictly monotone, the weak solution is unique; dependence on the solution value requires stronger monotonicity and Lipschitz continuity in that variable.Without these stronger conditions, the solution is usually not unique.Source: §2 · p. 4

3 Discrete setting

The discrete setting specifies admissible polytopal mesh sequences and local polynomial spaces, then establishes stability and approximation properties for L2-projectors, including trace estimates.

3.1 Assumptions on the mesh

  • Mesh faces are classified as interfaces or boundary faces, with element-face incidence and outward normals defining the local mesh structure.The mesh sequence is refined with maximum element diameter h tending to zero.Source: §3.1 · p. 4
  • Admissible meshes admit a matching shape-regular simplicial submesh, with simplex sizes controlled relative to their parent elements; the submesh is theoretical, not part of the discretization.The mesh assumptions also bound face counts and element-neighbor counts in terms of the regularity parameter.Source: §3.1 · pp. 4–5

3.2 Basic results on local polynomial spaces

  • Local polynomial spaces comprise polynomials of total degree at most l on an element or face, and the framework also allows curved-boundary subsets.The L2-projector maps integrable functions to these spaces through polynomial moment matching.Source: §3.2 · p. 5
  • For every p in [1, infinity], the L2-projector is Lp-stable under a geometric regularity assumption, with a constant depending on dimension, regularity, degree, and p.The stated bound is ||π_l g||_Lp(U) ≤ C||g||_Lp(U).Source: §3.2 · p. 5
  • The L2-projector has optimal W^{s,p}-approximation properties for s from 1 to k+1, and the result extends to finite unions of suitably star-shaped domains.This extension lets the approximation result apply to admissible mesh elements formed from a finite number of simplices.Source: §3.2 · pp. 5–6
  • For element traces, the projector error is bounded in broken face norms by h_T^{s-m}|v|_{W^{s,p}(T)} for m from 0 to s−1 under the stated simplicial-union geometry.The trace estimate requires stronger geometric assumptions than the volume approximation result.Source: §3.2 · p. 6
  • The projector is also W^{s,p}-stable on U and on element-face traces under the respective volume and mesh-element assumptions.The trace bound combines the element seminorm with the broken face seminorm of order s−1.Source: §3.2 · p. 6

4 The Hybrid High-Order method

The method combines local gradient and potential reconstructions with stabilization in a hybrid polynomial discretization. The section states existence and convergence results and reports numerical tests across polynomial degrees and mesh families.

4.1 Local degrees of freedom, interpolation and reconstructions

  • The local degrees of freedom pair element polynomials with face polynomials, from which the method builds gradient and potential reconstructions.The local interpolation operator is defined for W^1,1 functions, and the gradient reconstruction has a commuting property and optimal approximation properties.Source: §4.1 · pp. 6–7

4.2 Global degrees of freedom, interpolation and reconstructions

  • Global degrees of freedom patch the interface face values, while global reconstructions and a broken polynomial representation are assembled over the mesh.The global interpolation operator is defined on W^1,1(Ω).Source: §4.2 · p. 7

4.3 Discrete problem and main results

  • The discrete formulation combines a reconstructed-gradient contribution with stabilization designed to vanish on interpolated polynomials of degree at most k+1.A more naive face-to-element difference penalty would vanish only through degree k and could prevent optimal linear-case convergence orders.Source: §4.3 · p. 8
  • The simplification comes at the price of more element-based DOFs, making local operator-reconstruction computations and static condensation of element-based unknowns more onerous.For p=4, the method corresponding to l=k+1 shows no significant convergence advantage from taking element-based DOFs in Pk+1(T).Source: §4.3 · p. 9, §4.4 · p. 10
  • The discrete problem has at least one solution, and its approximations converge subsequentially to a weak solution under the stated assumptions.With strict monotonicity, the theorem gives additional convergence of the reconstructed gradient; when a is independent of s, strictly monotone, the discrete and continuous solutions are unique and the whole sequence converges.Source: §4.3 · p. 9

4.4 Numerical examples

  • Tests use the p-Laplace problem on the unit square with polynomial degrees k=0,…,4 and triangular, Cartesian, and distorted hexagonal mesh families.The analytical solution is u(x)=exp(x1+πx2), with non-homogeneous Dirichlet data inferred from that solution.Source: §4.4 · p. 9
  • For p=3 and p=4, increasing k improves convergence rates; rates are not expected to be optimal except for p=2 and depend on solution regularity and p.The l=k+1 variant for p=4 shows no significant convergence advantage over the method in Figure 4.Source: Figure 3 · p. 10, Figure 4 · p. 10, §4.4 · p. 10

5 Discrete functional analysis tools in hybrid polynomial spaces

This section develops discrete functional analysis tools for the convergence analysis of hybrid polynomial spaces, including norm equivalences, discrete Sobolev embeddings, and compactness results. It completes the compactness argument by establishing strong convergence results for interpolants, including the discrete gradient and stabilization contribution.

5.1 Discrete W 1,p-norms

  • The discrete W^1,p norm is uniformly equivalent to local seminorms based on the discrete gradient and potential reconstructions, together with the face penalty contribution.The equivalence constants do not depend on the mesh, element, or local degrees of freedom.Source: §5.1 · p. 11

5.2 Discrete Sobolev embeddings

  • The discrete Sobolev embedding bounds hybrid functions in L^q(Ω) for 1 ≤ q ≤ p* when p < d, and for every finite q when p ≥ d.Taking q = p yields a discrete Poincaré inequality.Source: §5.2 · p. 13

5.3 Compactness

  • A sequence bounded in the discrete W^1,p norm has a subsequence converging strongly in L^q(Ω) for every q < p* to a limit in W^1,p_0(Ω), for both the hybrid functions and their potential reconstructions.The reconstructed gradients converge weakly in L^p(Ω)^d to the limit’s gradient.Source: §5.3 · pp. 13, 15
  • For broken polynomial functions with zero average on each element, a Poincaré–Wirtinger–Sobolev inequality holds for p ≤ q ≤ p*, with a positive mesh exponent when q < p* and zero at q = p*.This inequality supports comparison between the hybrid function and its potential reconstruction.Source: §5.3 · pp. 13–14
  • In the compactness argument, the terms T1 and T2 tend to zero as h tends to zero.The passage states boundedness of the discrete function and its gradient for T1; it says T2 is handled similarly.Source: §5.3 · p. 15

5.4 Strong convergence of the interpolants

  • The interpolant’s discrete gradient and stabilization contribution converge strongly when its argument is a smooth function.Proposition 5.11 states the result for admissible mesh sequences and p in [1, ∞], in a framework broader than the smooth-function case needed for Theorem 4.6.Source: §5.4 · pp. 15–16

6 Convergence analysis

The convergence analysis establishes uniform estimates and uses discrete compactness to extract a weak solution limit. A discrete Minty argument identifies that limit, while strict monotonicity yields strong convergence of reconstructed gradients.

Existence and a priori estimates

  • The discrete problem has at least one solution, and its solution satisfies an a priori estimate uniform in h.Existence follows by applying a finite-dimensional surjectivity argument to the discrete operator.Source: §6 · p. 17

Convergence proof

  • Uniform estimates and discrete compactness yield a subsequence converging strongly in L^q for q<p* and weakly in W^1,p to a candidate limit.The limit lies in W^1,p_0(Ω).Source: §6 · p. 17
  • A discrete Minty argument identifies the candidate limit as a solution of the weak formulation.The proof passes to the limit in the discrete relation using projector approximation and strong convergence properties.Source: §6 · pp. 17–18
  • If a is strictly monotone, the reconstructed gradients converge strongly in L^p to the weak solution's gradient.The proof first identifies the pointwise limit, then uses equi-integrability and Vitali's theorem.Source: §6 · pp. 18–19

7 Other boundary conditions

The section extends the HHO scheme and convergence analysis to non-homogeneous Dirichlet and homogeneous Neumann boundary conditions. The Neumann analysis uses discrete functional inequalities and compactness for zero-average functions.

7.1 Non-homogeneous Dirichlet boundary conditions

  • For non-homogeneous Dirichlet data, lifting the boundary condition yields subsequential convergence of the discrete solution and reconstructed gradient to a weak solution.The limit satisfies the prescribed trace condition.Source: §7.1 · p. 19
  • The interpolant of a W^1,p function has a discrete norm bounded by that function’s W^1,p norm, element by element.The estimate is stated for every element of an admissible mesh sequence.Source: §7.1 · p. 19

7.2 Homogeneous Neumann boundary conditions

  • The Neumann scheme imposes zero global average, and its analysis decomposes discrete functions into piecewise-constant parts and higher-order variation.The decomposition underpins the discrete inequality and compactness arguments.Source: §7.2 · p. 20
  • For homogeneous Neumann conditions, a discrete Poincaré–Wirtinger–Sobolev inequality and a zero-average compactness result support the convergence analysis.The compactness result gives strong L^q convergence for q < p* along a subsequence.Source: §7.2 · pp. 20–21

8 Conclusion

The conclusion presents the HHO method as convergent for fully nonlinear Leray–Lions equations, including the p-Laplace model, and reports approximation tests across mesh types and orders.

8 Conclusion

  • The HHO method is extended to fully nonlinear Leray–Lions equations, including the p-Laplace model, with convergence proved without restrictive operator assumptions or unrealistic solution regularity.The lowest-order method belongs to the mixed-hybrid Mimetic Finite Differences, Hybrid Finite Volumes, and Mixed Finite Volumes family.Source: §8 · p. 21
  • The analysis develops discrete functional tools for Lp- and W^s,p-stability and approximation properties of L2-projectors on broken polynomial spaces.Source: §8 · p. 21
  • Numerical results demonstrate approximation properties on varied meshes and at both low and high orders.Source: §8 · p. 21

A Discrete functional analysis in local polynomial spaces

The appendix gathers discrete functional analysis results for local polynomial spaces, intended for broader use in polynomial-based discretizations.

A Discrete functional analysis in local polynomial spaces

A.1 Estimates in local polynomial spaces

This subsection presents estimates for local polynomial spaces, including direct and reverse Lebesgue and Sobolev embeddings, with stated geometric and degree-related boundaries.

A.1 Estimates in local polynomial spaces

  • Direct and reverse Lebesgue embeddings compare L^q and L^m norms of polynomials on sets satisfying the stated geometric condition.The reverse case beyond the classical q ≤ m embedding holds because the functions are polynomials.Source: §A.1 · p. 22
  • The reverse Sobolev embedding result is conditional: its stated range requires m ≥ r for open sets.The passage also identifies a failure case when m < r and m ≤ k.Source: §A.1 · p. 22
  • The local polynomial norm comparison is justified by mapping the set to a normalized geometry and using equivalence of norms on a finite-dimensional polynomial space.The resulting constant depends on δ, not on the geometry of the normalized set.Source: §A.1 · p. 22

A.2 Lp-stability and W s,p-approximation properties of L2-projectors

This section proves Lp-stability and W^{s,p}-approximation estimates for local L2-projectors, including approximation of their traces on polytopal cells.

A.2 Lp-stability and W s,p-approximation properties of L2-projectors

  • The local L2-projector is Lp-stable for every p in [1, infinity].For p = 2, orthogonal projection gives stability with constant 1.Source: §A.2 · p. 23
  • For star-shaped domains, L2-projectors onto degree-k polynomial spaces have W^{s,p}-approximation properties for 1 <= s <= k+1 and 1 <= p <= infinity.The domain is assumed star-shaped with respect to a ball whose radius is proportional to its diameter.Source: §A.2 · p. 23
  • The bulk approximation proof combines averaged Taylor polynomials with projector stability and bounds on the Taylor remainder.The remainder is controlled in W^{r,p} by a factor proportional to h_U^{s-r}.Source: §A.2 · p. 23
  • For polytopal cells formed from disjoint, shape-regular simplices, projected traces satisfy W^{m,p} approximation estimates for 0 <= m < s.The stated bound scales as h_T^{s-m} times the W^{s,p}(T) seminorm of the function.Source: §A.2 · p. 24
  • The trace approximation proof uses a continuous trace inequality valid for general p on cells triangulated by simplices comparable in size to the cell.The proof maps each simplex to a reference simplex and transfers the estimate by changes of variables.Source: §A.2 · p. 24

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