Residual-based a posteriori error estimates for hp-discontinuous Galerkin discretisations of the biharmonic problem

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Residual-based a posteriori error estimates for hp-discontinuous Galerkin discretisations of the

biharmonic problem

Zhaonan Dong, Lorenzo Mascotto, Oliver Sutton

To cite this version:

Zhaonan Dong, Lorenzo Mascotto, Oliver Sutton. Residual-based a posteriori error estimates for hp-discontinuous Galerkin discretisations of the biharmonic problem. SIAM Journal on Nu- merical Analysis, Society for Industrial and Applied Mathematics, 2021, 59 (3), pp.1273–1298.

�10.1137/20M1364114�. �hal-03107210v2�

hp-DISCONTINUOUS GALERKIN DISCRETISATIONS OF THE BIHARMONIC PROBLEM

ZHAONAN DONG, LORENZO MASCOTTO, AND OLIVER J. SUTTON

Abstract. We introduce a residual-baseda posteriori error estimator for a novelhp-version interior penalty discontinuous Galerkin method for the biharmonic problem in two and three dimensions. We prove that the error estimate provides an upper bound and a local lower bound on the error, and that the lower bound is robust to the local mesh size but not the local polynomial degree. The suboptimality in terms of the polynomial degree is fully explicit and grows at most algebraically. Our analysis does not require the existence of aC¹-conforming piecewise polynomial space and is instead based on an elliptic reconstruction of the discrete solution to theH² space and a generalised Helmholtz decomposition of the error. This is the firsthp-version error estimator for the biharmonic problem in two and three dimensions. The practical behaviour of the estimator is investigated through numerical examples in two and three dimensions.

AMS subject classification: 65N12, 65N30, 65N50.

Keywords: discontinuous Galerkin methods; adaptivity;hp-Galerkin methods; polynomial in- verse estimates; fourth order PDEs; a posteriori error analysis.

1. Introduction

Fourth-order problems are prominent in the theory of partial differential equations (PDEs), modelling physical phenomena such as the control of large flexible structures, bridge suspension, microelectromechanical systems, thin-plate elasticity, the Cahn-Hilliard phase-field model, and hyperviscous effects in fluid models. A prototypical fourth-order problem is the biharmonic problem, which arises in modelling the isotropic behaviour of thin plates.

Since the introduction of the globally C¹-conforming Argyris element in the 1960s [3], the biharmonic problem has been widely studied in the context of finite element methods; see also [25].

However, partly due to the sheer technicality of implementing C¹-conforming elements, several mixed and nonconforming approaches have been developed over the years. These impose lower smoothness requirements on the discrete function spaces, typically at the expense of larger or less well conditioned linear systems. For instance, families of C⁰-elements for Kirchhoff plates were developed in [10, 16, 26]; see also [13, 23] and the references therein. Discontinuous Galerkin (dG) methods have been employed, also inhpform, in e.g. [6, 24, 27, 29, 42–44].

Computablea posteriori error estimates and adaptivity for fourth order problems have received increasing attention over the last twenty years. For instance, we recall the conforming approximations of problems involving the biharmonic operator of [46], the treatment of Morley plates [9, 33], quadratic C⁰-conforming interior penalty methods [14] and general order dG methods [28] for the biharmonic problem, continuous and dG approximations of the Kirchhoff-Love plate [30], the dichotomy principle in a posteriori error estimates for fourth order problems [2], and the Ciarlet- Raviart formulation of the first biharmonic problem [20].

The central difficulty in employing conventional techniques to derive a posteriori error estimates for dG andC⁰-conforming interior penalty methods for the biharmonic problem lies in constructing an averaging operator to aC¹-conforming finite element space. Such an operator, which must satisfy optimal hp-approximation properties, is required to enable the stability of the continuous PDE operator to be applied to the error. An hp-version a posteriori error estimator for biharmonic problems has been presented in [7], relying on the assumption of the existence of the above aver- aging operator with optimalhp-approximation properties. In 2D, the averaging operator may be constructed for arbitrary polynomial degrees, based on conventional macro elements, see e.g. [16,27], while on tetrahedral meshes this is only possible forp= 3. However, also in 2D, an explicit analysis of optimalhp-approximation estimates is not available.

1

An alternative approach, recently proposed in [35] in the context of nonlinear PDEs in nondiver- gence form, is to reconstruct the solution intoC¹-conforming spaces introduced in [17, 45]. While this allows us to avoid problems with element geometries, it introduces the disadvantage in the current context that the resulting error estimate would gain an additional suboptimality of orderp^d indspatial dimensions, due to the repeated application of a polynomial inverse estimate apparently necessary for the analysis. This approach is discussed further in Remark 4.5 below.

The contribution of this paper is to give an explicit analysis of a residual-based a posteriori error estimator for a novelhp-version dG discretisation of the biharmonic problem. In particular, our analysis does not require aC¹-averaging operator, simultaneously addresses both 2D and 3D, and incorporates arbitrary polynomial degrees which may be variable over simplicial and tensor product meshes. Instead, the proof of the fact that the estimator forms an upper bound on the error is based on an elliptic reconstruction of the dG solution toH²and a generalised Helmholtz decomposition of the error, as used in error estimates for classical nonconforming elements [9, 19].

We further prove that the estimator forms a local lower bound on the error, using severalhp-explicit polynomial inverse estimates involving bubble functions and extension operators inspired by those of [40, 41]. The resulting lower bound is algebraically suboptimal with respect to the polynomial degree, guaranteeing that the estimator retains the same exponential convergence properties as the error for problems with point or edge singularities.

The analysis focusses on 2D and 3D meshes without hanging nodes, although the case of paral- lelogram or parallelepiped elements with hanging nodes is addressed in Remark 4.6. However, our analysis does not appear to directly extend to the case of simplicial meshes with hanging nodes due to certain missing technical results regarding the influence of hanging nodes onC⁰-conforming hp-version quasi-interpolation operators for H²-functions. Instead, we numerically demonstrate that the presence of hanging nodes has apparently little effect on the resulting scheme or estimator.

Arguments similar to those presented in this paper may be used to prove upper and lower bounds for the estimator for theC⁰-conforming interior penalty methods in [14, 16]; see Remark 4.9 below.

Outline of the paper.The formulation of the biharmonic problem and its discretisation via an interior penalty dG scheme is presented in Section 2. Section 3 contains certain hp-explicit approximation results and polynomial inverse and extension results required to derive the error estimate. The derivation of a computable error estimator that provides a local bound on the error, which is explicit in terms of the polynomial degree, is the topic of Section 4. We present 2D and 3D numerical results in Section 5 and draw some conclusions in Section 6.

Notation.We adopt standard notation for Sobolev spaces; see e.g. [1]. Given D ⊂ R^d, d = 2 or 3, we denote the Sobolev space of order s ∈ R over D by H^s(D), and let (·,·)s,D, k · ks,D, and| · |s,D,denote its associated inner product, norm and seminorm, respectively.

Let∇denote the gradient operator, and define the Laplacian ∆ =∇ · ∇, the bilaplacian ∆², and the Hessian matrixD²=∇∇^> operators. Givenφ∈H¹(Ω) andψ∈[H¹(Ω)]³, the vector-valued curl operator is defined as

curl(φ) = (−∂yφ, ∂xφ)^>, curl(ψ) = (∂yψ3−∂zψ2, ∂zψ1−∂xψ3, ∂xψ2−∂yψ3)^>.

Forv∈[H¹(Ω)]² with v= (v1, v2)^> andw ∈[H¹(Ω)]^3×3with rows w1,w2, andw3, the matrix- valuedcurloperator is defined as

curl(v) =

curl(v1), curl(v2)>

, curl(w) =

curl(w₁^>), curl(w₂^>), curl(w^>₃)>

. Throughout, c denotes a generic positive constant, which is independent of any discretisation parameters, but may depend on the dimension and shape-regularity constants of the mesh.

2. An interior penalty dG method for the biharmonic problem

We present the formulation of the biharmonic problem in Section 2.1, and introduce a novel interior penalty dG (IPdG) scheme in Section 2.3. The scheme is based on a mesh satisfying certain assumptions, which are discussed in Section 2.2.

2.1. The biharmonic problem. Let Ω⊂R^d, d= 2 or 3, be a bounded polygonal/polyhedral domain andf ∈L²(Ω). The biharmonic problem reads: find a sufficiently smoothu: Ω→Rsuch

Figure 1. Kites for triangular meshes. The continuous thick lines form the two elements, and the dashed line denotes the perimeter of the kite.

that

(∆²u=f in Ω

u=n_Ω· ∇u= 0 on∂Ω, (2.1)

wheren_Ωdenotes the unit outward normal vector on∂Ω. Define V :=H₀²(Ω), B(u, v) :=

Z

Ω

D²u:D²v,

where : denotes tensor contraction. A weak formulation of (2.1) reads: findu∈V such that

B(u, v) = (f, v)0,Ω ∀v∈V. (2.2)

The well posedness of problem (2.2) is proven e.g. in [15, Section 5.9]. Inhomogeneous boundary data can be addressed following [11]; see also Remark 4.10 below.

2.2. Meshes and polynomial degree distribution. We consider sequences of decompositionsTn

of Ω into disjoint shape-regular triangles or parallelograms in 2D, and tetrahedra or parallelepipeds in 3D. The set of faces ofTn is denoted byEn, and is split into a set of boundary facesE_n^B, which lie on∂Ω, and internal facesE_n^I =En\ E_n^B. The shape-regularity assumption implies that meshTn

is locally quasi-uniform, i.e., there exists a constantc_mesh≥1 such that, for elementsK₁, K₂∈ Tn

withK₁∩K₂6=∅,

c⁻¹_meshhK₁ ≤hK₂ ≤cmeshhK₁. (2.3) Here,hK and hF denote the diameter of the elementK ∈ Tn, and the faceF ∈ En, respectively.

These local mesh sizes form the piecewise constant mesh size functionh: Ω→R⁺ given by h(x) :=

(h_K ifx∈K for someK∈ T_n hF ifx∈F for someF ∈ En.

The meshes are assumed to contain no hanging nodes, although a technical argument outlined in Remark 4.6 below extends our results to cover hanging nodes in parallelogram or parallelepiped meshes.

Another consequence of the shape-regularity of Tn is the existence of a shape-regular kite Ke associated with each internal face F ∈ E_n^I such that Ke ⊂K1∪K2, where K1, K2 ∈ Tn are the elements meeting at F, and Ke is symmetric with respect to F. An example of such a kite is illustrated in Figure 1 for triangular elements. A 3D kite may be constructed analogously, as a hexahedron for tetrahedral meshes or an octahedron for cubic meshes.

The numerical scheme requires an integer polynomial degreep_K ≥2 associated with eachK∈ T_n. We suppose that there exists a constantcp≥1 such that, for allK1, K2∈ Tn withK1∩K26=∅,

c⁻¹_p pK₁ ≤pK₂ ≤cppK₁. (2.4) The local polynomial degrees are collected by the piecewise constant functionp: Ω→R⁺ with

p(x) :=







p_K ifx∈K for someK∈ Tn,

max(pK₁, pK₂) ifx∈F ∈ E_n^I, whereK1, K2∈ Tn meet atF, p_K ifx∈F ∈ E_n^B, whereF is a face ofK.

Finally, we define face jump and average operators. Letv be a scalar-, vector-, or matrix-valued function on Ω, smooth on eachK∈ Tn but possibly discontinuous across eachF ∈ E_n^I. ForF∈ E_n^I, letK⁺ andK⁻ be the two mesh elements meeting atF, and letv⁺ andv⁻ denote the restriction ofvto K⁺ andK⁻, respectively. The face average and jump operators onF are given by

{v}(x) :=1

2(v⁺(x) +v⁻(x)), JvK(x) :=v⁺(x)−v⁻(x) ∀x∈F,

respectively. These definitions are extended to boundary faces where{v}(x) =JvK(x) =v(x).

Remark 2.1. To simplify the notation, we avoid considering mixed meshes of simplicial and tensor product elements. Although we do not expect such meshes to pose significant difficulties, this assumption allows us to simplify the presentation of the quasi-interpolant in Proposition 3.3 below.

2.3. The interior penalty dG scheme. Given a meshTnand a polynomial degree distributionp, we introduce the dG space of discontinuous piecewise polynomial functions overTn as

V_n:={q_p∈L²(Ω) :q_p|K ∈PpK(K) for each K∈ T_n}.

For future convenience we introduce the following broken norms: givens >0, k · k²_s,Tn:= X

K∈Tn

k · k²_s,K. (2.5)

We define the lifting operatorL:Vn+V →[Vn]^d×d,d= 2, 3, as Z

Ω

L(un) :vn:=

Z

En

{n·(∇ ·vn)}JunK− Z

En

{(vn)n} ·J∇unK ∀vn∈[Vn]^d×d, (2.6) wherendenotes the unit normal vector to a face, with arbitrary orientation on internal faces and directed outward on boundary faces. LetD_n² denote the elementwise Hessian matrix operator, given by (D_n²v)_|K =D²(v_|K) on eachK∈ Tn. We also introduce the piecewise constant dG penalisation parametersσ :En →R⁺ and τ : En →R⁺. Then, we construct the interior penalty dG bilinear formBn :Vn×Vn→Ras

Bn(un, vn) :=

Z

Ω

D²_nun:D²_nvn+ Z

Ω

L(un) :D²_nvn+L(vn) :D²_nun

+ Z

En

σJunKJvnK+τJ∇unK·J∇vnK

.

Observe that the terms in the dG bilinear form involving the lifting operators are equivalent to Z

Ω

L(un) :D_n²vn+L(vn) :D_n²un

= Z

En

{n·(∇∆vn)}JunK− Z

En

{(D_n²vn)n} ·J∇unK +

Z

E_n

{n·(∇∆u_n)}Jv_nK− Z

E_n

{(D²_nu_n)n} ·J∇v_nK. We pose the following interior penalty dG scheme for approximating solutions to the biharmonic problem (2.2): findun∈Vn such that

Bn(un, vn) = (f, vn)0,Ω ∀vn∈Vn. (2.7) Forv∈V_n+V, we define the dG norm associated with the bilinear formB_n(·,·) as

kvk²_dG:=kD²_nvk²_0,Ω+ X

F∈En

kτ¹²J∇vKk²_0,F + X

F∈En

kσ¹²JvKk²_0,F. (2.8) The stability ofB_n in this norm follows from the properties ofLand a suitable choice ofσ andτ, as encapsulated in the following result, which may be proven by arguing as in [27, Lemma 5.1].

Lemma 2.2(Stability of the scheme). There exists a constantcs>0 such thatL satisfies kL(v_n)k²_0,Ω≤c_s

p⁶ h³

¹₂ Jv_nK

2 0,En

+

p² h

¹₂ J∇v_nK

2 0,En

.

Consequently, if we pick the dG penalisation parameters as σ=cσ

p⁶

h³, τ =cτ

p²

h, (2.9)

with cσ,cτ ≥2cs+¹₂, then the dG bilinear form satisfies Bn(v, v)≥ 1

2kvk²_dG, Bn(u, v)≤2kukdGkvkdG, for allu,v∈V +V_n, and discrete problem (2.7)is well posed.

Remark 2.3. The dG method (2.7)is based on the Hessian weak formulation of the biharmonic problem, rather than the Laplacian formulation used e.g in [27]. In particular, it may be viewed as an extension of the formulation used for the C⁰-conforming interior penalty method in [16], with additional face and penalisation terms to account for the fully discontinuous trial and test functions.

3. hp-explicit polynomial inverse and extension results

In this section, we present a variety ofhp-explicit approximation results and polynomial inverse and extension estimates, which are required for the error estimate of Section 4. Throughout, we suppose thatp ∈N, and we useb_K to denote the standard bubble function constructed on the polygon or polyhedronK as the product of the affine functions vanishing on each face ofK.

First, we recall the standard trace inequality from e.g. [15, Theorem (1.6.6)]. For this, we introduce the concept of chunkiness parameter of a domainω⊂R^d. We set

γ:= diam(ω)

ρ_max , (3.1)

whereρ_max denotes the maximum over the diameter of all possible balls contained inω.

The boundedness of chunkiness parameter (3.1) of an elementK∈ Tn is a consequence of the shape-regularity assumption in Section 2.2.

Proposition 3.1(Trace inequality). Given a bounded Lipschitz domain ω⊂R^d with diameterh and bounded chunkiness parameter (3.1), there exists a constant c >0depending only on ω such that

kvk²_0,∂ω ≤c h⁻¹kvk²_0,ω+kvk0,ω|v|1,ω

∀v∈H¹(ω). (3.2)

The following hp-explicit inverse estimates are well known. Estimate (3.3) was proven in [31, Theorem 4] with explicit constants. On the other hand, the 2D variant of (3.4) may be found in [48, Theorem 4.76], and the 3D case follows analogously.

Proposition 3.2(hp-explicit inverse estimates). Let K be a shape-regular triangle, parallelogram, tetrahedron, or parallelepiped, with diameterh, and letF be a face of K with diameter scaling ash.

Then, there exists a constant c >0independent of hor psuch that, for all qp∈P^p(K)if K is a triangle or tetrahedron, or qp∈Qp(K) ifK is a parallelogram or parallelepiped,

kq_pk_0,F ≤cph⁻¹²kq_pk_0,K. (3.3) and

|qp|1,K ≤cp²h⁻¹kqpk0,K. (3.4) The analysis in Section 4 below requires aC⁰-conforminghp-quasi-interpolant for functions that are not necessarily smooth. For this, we use a generalisation of Babuˇska-Suri operator [5, Lemma 4.5], constructed by combining it with the Karkulik-Melenk smoothing techniques from [34, Section 2].

Proposition 3.3(Karkulik-Melenk generalisation of the Babuˇska-Surihp-quasi-interpolant). Given a domain Ω ⊂ R^d partitioned into a mesh Tn of triangles, parallelograms, tetrahedra or paral- lelepipeds, there exists an operatorI:H^s(Ω)→Vn∩ C⁰(Ω) such that for all 0≤q≤s,

kv−Ivkq,T_n≤c

hmin(p+1,s)−q

p^s−q v _s,T

n

∀v∈H^s(Ω), (3.5)

where the constantc >0is independent ofhandp, and the broken norms above are defined in (2.5).

Now, we focus onhp-explicit inverse inequalities involving bubble functions. Their proof is based on the following two technical lemmata. The first may be proven by arguing as in [12].

Lemma 3.4(hp-explicit polynomial inverse estimates with bubble functions in 1D).LetIb= (−1,1).

Given0≤α≤β, there exists a constantc >0depending on αandβ but notpsuch that, for all qp∈Pp(I),b

Z

Ib

(1−x²)^αq_p(x)²≤cp^2(β−α) Z

Ib

(1−x²)^βq_p(x)², (3.6) and

Z

bI

(1−x)^αq_p(x)²≤cp^2(β−α) Z

Ib

(1−x)^βq_p(x)², Z

Ib

(1 +x)^αq_p(x)²≤cp^2(β−α) Z

Ib

(1 +x)^βq_p(x)². (3.7) The second technical lemma extends the above two inequalities to a quasi-1D result on the trapezoid

D=D(a, b, d) ={(x, y)∈R²:y∈[0, d], −1 +a y≤x≤1 +b y},

whered∈(0,1) anda,b∈Rsatisfy−1 +a d <1 +b d. Analogous arguments directly extend this result to 3D trapezoidal polyhedra.

Lemma 3.5(hp-polynomial inverse estimate with bubbles in quasi-1D trapezoids). Assume thatD has diameter hD ≈1. To each y^∗ ∈[0, d], associate the segment I(y^∗) =I^∗ = [ay^∗−1,1 +b y^∗], and let F :I^∗ →[−1,1] be affine withF(ay^∗−1) =−1 andF(1 +b y^∗) = 1. Introduce ψF(x) = 1−(F(x))²:I^∗→[0,1], and defineΦ∈ C⁰(D) such that, for somes∈N,

Φ(·, y^∗)∈P2s(I^∗), c1ψ_F^s(x)≤Φ(x, y^∗)≤c2ψ_F^s(x) ∀x∈I^∗, where the constants c₁,c₂>0 depend only on a,b,Φ, andy^∗.

Then, there exists a constant c >0, depending only onα,β, andΦ, such that, for allβ > α≥0, kΦ^α2q_pk0,D ≤c p^s(β−α)kΦ^β2q_pk0,D ∀qp∈Pp(D).

Proof. The proof is based on [39, Lemma D.2]. By assumption, Φ is a continuous function of y^∗. Therefore,c₁ andc₂ depend continuously ony^∗. Sincey^∗∈[0, d], c₁ andc₂attain their extremal valuesc₁= min_y∗∈[0,d](c₁(y^∗)) andc₂= max_y∗∈[0,d](c₂(y^∗)), which satisfy 0< c₁≤c₂, and

c1ψ^s_F(x)≤Φ(x, y^∗)≤c2ψ_F^s(x) ∀x∈I^∗. (3.8) Bounds (3.6) and (3.8) imply that

Z

I^∗

Φ(x, y^∗)^αqp(x, y^∗)²≤c^α₂ Z

I^∗

ψ_F^sα(x)qp(x, y^∗)²≤cc^α₂p^s(β−α) Z

I^∗

ψ_F^sβ(x)qp(x, y^∗)²

≤cc^α₂

c^β₁(p+ 1)^s(β−α) Z

I^∗

Φ(x, y^∗)^βqp(x, y^∗)²,

wherec is independent ofy^∗. The assertion follows by integrating overy^∗∈[0, d].

These two technical lemmata enable us to prove an inverse estimate for bubble functions on 2D and 3D elements, using a partitioning argument introduced by Melenk and Wohlmuth [41]; see also [39, 40].

Proposition 3.6 (hp-polynomial inverse estimate with bubbles). LetK ⊂R^d, d= 2 or 3, be a triangle, parallelogram, tetrahedron, or parallelepiped. Then, there exists a constant c >0, indepen- dent of handp, such that, for all qp∈Pp(K) ifK is a triangle or tetrahedron, orqp∈Qp(K)if

K is a parallelogram or parallelepiped,

kb_K^α2q_pk_0,K≤cp^d(β−α)kb_K^β2q_pk_0,K, −1

2 < α≤β. (3.9) Proof. The proof is similar to that of [39, Theorem D2] and for this reason we only sketch it. If K is a parallelogram or parallelepiped, the result follows from Lemma 3.5.

SupposeKis the triangle with vertices{(0,0),(1,0),(0,1)}. SplitKinto the overlapping subsets K= ∪⁶_i=1Di

∪ ∪³_i=1Pi

∪R,

whereDi,i= 1, . . . ,6 are the trapezoids depicted in Figure 2, andPi,i= 1,2,3, are the parallel- ograms shown in Figure 3. The remainderR, illustrated in Figure 4, is separated from∂K. The assertion follows by applying Lemma 3.5 on each trapezoidDi, using (3.7) and a tensor product argument on each parallelogram, and observing thatbK≈1 on the remainderR.

0,₁₂⁵

0,₁₂⁷

5 12,0

7 12,0

₁₉ 24,₂₄⁵ 17

24,₂₄⁷ ₅

24,¹⁹_{24 7}

24,¹⁷₂₄

D1

D2

₇ 12,₁₂⁵ ₅

12,₁₂⁷

₅ 12,0 ₇

12,0

0,₂₄⁵

0,₂₄⁷

0,¹⁷₂₄

0,¹⁹₂₄

D3

D4

7 12,₁₂⁵ ₅

12,₁₂⁷

5 24,0

7

24,0

5 24,0

0,₂₄⁷

0,¹⁷₂₄

17 24,0

D5

D6

Figure 2. TrapezoidsDi,i= 1, . . . ,6.

11 24,0 ₁₁

24,¹¹₂₄

0,¹¹₂₄

(0,0)

P1

11 24,¹_{2 11}

24,¹³₂₄ (0,1)

0,¹³₂₄

P2

₁₁ 24,0

₁₃ 24,¹¹₂₄

(1,0) ₁

12,¹¹₂₄

P3

Figure 3. ParallelogramsPi,i= 1,2,3.

Figure 4. The three white “small holes” inside triangleKdenote the remainderR.

When Kis a tetrahedron, we construct a similar overlapping decomposition consisting of paral- lelepipeds, trapezoidal polyhedra, and a remainder, and apply analogous arguments on each. The powerd(β−α) in (3.9) is due to the application of the tensor product version of inverse estimate (3.7) when dealing with the parallelepipeds.

In Proposition 3.6, we deduced the same hp-polynomial inverse estimate as in [49, Proposi- tion 3.45]. However, our proof extends to polygonal and polyhedral elements as well.

The followinghp-explicit polynomial weighted inverse estimate is proven in [49, Propositions 3.85, 3.86] with explicit constants in both 2D and 3D.

Proposition 3.7(hp-explicit polynomial weightedH¹toL²inverse estimate). LetKbe a triangle, parallelogram, a tetrahedron or parallelepiped with diameterh. Then, there exists a constant c >0, independent ofhandp, such that for allqp∈Pp(K)if K is a triangle or tetrahedron, and for all q_p∈Qp(K)ifK is a parallelogram or parallelepiped,

k∇(b_Kqp)k0,K≤cp

hkqpk0,K. (3.10)

Next, we prove anhp-explicit polynomial extension stability result.

Proposition 3.8(ε-weightedhp-explicit polynomial extension stability result). Let Kbe a shape- regular triangle, parallelogram, tetrahedron, or parallelepiped with diameter hK, and letF be a face

of K. The shape-regularity implies thathF scales likehK. Associated with face F, define

ΦF =

(bF if K is a triangle/tetrahedron

1 if K is a parallelogram/parallelepiped,

where b_F denotes the standard bubble function associated with face F. Then, in 2D and in 3D when F is a triangle, there exists an extension operator E : Pp(F) → H²(K) such that, for all qp ∈ P^p(F) and for all ε > 0 sufficiently small, there exists a constant c > 0, independent of h andp, such that

E(qp)_|F = ΦFq_p|F on F, (3.11)

kE(qp)k0,K≤ch

1 2

Kε¹²kqpk0,F, (3.12)

|E(qp)|1,K≤ch⁻_K¹²(εp⁴+ε⁻¹)¹²kqpk0,F, (3.13) kD²E(qp)k0,K ≤ch⁻

3 2

K (εp⁸+ε⁻³+ε⁻¹p⁴)¹²kqpk0,F. (3.14) In 3D whenF is a parallelogram, the above bounds are valid substitutingPp(F)withQp(F).

Proof. We suppose thatK=Kb is the reference element with diameter h

Kb = 1, and thatF =Fb has size 1. The general case then follows by a scaling argument.

When Kb is a parallelogram, the proof is based on that of [41, Lemma 2.6]. For the sake of completeness, we sketch the proof whenKb = [0,1]² andFb= [0,1]× {0}.

Introducing the extension operator

E(qp) =qp(1−y)e^−yε,

the properties (3.11) and (3.12) are immediate. To show (3.13), Lemma 3.7 implies that k∂_xE(q_p)k²

0,Kb =k∂_xq_pk²

0,bFk(1−y)e^−yεk²_0,[0,1]≤cp⁴εkq_pk²

0,Fb, and moreover

k∂yE(qp)k²_0,

Kb ≤ckqpk²_0,

Fb

ke^−yεk²_0,[0,1]+ 1

ε²ke^−yεk²_0,[0,1]

≤cε⁻¹kqpk²_0,bF.

We prove (3.14) analogously, observing that Lemma 3.7 provides k∂xxE(qp)k²_0,

Kb ≤cp⁸εkqpk²_0,

Fb, and we directly obtain

k∂yyE(qp)k²_0,

Kb =kqpk²_0,

Fbk∂yy((1−y)e^−yε)k²_0,[0,1]≤cε⁻³kqpk²_0,

Fb, and

k∂xyE(qp)k²_0,

Kb =k∂xqpk²_0,bFk∂y((1−y)e^−yε)k²_0,[0,1]≤cp⁴εkqpk²_0,

Fb.

Next, supposeKb is the triangle with vertices{(0,0),(1,0),(0,1)}, and, without loss of generality, takeFb= [0,1]× {0}. Defining the extension operator

E(qp) :=qpx(1−x−y)e^−yε, property (3.11) is once again immediate and (3.12) is valid because

kE(q_p)k²

0,Kb = Z 1

0

Z 1−x 0

q_p(x)²x²(1−x−y)²e^−2yεdy dx≤ Z 1

0

q_p²(x)dx Z 1

0

e^−2yεdy≤cεkq_pk²

0,Fb. To show (3.13), denote the derivative ofqp with respect to the local coordinate system onFbbyq⁰_p. Expanding the integral as before and applying Lemma 3.2 implies that

k∂xE(qp)k²_0,

Kb ≤ Z 1

0

q_p⁰(x)² Z 1−x

0

e^−2yεdy dx+ 2 Z 1

0

qp(x)² Z 1−x

0

e^−2yεdy dx

≤cε(kq_p⁰k²

0,Fb+kqpk²

0,bF)≤cεp⁴kqpk²

0,Fb, and similarly

k∂_yE(q_p)k²

0,Kb ≤ Z 1

0

(1 +ε⁻²)q_p(x)² Z 1−x

0

e^−2yεdy dx≤c(ε+ε⁻¹)kq_pk²

0,Fb≤cε⁻¹kq_pk²

0,Fb.

Finally, to prove (3.14) we note that k∂xxE(qp)k²_0,

Kb ≤c Z 1

0

(q⁰⁰_p(x)²+q_p⁰(x)²+qp(x)²) Z 1−x

0

e^−2yεdy dx

≤cε p⁸+p⁴+ 1 kq_pk²

0,Fb≤cεp⁸kq_pk²

0,Fb, and

k∂_yyE(q_p)k²

0,Kb ≤c Z 1

0

q_p(x)² Z 1−x

0

(ε⁻²(1−x−y)e^−yε)²dydx≤cε⁻³kq_pk²

0,bF. Similarly,

k∂_xyE(q_p)k²

0,Kb ≤c Z 1

0

Z 1−x 0

(1 +ε⁻¹(1−x−y))(q(x) +xq⁰(x)) +ε⁻¹xq(x)²

e^−2yεdydx

≤cε⁻¹ p⁴kqpk²_0,bF+kqpk²_0,

Fb

.

and the assertion follows. The 3D case follows by extending the arguments above.

Selecting ε=p⁻², the following result is an immediate consequence of Proposition 3.8.

Corollary 3.9(hp-polynomial extension stability result). Using the same notation and under the same assumptions as in Proposition 3.8, there exists an extension operator E:Pp(F)→H²(K) such that there exists a constantc >0, independent ofhandp, such that, for allqp∈Pp(F),

E(qp)_|F = ΦFq_p|F onF, (3.15)

kE(qp)k0,K+h¹_Kp⁻²|E(qp)|1,K+h²_Kp⁻⁴kD²E(q_p)k0,K≤ch_K²¹p⁻¹kqpk0,F. (3.16) In 3D whenF is a parallelogram, the spacePp(F)is replaced byQp(F).

4. Error estimator and a posteriori error analysis

In this section, we introduce a computable error estimator, which provides an upper bound and a local lower bound on the error measured in the dG norm in (2.8). These are the results of Theorems 4.3 and 4.7 below, respectively.

Definition 4.1(Error estimator). We introduce the error estimator η²:= X

K∈Tn

η²_K with η²_K :=η_K,1² +η_K,2² +η_K,3² +η_K,4² +η_K,5² +η_K,6² , where

η_K,1² :=

h p

2

(f−∆²un)

2 0,K

, η²_K,2:= 1

2 X

F∈E^K∩E_n^I

h p

³₂

Jn· ∇∆unK

2 0,F

,

η_K,3² :=1 2

X

F∈E^K∩E_n^I

h p

¹₂

J(D²un)nK

2

0,F, η²_K,4:= 1 2

X

F∈E^K

αF

h p

¹₂

J(D²un)tK

2 0,F,

η_K,5² :=1 2

X

F∈E^K

αFkp¹²τ¹²J∇unKk²_0,F, η²_K,6:= 1 2

X

F∈E^K

αFkσ¹²JunKk²_0,F,

with αF = 2 forF ∈ E_n^B andαF = 1 otherwise.

The notation for the tangential Hessian (D²un)t and tangential gradient (∇un)·t must be defined separately in 2D and 3D. In 2D, the unit tangential vector on a given face uniquely (up to its sign) satisfies t·n = 0. Hence, for v ∈ R² and M ∈ R^2×2, the termsv·t and Mt have their usual linear algebraic meaning. In 3D, where faces are spanned by two tangential vectors, we commit an abuse of notation and define the action of the tangent onv∈R³andM ∈R^3×3 as

v·t=v×n, Mt= [M₁^>×n, M₂^>×n, M₃^>×n]^>, (4.1) whereMi denotes rowiof M.

To avoid requiringC¹-conforming piecewise polynomial spaces, the analysis revolves around a variant of the elliptic reconstruction operator [38] and a Helmholtz decomposition.

We define the elliptic reconstructionuc∈V of the dG solutionun∈Vn to satisfy

B(u_c, v) =B_n(u_n, v) ∀v∈V. (4.2) The elliptic reconstruction is well defined for any un ∈ Vn due to the coercivity of B(·,·). It equivalently satisfies

Z

Ω

D_n²(u_c−u_n) :D²v= Z

Ω

L(un) :D²_nv ∀v∈V. (4.3) We shall combine this with the following Helmholtz decomposition, which is shown in [9, Lemma 1]

in 2D and [33, Lemma 5.2] in 3D.

Lemma 4.2(Helmholtz decomposition). LetΩ⊂R^dandΣ∈L²(Ω,R^d×d). There existξ∈H₀²(Ω), ρ∈L²₀(Ω) or(ρ_a, ρ_b, ρ_c)^>∈L²(Ω,R³), andΨ₂∈[H¹(Ω)]² orΨ₃∈H¹(Ω,R^3×3), such that

Σ=D²ξ+ρ_d+curlΨ_d, where ρ₂=

0 −ρ ρ 0

, ρ₃=





0 ρ_c −ρ_b

−ρ_c 0 ρ_a ρ_b −ρ_a 0



.

Moreover, there exists a positive constantcΩ, depending only onΩ, such that

kD²ξk0,Ω+kρ_dk0,Ω+kΨdk1,Ω≤cΩkΣk0,Ω. (4.4) Recall the following identity from [4]:

X

K∈T_n

Z

∂K

v·nσ= Z

En

{v·n}JσK+ Z

E_n^IJv·nK{σ} ∀v,σ∈[Vn+V]^d. (4.5) 4.1. Upper bound. Here, we show that the estimator of Definition 4.1 forms an upper bound of the error, using the elliptic reconstruction (4.2) and the Helmholtz decomposition of Lemma 4.2. For technical simplicity, we suppose thatf ∈Vn. More general cases may be treated by proceeding as in [21, 36], resulting in an additional data approximation term, which may dominate the estimator.

Theorem 4.3 (A posteriori error estimate for the hp-version dG scheme). Let u∈ H₀²(Ω) and un∈Vn solve the biharmonic problem (2.2)and the dG scheme (2.7), respectively, and letη be the error estimator of Definition 4.1. Then, there exists a constantc >0, independent ofhandp, such that

ku−unk²_dG≤cη².

Proof. We use the elliptic reconstruction uc in (4.2) to split the error into a conforming error ec:=u−uc and anonconforming error enc:=uc−un, which we estimate separately.

Estimate of conforming error ec.Recalling (4.2), the elliptic reconstructionuc satisfies kD²(u−u_c)k_0,Ω² =B(u−u_c, u−u_c) = (f, u−u_c)_0,Ω−B_n(u_n, u−u_c), and the dG scheme (2.7) implies that

kD²e_ck²_0,Ω= (f, e_c−v_n)_0,Ω−B_n(u_n, e_c−v_n) ∀vn∈V_n.

Choosingvn =Iec and denotingηn :=ec−Iec, whereIec∈Vn∩ C⁰(Ω) is the quasi-interpolant from Corollary 3.3 withs= 2, integrating by parts twice, applying the definition (2.6) of the lifting operatorL, and using the continuity ofηn in the relation (4.5) provide the error relation

kD²eck²_0,Ω= Z

Ω

(f−∆²_nun)ηn− Z

E_n^IJ(D²un)nK· {∇ηn}+ Z

E_n^IJn· ∇∆unKηn

− Z

Ω

L(u_n) :D²_nη_n− Z

E_n

τJ∇u_nK·J∇η_nK=:

5

X

j=1

T_j. We proceed with the estimate by treating each termTj separately.

Estimate (3.5) provides T₁≤

h p

²

(f−∆²_nu_n) _0,Ω

h p

−2

η_n

_0,Ω≤c

h p

²

(f −∆²_nu_n)

_0,ΩkD²_ne_ck0,Ω, and trace inequality (3.2) and estimate (3.5) give

T₂≤c

h p

¹₂

J(D²u_n)nK _0,EI

n

kD²e_ck0,Ω, T₃≤c

h p

³₂

Jn· ∇∆unK _0,EI

n

kD²e_ck0,Ω.

Similarly, recalling Lemma 2.2, trace inequality (3.2), and estimate (3.5), we find that T₄≤c(kσ¹²Ju_nKk²_0,En+kτ¹²J∇unKk²_0,En)kD²e_ck0,Ω, T₅≤c

p¹²τ¹²J∇unK

0,EnkD²e_ck0,Ω. Collecting the estimates for the individual termsT_j, we deduce that

kD²e_ck_0,Ω≤c

h p

2

(f−∆²_nu_n) _0,Ω+

h p

¹₂

J(D²u_n)nK _0,EI

n

+

h p

³₂

Jn· ∇∆unK _0,EI

n

+

p¹²τ¹²J∇unK

0,En+kσ¹²JunKk_0,En .

(4.6)

Estimate of nonconforming errore_nc.Applying the Helmholtz decomposition of Lemma 4.2 toΣ=D²_ne_nc, the skew symmetry of termρ implies that

kD²_nenck²_0,Ω= Z

Ω

D²_nenc:D²ξ+ Z

Ω

D_n²enc:curl Ψ. (4.7) To estimate the first term of (4.7), we use the smoothness ofξ∈H₀²(Ω) with the property (4.3), Lemma 2.2, and the stability of the Helmholtz decomposition (4.4) to find that

Z

Ω

D²_n(u_c−u_n) :D²ξ≤ kL(un)k0,ΩkD²ξk0,Ω≤(c_s)¹²(kσ²¹Ju_nKk²_0,En+kτ¹²J∇unKk²_0,En)¹²kD²ξk0,Ω

≤c(kσ¹²JunKk²_0,En+kτ¹²J∇unKk²_0,En)¹²kD_n²(uc−un)k0,Ω. (4.8) As for the second term of (4.7), we insert the vector-valued version I of the quasi-interpolant introduced in Corollary 3.3 withs= 1, recall that D_n² =∇∇^>, integrate by parts twice and use properties of elementary differential operators to obtain

Z

Ω

D²_ne_nc:curl Ψ= X

K∈Tn

Z

∂K

(D²_ne_nc)t

: (Ψ−IΨ) + Z

∂K

∇enc·((curl IΨ)n)

, (4.9) observing our notational convention (4.1) for the tangential component (D_n²enc)tof the Hessian.

To estimate the first term of (4.9), the relation (4.5), the continuity of Ψ−IΨ, and the fact thatuc ∈H₀²(Ω) give

X

K∈Tn

Z

∂K

(D_n²e_nc)t

: (Ψ−IΨ) = Z

En

J(D²e_nc)tK:{Ψ−IΨ}=− Z

En

J(D²u_n)tK: (Ψ−IΨ)

≤

h p

¹₂

J(D²un)tK 0,En

p h

¹₂

(Ψ−IΨ) 0,En

.

Applying (3.5), and using (3.4) and the stability of the Helmholtz decomposition (4.4) then imply ec

h p

¹₂

J(D²un)tK 0,En

kΨk1,Ω≤c

h p

¹₂

J(D²un)tK 0,En

kD²_nenck0,Ω. (4.10) To estimate the second term of (4.9), we note thatJ(curl IΨ)nKE_n^I = 0 since each entry ofΨ is inH¹(Ω). Consequently, each entry of IΨis also in H¹(Ω), implying curl IΨ∈ H(curl,Ω); see also [22] and [8, proof of Lemma 3.3]. Combining this with (4.5) andu_c∈H₀²(Ω), we find

X

K∈Tn

Z

∂K

∇enc·((curl IΨ)n) =−

Z

En

J∇unK·((curl IΨ)n)≤ kτ¹²J∇unKk_0,Enkτ⁻¹²(curl IΨ)nk_0,En. Trace inverse estimate (3.3), definition (2.9) ofτ, the stability ofI, which follows from (3.5), and the Helmholtz decomposition further yield

kτ⁻¹²(curl IΨ)nk0,E_n≤c1kcurl IΨk0,Ω≤c2kIΨk1,Ω≤c3kΨk1,Ω≤c4kD²_nenck0,Ω. Combined with estimates (4.7), (4.8), (4.9), and (4.10), this provides

kD²_nenck0,Ω≤c

kσ¹²JunKk_0,En+kτ¹²J∇unKk_0,En+

h p

¹₂

J(D²un)tK _0,E

n

. (4.11)

The result follows by combining (4.11) with (4.6), and recalling definition (2.8).