General theory of interpolation error estimates on anisotropic meshes

We propose a general theory of estimating interpolation error for smooth functions in two and three dimensions. In our theory, the error of interpolation is bound in terms of the diameter of a simplex and a geometric parameter. In the two-dimensional case, our geometric parameter is equivalent to the circumradius of a triangle. In the three-dimensional case, our geometric parameter also represents the flatness of a tetrahedron. Through the introduction of the geometric parameter, the error estimates newly obtained can be applied to cases that violate the maximum-angle condition.

Authors

• 5 publications
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1 Introduction

It is challenging to construct accurate and efficient finite element schemes for solving partial differential equations in various domains. Estimations of interpolation error are important in terms of ensuring the validity of schemes and their accuracy sometimes depends on geometric conditions of meshes of the domain. Many studies have imposed the condition of shape regularity to a family of meshes

Bra07 ; BreSco08 ; Cia02 ; ErnGue04 ; Ran17 ; i.e., triangles or tetrahedra cannot be too flat in a shape-regular family of triangulations.

In BabAzi76 , the shape regularity condition was relaxed to the maximum-angle condition, which refers to the maximum angle of each triangle in meshes being smaller than a constant . A family of triangulations under the maximum-angle condition allows the use of anisotropic finite element meshes. Anisotropic meshes have different mesh sizes in different directions, and the shape regularity assumption on triangulations is no longer valid on these meshes.

The question arises whether the maximum-angle condition can be relaxed further. The answer was given by HanKorKri12 ; KobTsu14 ; KobTsu15 ; KobTsu20 ; i.e., it is known that the maximum-angle condition is not necessarily needed to obtain error estimates.

The present paper proposes a general theory of interpolation error estimates for smooth functions that can be applied to, for example, Lagrange, Hermite, and Crouzeix–Raviart interpolations. For a -simplex , we introduce a new geometric parameter in Section 3.7 and the error of interpolations is bounded in terms of the diameter of and . We emphasize that we do not impose the shape regularity condition and the maximum-angle condition for the mesh partition.

Using the new parameter , we also propose error estimates for the Raviart–Thomas interpolation. The Raviart–Thomas interpolation error estimates on anisotropic meshes play an important role in first-order Crouzeix–Raviart finite element analysis. In AcoDur99 , the interpolation error analysis in the lowest-order case was given under the maximum-angle condition for triangles and tetrahedra. In AcoApe10 , the authors extended the results to the Raviart–Thomas interpolation with any order in two- and three-dimensional cases.

Meanwhile, in KobTsu18 , the lowest-order Raviart–Thomas interpolation error analysis under a condition weaker than the maximum-angle condition was introduced in the two-dimensional case. The analysis was based on the technique of Babuka and Aziz BabAzi76 . The technique requires a Poincaré-like inequality on reference elements. However, it is not easy to deduce the inequality in the three-dimensional case. To overcome this difficulty, we use the component-wise stability estimates of the Raviart–Thomas interpolation in reference elements introduced in AcoApe10 . We consequently have the Raviart–Thomas interpolation error estimates of any order in two- and three-dimensional cases under the relaxed mesh condition.

The remainder of the paper is organized as follows. Section 2 introduces notations and basic concepts of the Raviart–Thomas finite element. Section 3 introduces standard positions and the new geometric parameter. Further, we propose affine mappings and Piola transformations on standard positions and present the finite element generation. Section 4 proves interpolation error estimates of smooth functions that can be applied to, for example, Lagrange, Hermite, and Crouzeix–Raviart interpolations. Section 5 proves the Raviart–Thomas interpolation error estimate. Our main theorems are presented as Theorem 2 and Theorem 3.

2 Preliminaries

2.1 Function Spaces

Let . Let denote the set of non-negative integers. Let be a multi-index. For the multi-index , let

 ∂β:=(∂∂x1)β1…(∂∂xd)βd=∂|β|∂xβ11…∂xβddwith|β|:=β1+…+βd.

Let be an open domain of . Let be a nonnegative integer and with . We define the Sobolev space

 Wℓ,p(Ω):={φ∈Lp(Ω); ∂βφ∈Lp(Ω), 0≤|β|≤ℓ},

equipped with the norms

 ∥φ∥Wℓ,p(Ω) :=⎛⎝∑0≤|β|≤ℓ∥∂βφ∥pLp(Ω)⎞⎠1/p if 1≤p\textless∞, ∥φ∥Wℓ,∞(Ω) :=max0≤|β|≤ℓ(ess.supx∈Ω|∂βφ(x)|).

We use the semi-norms

 |φ|Wℓ,p(Ω) :=⎛⎝∑|β|=ℓ∥∂βφ∥pLp(Ω)⎞⎠1/p if 1≤p\textless∞, |φ|Wℓ,∞(Ω) :=max|β|=ℓ(ess.supx∈Ω|∂βφ(x)|).

If , we use the notation

 Hℓ(Ω):=Wℓ,2(Ω).

We set . The space is a Hilbert space equipped with the scalar product

 (φ,ψ)Hℓ(Ω):=∑|β|≤ℓ(∂βφ,∂βψ)L2(Ω),

where denotes the -inner product, which leads to the norm and semi-norm

 ∥φ∥Hℓ(Ω):=⎛⎝∑|β|≤ℓ∥∂βφ∥2L2(Ω)⎞⎠1/2, |φ|Hℓ(Ω):=⎛⎝∑|β|=ℓ∥∂βφ∥2L2(Ω)⎞⎠1/2.

The dual space of is defined and denoted by . is a Banach space with norm

 ∥χ∥Wℓ,p(Ω)′:=supv∈Wℓ,p(Ω)|χ(v)|∥v∥Wℓ,p(Ω)∀χ∈Wℓ,p(Ω)′.

For any , the norm is defined by

 ∥v∥Wℓ,p(Ω)d:=(d∑i=1∥vi∥2Wℓ,p(Ω))1/2.

We introduce the function space

 H(div;Ω):={v∈L2(Ω)d; divv∈L2(Ω)},

with the norm

 ∥v∥H(div;Ω):=(∥v∥2L2(Ω)d+∥divv∥2)1/2.

Let be a matrix, and denote an operator norm as

 ∥A∥2:=sup0≠x∈Rd|Ax||x|,

where for .

2.2 Raviart–Thomas Finite Element on Simplices

For any , let be the space of polynomials with degree at most . is spanned by the restriction to of polynomials in , where is a closed domain. Let be a -simplex. The local Raviart–Thomas polynomial space of order is defined by

 RTk(T):=Pk(T)d+xPk(T),x∈Rd. (2.1)

For

, the local degrees of freedom are given as

 χ1i,j(v) :=∫Fiv⋅nFipjds,∀pj∈Pk(Fi),Fi⊂∂T, (2.2)

and if ,

 χ2ℓ(v) :=∫Tv⋅qℓdx,∀qℓ∈Pk−1(T)d. (2.3)

Here,

denotes the outer unit normal vector of

on the face . For the simplicial Raviart–Thomas element in , it holds that

 dimRTk(T)={(k+1)(k+3)if % d=2,12(k+1)(k+2)(k+4)if d=3. (2.4)

It is known that the Raviart–Thomas finite element with the set of linear forms is unisolvent; e.g., see (BofBreFor13, , Proposition 2.3.4). The triple is then a finite element.

We set the domain of the local Raviart–Thomas interpolation as ; e.g., see also (ErnGue04, , p. 27).

The local Raviart–Thomas interpolation is then defined as follows. For any ,

 ∫FIRTTv⋅nFpkds =∫Fv⋅nFpkds∀pk∈Pk(F), F⊂∂T, (2.5)

and if ,

 ∫TIRTTv⋅qk−1dx =∫Tv⋅qk−1dx∀qk−1∈Pk−1(T)d. (2.6)

Let with be the Raviart–Thomas finite element. Let , be an affine mapping such that with a regular matrix and . The Piola transformation is defined by

 ˆΨ:L2(ˆT)∋^v(^x) ↦v(x):=ˆΨ(^v)(x):=1|det(ˆA)|ˆA^v(^x)∈L2(T).

The following lemmata introduce the fundamental properties of the Piola transformation.

Lemma 1

For , , we define and . Then,

 ∫Tdivvφdx =∫ˆTˆdiv^v^φd^x, ∫Tv⋅∇xφdx =∫ˆT^v⋅ˆ∇^x^φd^x, ∫∂Tv⋅nTφds =∫∂ˆT^v⋅^nˆT^φd^s. (2.7)

Here, and are respectively the unit outward normal vectors of and .

Proof

See, for example, (BofBre08, , Lemma 3.3). ∎

By applying (2.7), we can prove the invariance of the Raviart–Thomas interpolation under the Piola transform; e.g., see (BofBre08, , Lemma 3.4).

Lemma 2

For , we have

 IRTˆT^v=ˆΨ−1IRTTˆΨ^v.

That is to say, the diagram

 H1(ˆT)dˆΨ−−−−→H1(T)dIRTˆT⏐⏐↓⏐⏐↓IRTTRTk(ˆT)ˆΨ−−−−→RTk(T)

commutes.

3 Standard Positions and Reference Elements

This section introduces the Jacobian matrix proposed in KobTsu20 for the three-dimensional case and that proposed in KobTsu14 ; KobTsu15 ; LiuKik18 for the two-dimensional case.

Let us first define a diagonal matrix as

 ˆA(d):=diag(α1,…,αd),αi∈R. (3.1)

3.1 Two-dimensional case

Let be the reference triangle with vertices , , and .

Let be the family of triangles

 ˜T=ˆA(2)(ˆT),

with vertices , , and .

We next define the regular matrices by

 ˜A:=(1s0t), (3.2)

with parameters

 s2+t2=1,t\textgreater0.

For , let be the family of triangles

 T =˜A(˜T),

with vertices . We then have , .

3.2 Three-dimensional cases

There are two three-dimensional cases

Let and be reference tetrahedrons with the following vertices.

(i)

has the vertices , , .,

(ii)

has the vertices , , ..

Let , , be the family of triangles

 ˜Ti=ˆA(3)(ˆTi),i=1,2

with vertices

(i)

, , , and ,

(ii)

, , , and .

We next define the regular matrices by

 ˜A1:=⎛⎜⎝1s1s210t1s2200t2⎞⎟⎠, ˜A2:=⎛⎜⎝1−s1s210t1s2200t2⎞⎟⎠ (3.3)

with parameters

 {s21+t21=1, s1\textgreater0, t1\textgreater0, α2s1≤α1/2,s221+s222+t22=1, t2\textgreater0, α3s21≤α1/2.

For , , let , be the family of triangles

 Ti =˜Ai(˜Ti),i=1,2

with vertices

 x1:=(0,0,0)T, x2:=(α1,0,0)T, x4:=(α3s21,α3s22,α3t2)T, {x3:=(α2s1,α2t1,0)Tfor the case (i),x3:=(α1−α2s1,α2t1,0)Tfor the % case (ii).

We then have , , and

 α2={|x1−x3|\textgreater0for the case (i),|x2−x3|\textgreater0for the case (ii).

3.3 Standard Positions

In what follows, we impose conditions for in the two-dimensional case and in the three-dimensional case.

Condition 1 (Case that d=2)

Let with vertices () introduced in Section 3.1. We assume that is the longest edge of ; i.e., . Recall that and . We then assume that . Note that .

Condition 2 (Case that d=3)

Let with vertices () introduced in Section 3.2. Let () be edges of . We denote by the edge of with minimum length; i.e., . Among the four edges that share an end point with , we take the longest edge . Let and be end points of the edge . We thus have

 α1=|L(min)max|=|x1−x2|.

Consider cutting with the plane that contains the midpoint of the edge and is perpendicular to the vector . We then have two cases:

(i)

and belong to the same half-space;

(ii)

and belong to different half-spaces.

In each case, we respectively set

(i)

and as the end points of , that is ;

(ii)

and as the end points of , that is .

Finally, recall that . Note that we implicitly assume that and belong to the same half space. Also note that and , where denotes the diameter of .

Each -simplex is congruent to the unique satisfying Condition 1 or Condition 2. is therefore called the standard position of the -simplex. See Figure 1 and 2.

3.4 Affine Mappings and Piola Transforms

The present paper adopts the following affine mappings and Piola transformations.

Definition 1

Let satisfy Condition 1 or Condition 2. Let , and be the simplices defined in Sections 3.1 and 3.2. That is to say,

 ˜T=ˆΦ(ˆT),T=˜Φ(˜T)with~x:=ˆΦ(^x):=ˆA(d)^x,x:=˜Φ(~x):=˜A~x.

We define the affine mapping by

 Φ:=˜Φ∘ˆΦ:Rd→Rd, x:=Φ(^x):=A^x,A:=˜AˆA(d). (3.4)

Let and be the Piola transformations with respect to and , respectively. We define by , which is the Piola transformation with respect to .

3.5 Finite Element Generation on Standard Positions

We follow the procedure described in (ErnGue04, , Section 1.4.1 and 1.2.1).

For the reference element defined in Sections 3.1 and 3.2, let be a fixed reference finite element, where is a vector space of functions for some positive integer (tipically or ) and is a set of linear forms such that

 ˆP∋^p↦(^χ1(^p),…,^χn0(^p))T∈Rn0

is bijective; i.e., is a basis for . Further, we denote by in the local (-valued) shape functions such that

 ^χi(^θj)=δij,1≤i,j≤n0.

Let be a normed vector space of functions such that and the linear forms can be extended to . The local interpolation operator is then defined by

 IˆT:V(ˆT)∋^v↦n0∑i=1^χi(^v)^θi∈ˆP. (3.5)

Let , , and be the affine mappings defined in (3.4). For , we first define a Banach space of -valued functions that is the counterpart of and define a linear bijection mapping by

 ψT:=ψˆT∘ψ˜T:V(T)∋v↦^v:=ψT(v):=v∘Φ∈V(ˆT),

with two linear bijection mappings:

 ψ˜T:V(T)∋v↦~v:=ψ˜T(v):=v∘˜Φ∈V(˜T), ψˆT:V(˜T)∋~v↦^v:=ψˆT(~v):=~v∘ˆΦ∈V(ˆT).

Furthermore, the triple is defined by

 ⎧⎪ ⎪⎨⎪ ⎪⎩˜T=ˆΦ(ˆT);˜P={ψ−1ˆT(^p); ^p∈ˆP};˜Σ={{~χi}1≤i≤n0; ~χi=^χi(ψˆT(~p)),∀~p∈˜P,^χi∈ˆΣ},

while the triple is defined by

 ⎧⎪ ⎪⎨⎪ ⎪⎩T=˜Φ(˜T);P={ψ−1˜T(~p); ~p∈˜P};Σ={{χi}1≤i≤n0; χi=~χi(ψ˜T(p)),∀p∈P,~χi∈˜Σ}.

and are then finite elements. The local shape functions are and , , and the associated local interpolation operators are respectively defined by

 I˜T:V(˜T)∋~v↦I˜T~v :=n0∑i=1~χi(~v)~θi∈˜P, (3.6) IT:V(T)∋v↦ITv :=n0∑i=1χi(v)θi∈P. (3.7)

The diagrams

commute.

Proof

See (ErnGue04, , Proposition 1.62). ∎

Example:

Let be a finite element.

1. For the Lagrange finite element of degree , we set .

2. For the Hermite finite element, we set .

3. For the Crouzeix–Raviart finite element with , we set .

3.6 Raviart–Thomas Finite Element on Standard Positions

For the reference element defined in Section 3.1 and 3.2, let be the Raviart–Thomas finite element with . Let , , and be the affine mappings defined in (3.4). Let , , and be the Piola transformations defined in Definition 1.

We then define and by

 ⎧⎪ ⎪⎨⎪ ⎪⎩˜T=ˆΦ(ˆT);RTk(˜T)={ˆΨ(^p); ^p∈RTk(ˆT)};˜Σ={{~χi}1≤i≤n0; ~χi=^χi(ˆΨ−1(~p)),∀~p∈RTk(˜T),^χi∈ˆΣ};

and

 ⎧⎪ ⎪⎨⎪ ⎪⎩T=˜Φ(˜T);RTk(T)={˜Ψ(~p); ~p∈RTk(˜T)};Σ={{χi}1≤i≤n0; χi=~χi(˜Ψ−1(p)),∀p∈RTk(T),~χi∈˜Σ}.

and are then the Raviart–Thomas finite elements. Furthermore, let

 IRT˜T:Vdiv(˜T)→RTk(˜T) (3.8)

and

 IRTT:Vdiv(T)→RTk(T) (3.9)

be the associated local Raviart–Thomas interpolation defined in (2.5) and (2.6), respectively.

3.7 Parameter Ht and Mesh

We first propose a new parameter .

Definition 2

Let satisfy Condition 1 or Condition 2. Furthermore, let be defined in Condition 1 or Condition 2. We then define the parameter as

 HT:=∏di=1αi|T|hT,

where .

In the sequel of this paper, the interpolation errors are bounded in terms of and . However, the parameters and proposed below might be more convenient for the practical computation of finite element methods.

We assume that is a bounded polyhedral domain. Let be a simplicial mesh of , made up of closed -simplices, such as

 ¯¯¯¯Ω=⋃T0∈T