## 1 Introduction

Let , , be a bounded polyhedral domain. Furthermore, we assume that is convex if necessary. We consider the Poisson problem as follows. Find such that

(1.1) |

where is a given function. Let be a nonnegative integer. is a Hilbert space with scalar product and norm . We set with and . This paper gives error estimates for the first-order Crouzeix–Raviart (CR) finite-element approximation on anisotropic meshes in three dimensions. Anisotropic meshes have different mesh sizes in different directions. The shape regularity assumption on triangulations is no longer valid on these meshes; see for example Ape99 . Furthermore, we do not impose the maximum-angle condition proposed in BabAzi76 during mesh partitioning. In many instances, the discussion also relates to two dimensions. We therefore discuss the problem here as uniformly valid in an arbitrary number of dimensions.

CR finite error estimates for the non-homogeneous Dirichlet Poisson problem are known. Let be the CR finite-element space, to be defined in Section 2.3. Let and be the exact and CR finite-element solutions, respectively. In (Gud10, , Corollary 2.2), adopting medius analysis, the estimate

(1.2) |

is given, where denotes the broken (piecewise) -semi norm defined in Section 2.2, and a positive constant independent of . Here, the oscillation is expressed as

where denotes the piecewise constant space in . Suppose that and oscillation vanishes. Let

be the nodal interpolation of

at the midpoints of the faces. Then, from the standard interpolation error estimate (see for example (ErnGue04, , Corollary 1.109)), we havewhere represents a positive constant independent of and but dependent on the parameter of the simplicial mesh; see for example (ErnGue04, , Definition 1.107). This parameter is bounded if the simplicial mesh sequence is shape regular. However, the situation is different without the shape-regular condition. The aim of the present paper is to deduce an analogous error estimate on anisotropic finite-element meshes. Note that very flat elements might be included in the mesh sequence. In many papers reporting on such investigations, the maximum-angle condition instead of the shape-regular condition is imposed. However, the maximum-angle condition is not necessarily needed to obtain error estimates. Recently, in the two-dimensional instance, the CR finite-element analysis of the non-homogeneous Dirichlet-Poisson problem has been investigated under a more relaxed mesh condition, KobTsu18b . The present paper extends previous research to a three-dimensional setting.

However, it may not be easy to use the estimate (1.2) on anisotropic finite-element meshes. The CR finite-element space is not in . Hence, an error between the exact solution and the CR finite-element approximation solution with a -broken seminorm is divided into two parts. One is an approximation error that measures how well the exact solution is approximated by the CR finite-element functions. The other is a nonconformity error term. For the former, the CR interpolation error estimates are often used; in the latter, the standard scaling argument is often used to obtain the error estimates. However, in this way, we are unable to derive the correct order on anisotropic meshes. To overcome this difficulty, we shall use the lowest-order Raviart–Thomas (RT) interpolation error estimates on anisotropic meshes. By this technique, we consequently have the error estimates in the -broken seminorm (Theorem 4) and the norm (Theorem 5) on anisotropic meshes. Furthermore, we present an error estimate for the first-order RT finite-element approximation of the Poisson problem (1.1) based on the dual mixed formulation (Theorem 7). We again emphasise that we do not impose either the shape-regular or the maximum-angle condition during mesh partitioning.

We next present the equivalence of the enriched piecewise linear CR finite-element method introduced by HuMa15 and the first-order RT finite-element method. In two dimensions, the work ArnBre85 represents pioneering research. Marini Mar85 further found an expression relating RT and CR finite-element methods:

(1.3) |

where denotes a mesh element, () the vertices of triangle , the barycentre of such that , and and respectively denote the RT and CR finite-element solutions with a given external piecewise-constant function . It was recently proved HuMa15 that the enriched piecewise-linear CR finite-element method is identical to the first-order RT finite-element method for both the Poisson and Stokes problems in any number of dimensions. In the present paper, we extend Marini’s results to three dimensions (Lemma 10).

The remainder of the present paper is organised as follows. Section 2 introduces the weak form of the continuous problem (1.1), the finite-element meshes, and finite-element spaces. Furthermore, we propose a parameter . Section 3 introduces discrete settings of the CR finite-element method for (1.1) and proposes error estimates. Section 4 proves error estimates for the first-order RT finite-element method based on the dual mixed formulation of the Poisson problem. Section 5 gives the equivalence of the RT and CR finite-element problems. Finally, Section 6 presents numerical results obtained using the Lagrange P1 element and the first-order CR element.

## 2 Preliminaries

### 2.1 Weak formulation

The variational formulation for the Poisson problem (1.1) is then as follows. Find such that

(2.1) |

where denotes a bilinear form defined by

Here, we define as the closure of in the semi-norm . By the Lax–Milgram lemma, there exists a unique solution for any and it holds that

where is the Poincar constant depending on . Furthermore, if is convex, then and

(2.2) |

The proof can be found in, for example, (Gri11, , Theorem 3.1.1.2, Theorem 3.2.1.2).

### 2.2 Meshes, Mesh faces, Averages and Jumps

Let be a simplicial mesh of , made up of closed -simplices, such as

with , where . We assume that each face of any -simplex in is either a subset of the boundary or a face of another -simplex in . That is, is a simplicial mesh of without hanging nodes.

###### Definition 1

For any , we define the parameter as

where denotes edges of the triangle . Further, we define the parameter as

where denotes edges of the tetrahedra . Here, denotes the measure of . Furthermore, we set

We impose the following assumption.

###### Assumption 1

We assume that is a sequence of triangulations of such that

We adopt the concepts of mesh faces, averages and jumps in the analysis of RT and CR finite element method. Let be the set of interior faces and the set of the faces on the boundary . Let . For any , we define the unit normal to as follows: (i) If with , , let and be the outward unit normals of and , respectively. Then, is either of ; (ii) If , is the unit outward normal to .

Let be a positive integer. We then define the broken (piecewise) Sobolev space as

with the norm

Let . Suppose that with , . Set and . The jump and the average of across is then defined as

For a boundary face with , and . When is an -valued function, we use the notation

for the jump of the normal component of . For a boundary face with , and . Whenever no confusion can arise, we simply write and , respectively.

We here define a broken gradient operator as follows.

###### Definition 2

For , the broken gradient is defined by

Note that and the broken gradient coincides with the distributional gradient in .

### 2.3 Finite Element Spaces and Interpolations Error Estimates

This section introduce the RT, CR and piecewise-constant finite element spaces and the interpolation error estimates poroposed in IshKobTsu .

#### 2.3.1 RT finite element space

Let . For any , let be the space of polynomials with degree at most in .

The lowest order RT finite element space is defined by

The functionals are defined by, for any ,

(2.3) |

where

denotes the outer unit normal vector of

along . We set . Note that . The triple is then a finite element. We define the global RT finite element space byNote that .

We next define the local RT interpolation as

(2.4) |

using

(2.5) |

Further, we define the global RT interpolation by

(2.6) |

We give the local RT interpolation error estimate.

###### Theorem 1

We have the following estimate such that

(2.7) |

where is a positive constant independent of .

###### Proof

The proof can be found in (IshKobTsu, , Theorem 3). ∎

The global RT interpolation error estimate is obtained as follows.

###### Corollary 1

Let be a family of conformal meshes satisfying Assumption 1. Then, there exists , independent of , such that

(2.8) |

#### 2.3.2 CR finite element space

In introducing a nonconforming method, we define the following CR finite element space as

Using the barycentric coordinates , , we define the local basis functions as

(2.9) |

For , let be the face of and the barycentre of face . We then define the local CR interpolation operator as

(2.10) |

Because the trace of a function in is in , is meaningful. Further, it holds that

(2.11) |

We define the global CR interpolation by

(2.12) |

We give the local CR interpolation error estimate.

###### Theorem 2

We have the following estimates such that

(2.13) | ||||

(2.14) |

Here, and are positive constants independent of and .

###### Proof

The proof can be found in (IshKobTsu, , Theorem 2). ∎

###### Remark:

The inequality (2.14) can be improved by replacing with .

The global CR interpolation error estimates are obtained as follows.

###### Corollary 2

Let be a family of conformal meshes satisfying Assumption 1. Then, there exist , independent of and , such that

(2.15) | ||||

(2.16) |

#### 2.3.3 Piecewise-constant finite element space

We define the standard piecewise constant space as

The local -projection from into the space is defined by

(2.17) |

Note that is the constant function equal to . We also define the global -projection to the space by

(2.18) |

The error estimate of the -projection is as follows.

###### Theorem 3

We have the error estimate of the -projection such that

(2.19) |

Here, is a positive constant independent of .

###### Proof

The proof can be found in (IshKobTsu, , Theorem 2). ∎

The global error estimate of the -projection is obtained as follows.

###### Corollary 3

Let be a family of conformal meshes satisfying Assumption 1. Then, there exists , independent of , such that

(2.20) |

Between the RT interpolation and the -projection , the following relation holds:

###### Lemma 1

For any , it holds that

That is to say, the diagram

commutes.

###### Proof

The proof of this lemma is found in Bra07 . ∎

The following relation plays an important role in the CR finite element analysis on anisotropic meshes.

###### Lemma 2

It holds that

(2.21) |

###### Proof

For any and , using Green formula and the fact for any , we can derive

∎

### 2.4 Discrete Poincaré Inequality on Anisotropic Meshes

We propose the discrete Poincaré inequality on anisotropic meshes.

###### Lemma 3 (Discrete Poincaré inequality on anisotropic meshes)

Assume that is convex. If , there exists , independent of , , and the geometry of meshes, such that

(2.22) |

###### Proof

Let . We consider the dual problem. Find such that

We then have a priori estimates:

where is the Poincaré constant. We use the duality argument to show the target inequality. That is to say, we have

which leads to

We here used

where

∎

## 3 CR Finite Element Approximation

### 3.1 Finite Element Approximation

The CR finite element problem is to find such that

(3.1) |

where is defined by

This problem is nonconforming because .

### 3.2 Classical Error Analysis

The starting point for error analysis is the Second Strang Lemma, e.g. see (ErnGue04, , Lemma 2.25),

(3.2) |

The first term of the inequality (3.2) is estimated as follows. Using the CR interpolation error estimate (2.16), we have, for any ,

(3.3) |

From the standard scaling argument, we have a consistency error inequality, e.g., see (ErnGue04, , Lemma 3.36).

###### Lemma 4 (Asymptotic Consistency)

Let be the solution of the homogeneous Dirichlet Poisson problem (1.1). It then holds that

(3.4) |

where denotes the set of all faces of . Here, denotes the distance of the vertex of opposite to to the face.

###### Proof

We follow (ErnGue04, , Lemma 3.36).

Let . Because , we have

Because each face of an element located inside appears twice in the above sum, we have

with the mean value

Furthermore, we get

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