## 1 Introduction

A central question in the finite element approximation theory is to establish local upper bounds on the best-approximation error for functions that satisfy some minimal regularity assumptions typically quantified in the scale of the fractional-order Sobolev spaces. The goal of the present work is to derive some novel results in this context when the approximation is realized using Nédélec finite elements and Raviart–Thomas finite elements. Most of our developments focus on the Nédélec finite elements since they require more elaborate arguments. The corresponding results for the Raviart–Thomas finite elements only improve marginally the state of the art from the literature, and only a short discussion is provided in a specific section of the paper.

We are interested in approximating fields with a smoothness Sobolev index that is so low that one cannot invoke the canonical interpolation operators associated with the considered finite elements. In this case, the best-approximation error can be bounded by considering quasi-interpolation errors, such as those derived in [10]. However, the resulting upper bound cannot be localized to the mesh cells if the regularity of the target function is only measured in the scale of the fractional-order Sobolev spaces. The lack of localization is relatively mild if no boundary conditions are prescribed in the finite element spaces, since in this case the quasi-interpolation error can still be bounded by local contributions involving the fractional-order Sobolev seminorm of the target function over patches of mesh cells instead of just each mesh cell individually. The lack of localization is more significant if boundary conditions are additionally prescribed in the finite element spaces since in this case the upper bound on the quasi-interpolation error is global. This is not surprising as in this situation the target function has not enough smoothness to have a well-defined trace at the boundary. The main contribution of this work is to show that the localization becomes possible provided some (mild) additional assumptions are made on the integrability of the curl or the divergence of the target function. The main tool to achieve this result hinges on the face-to-cell lifting operators introduced in [15]. The additional assumptions on the curl or the divergence allow us to give a weak meaning to the trace of the target function in the dual space of a suitable fractional-order Sobolev space.

In this work, the space dimension is for the Nédélec finite elements and for the Raviart–Thomas finite elements. For , the results for the Raviart–Thomas elements can be transposed to the Nédélec elements by invoking a rotation of angle ; details are omitted for brevity (see, e.g., [13, Sec. 15.3.1]). We consider a polyhedral Lipschitz domain . Moreover, we use boldface for -valued fields and linear spaces composed of such fields. For instance, for real numbers and (we assume if ), denotes the (fractional-order) Sobolev space equipped with the Sobolev–Slobodeckij norm.

Let denote a shape-regular family of affine, matching, simplicial meshes such that each mesh covers exactly. Let denote the -conforming finite element space built on the mesh using the Nédélec finite element of degree (here, the superscript refers to the curl operator). Given a target field , with possibly very small, our goal is to establish localized upper bounds on the best-approximation error

(1) |

The first natural idea is to invoke the canonical interpolation operator for Nédélec finite elements, say . Since this operator can only act on those fields having an integrable tangential trace along all the mesh edges, invoking the standard trace theory in Sobolev spaces (see, e.g., [16]) shows that a suitable domain for the canonical interpolation operator is with (and if ). Assume that the polynomial degree is such that if and otherwise, and let if or if . Then, it is well-known (see, e.g., [18] or [13, Sec. 16.2]) that there is such that for all , all , and all , we have

(2) |

where denotes the diameter of the mesh cell . Here, the symbol denotes a generic positive constant whose value can change at each occurrence provided it only depends on the mesh shape-regularity, the space dimension, and the polynomial degree of the considered finite elements. Notice that is unbounded as if . For instance, in the Hilbert setting where , the minimal regularity requirement is with , and is unbounded as .

The requirement can be lowered to (with ) by invoking more sophisticated results on traces derived in [2] which, however, hinge on some additional integrability assumption on

. To use these results, the edge-based degrees of freedom of the Nédélec finite element are extended by defining them using edge-to-cell lifting operators and an integration by parts formula (see, e.g.,

[13, Sec. 17.3]). One can then show (see [6]; see also [1, 8, 3] for slight variants) that for all and all , there is such that for all with , all , and all , we have(3) |

with unbounded as or .

Unfortunately, the regularity assumption is often not realistic in applications. To go beyond this assumption, one can invoke the quasi-interpolation operators devised in [10]. Recall that the construction of these quasi-interpolation operators consists of first projecting the target field onto a fully discontinuous finite element space and then stitching together the projected field to recover the desired conformity property by averaging the canonical degrees of freedom of the projected field. Since the projected field is always piecewise smooth, this construction is always meaningful, regardless of the regularity of the target field . Let denote the quasi-interpolation operator thus constructed with the Nédélec finite elements. Then, [13, Thm. 22.6] shows that there is such that for all , all if and otherwise, all , all , and all , we have

(4) |

where and denotes the collection of mesh cells sharing at least one edge with . We notice that a slight loss of localization occurs in (4) since the Sobolev–Slobodeckij seminorm on the right-hand side is evaluated over the macroelement and not just over . In the present work, we show that provided some (mild) additional integrability assumption is made on , the estimate (4) can be localized to the mesh cells in ; see Theorem 2.1 and Corollary 2.2.

The loss of localization is more striking if one wants to additionally enforce a homogeneous boundary condition on the tangential component of the target field. We assume for simplicity that the condition is enforced over the whole boundary of . Recall that the tangential trace operator is defined through a global integration by parts formula (see, e.g., [13, Thm. 4.15]) and that we have whenever the field is smooth enough, where denotes the unit outward normal to . Then, setting , one is interested in establishing local upper bounds on the best-approximation error

(5) |

Let denote the quasi-interpolation operator with homogeneous boundary prescription associated with the Nédélec finite elements. Then, [13, Thm. 22.14]) shows that for all , there is such that for all , and all , we have

(6) |

where , is a characteristic (global) length scale associated with , and . Notice that the target field has not sufficient regularity to have a well-defined tangential trace on the boundary. The loss of localization in (6) arises when bounding the quasi-interpolation error over those mesh cells that have at least one edge located on the boundary (the upper bound (4) holds true for the other mesh cells). The presence of the global length scale and of the full Sobolev–Slobodeckij norm of instead of just the seminorm in (6) comes from the need to invoke a Hardy inequality near the boundary (see the proof of [10, Thm. 6.4]). In the present work, we show that provided some (mild) additional integrability assumption is made on , the estimate (6) can be fully localized to the mesh cells; see again Theorem 2.1 and Corollary 2.2.

## 2 Main results Nédélec finite elements

In this section, we first state our main results and then present their proofs.

### 2.1 Statement of the main results

Let us first observe that the domain of the tangential trace operator can be extended to for all . Indeed, for all , can still be defined by duality by setting for all ,

(7) |

where denotes a lifting of in . Indeed, owing to the Sobolev embedding theorem, and the fact that , we infer that with , so that the second term on the right-hand side of (7) is meaningful owing to Hölder’s inequality. We can now state our main result. For simplicity, we estimate the quasi-interpolation error only in the -norm.

###### Theorem 2.1 (Localized quasi-interpolation error estimate for Nédélec elements)

For all and all , there is such that for all with , all , and all , we have

(8) |

Moreover, assuming that , we also have

(9) |

Squaring the above inequalities, summing over the mesh cells, and observing that the cardinality of the set is uniformly bounded for all and all , we infer the following result.

###### Corollary 2.2 (Localized best-approximation error for Nédélec elements)

For all and all , there is such that for all with , and all , we have

(10) |

Moreover, assuming that , we also have

(11) |

###### Remark 2.3 (Exponent)

Observe that since . Moreover, we have for .

### 2.2 Preliminary: localizing the tangential trace to the mesh faces

This section collects some preliminary results needed in the proof of Theorem 2.1. These results are drawn from [15] and are briefly restated here for the reader’s convenience.

Let be a mesh cell, let be the collection of the faces of , and let . To define a tangential trace that is localized to the mesh face , we introduce the local functional space with (as above) and . We equip this space with the (dimensionally consistent) norm

(12) |

Let be such that (this is indeed possible since the function is increasing on ). Let be such that . We consider the (fractional-order) Sobolev space , equipped with the (dimensionally consistent) norm

(13) |

Let denote the dual space of . It is shown in [15, Equ. (5.5)] that upon introducing suitable face-to-cell lifting operators, it is possible to define a tangential trace operator localized to the mesh face through an integration by parts formula, namely , such that the following two properties hold: (i) whenever the field is smooth, where denotes the unit normal to pointing outward ; (ii) There is such that for all , all , and all ,

(14) |

Let be composed of the restriction to of the Nédélec polynomials of order . Let us set for all . In this work, we need to invoke the following inverse inequality.

###### Lemma 2.4 (Inverse inequality on )

Let . There is such that for all , all , all , and all ,

(15) |

###### Proof.

Let . Recalling the definition (13) of the -norm and invoking an inverse inequality on (see, e.g., [13, Sec. 12.1], and observe that , where is the geometric mapping from the reference -dimensional simplex to and where is composed of -valued, -variate polynomials of order at most ), we infer that

with such that . This implies that

Since , the assertion follows from the definition of the dual norm in and the identity . ∎

Let us set

(16) | ||||

(17) |

(Observe that the tangential trace operator is meaningful on since .) We notice that for all and all , we have . We also define the broken version of as follows:

(18) |

The collection of the mesh faces, , is split into the collection of the mesh interfaces, , and the collection of the mesh boundary faces, . For all , there are two distinct mesh cells such that . For all , there is one mesh cell such that . For every field , the jump of the tangential component across the mesh interface is defined as

(19) |

Moreover, for every mesh boundary face , we conventionally set

(20) |

###### Lemma 2.5 (Vanishing jumps and boundary traces)

(i) For all , we have for all . (ii) For all , we additionally have for all .

###### Proof.

Let and let . For all such that , is defined in [15, Equ. (5.5)] so that

for all ,
where is the
face-to-cell lifting operator from [15, Def. 5.1].

(i) Let .
We define the global lifting operator such that
and
otherwise. Summing the above identity for , we infer that

(21) |

for all . Since, by construction, has a zero trace at the boundary of , we conclude that

and since is arbitrary in
, this implies that .

(ii) If , we set
and
otherwise. This yields again (21), and invoking that
still gives
, whence the assertion.
∎

### 2.3 Proof of (8)

Let us start with a preliminary result of independent interest. For all , let denote the (local) -orthogonal projection onto (that is, is the mean value of over ).

###### Lemma 2.6 (Localized quasi-interpolation error in )

For all and all , there is such that for all , all , and all , we have

(22) |

###### Proof.

Let denote the global -orthogonal projection onto the broken Nédélec finite element space . Let denote the local -orthogonal projection onto , so that we have for all and all . Recall (see [10, Sec. 5]) that we have , where the operator is built by averaging the canonical degrees of freedom, see [10, Sec. 4.2]. The triangle inequality implies that

Since , standard properties of the -orthogonal projection and Hölder’s inequality (recall that ) imply that

Moreover, [10, Lem. 4.3] followed by Lemma 2.4 (with ) give (recall that the value of can change at each occurrence)

where denotes the collection of the mesh interfaces sharing at least an edge with . Observing that since (see Lemma 2.5(i)), we infer that

By definition of the jump operator, invoking the triangle inequality and recalling the definition of the set yields

Owing to the bound (14) and the shape-regularity of the mesh sequence, we infer that

(23) |

Invoking the triangle inequality and recalling the definition (12) of the norm equipping gives for all ,

where we used that is a constant field in . Invoking inverse inequalities on the first line, the triangle inequality and standard properties of the -orthogonal projection on the second line, and, finally, Hölder’s inequality (recall that ) and the regularity of the mesh sequence on the third line, we also have