Interpolation operators that approximate a given function with weak gradient, curl, or divergence by a piecewise polynomial of degree are fundamental in numerical analysis. Typically, this has to be done over a computational domain covered by a mesh with characteristic size
. Probably the most widespread are the canonical interpolation operators associated with the canonical degrees of freedom of the finite elements from the discrete de Rham sequence, which in particular include the Nédélec and Raviart–Thomas finite elements. The advantage of these operators is that they are local (that is, defined independently on each elementof the mesh ) and that they commute with the appropriate differential operators. They are also projectors, i.e., they leave the interpolated function invariant if it is already a piecewise polynomial, and they lead to optimal approximation error bounds with respect to the mesh size . However, the canonical interpolation operators have two main deficiencies. Firstly, these operators can act on a given function only if it possesses more regularity beyond the minimal , , and regularity. Secondly, they are not well-suited to derive approximation error bounds that are explicit in the polynomial degree .
1.1 Interpolation operators and -approximation
The projection-based interpolation operators, see Demkowicz and Buffa [Demk_Buf_q_opt_proj_int_05], Demkowicz [Demk_pol_ex_seq_int_06], and the references therein, lead to optimal approximation properties in the mesh size and quasi-optimal approximation properties in the polynomial degree (up to logarithmic factors). They were derived under a conjecture of existence of commuting and polynomial-preserving extension operators from the boundary of the given element to its interior which was later established by Demkowicz et al. in [Demk_Gop_Sch_ext_I_09, Demk_Gop_Sch_ext_II_09, Demk_Gop_Sch_ext_III_12]; the approximation results are summarized in [Demk_Gop_Sch_ext_III_12, Theorem 8.1]. Thus, these operators essentially lift the second drawback of the canonical interpolation operators described above (up to logarithmic factors), while still sharing the same important properties, i.e., they are defined locally, they are projectors, and they commute with the appropriate differential operators. However, these operators again require more regularity beyond the minimal , , and regularity, so that the first drawback remains.
In the particular case of , which constitutes the focus of the present work, the normal component of the interpolate on each mesh face is fully dictated by the normal component of the interpolated function on that face, which requires regularity with , which is slightly more than regularity. Some further refinements can be found in Bespalov and Heuer [Besp_Heue_H_div_hp_2D_11] in the case. Recently, building on [Demk_Buf_q_opt_proj_int_05, Demk_pol_ex_seq_int_06], a commuting projector that fully removes the second drawback above in that it has fully optimal -approximation properties (does not feature the logarithmic factors) has been devised in [Mel_Roj_com_p_interp_19]. To define the projector, though, higher regularity is needed, with in particular , , in the case of interest here.
The issue of constructing (quasi-)interpolation projectors under the minimal regularities , , and has been addressed before, cf., e.g., Clément [Clement_appr_75], Scott and Zhang [Scott_Zh_int_nonsm_90], and Bernardi and Girault [Ber_Gir_loc_reg_98] in the case, Nochetto and Stamm [Noch_Stamm_H_dv_19] in the case, and Bernardi and Hecht [Ber_Hecht_appr_Ned_07] in the case; see also the references therein. Stability and -optimal approximation estimates in any -norm, , has recently been achieved by Ern and Guermond in [Ern_Guer_quas_int_best_appr_17] in a unified setting for a wide range of finite elements encompassing the whole discrete de Rham sequence. The arguments used in [Ern_Guer_quas_int_best_appr_17] are somewhat different from those in the previous references: a projection onto the fully discontinuous (broken) piecewise polynomial space is applied first, followed by an averaging operator to ensure the appropriate , , or trace continuity. Unfortunately, all of the above quasi-interpolation projectors do not commute with the appropriate differential operators and, moreover, they are only shown to be optimal in but not in .
1.2 Stable local commuting projectors under minimal regularity
Constructing projectors applicable under the minimal regularities , , and that would in addition be commuting, stable, and locally defined represents a long-standing effort. Stability, commutativity, and the projection property were obtained by Christiansen and Winther in [Christ_Wint_sm_proj_08] by composing the canonical interpolation operators with mollification, following some earlier ideas in particular from Schöberl [Schob_com_quas_int_MFE_01, Schob_ML_dec_H_curl_05], cf. also Ern and Guermond [Ern_Guer_mol_de_Rham_16] for a shrinking technique avoiding the need of extensions outside of the domain and Licht [Licht_sm_proj_19] for essential boundary conditions only prescribed on the part of the boundary of . These operators are, however, not locally defined. This last remaining issue was finally remedied in [Falk_Winth_loc_coch_14], where a patch-based construction resembling that of the Clément operator [Clement_appr_75] is introduced. However, no approximation properties are discussed, and stability is achieved only in the graph space of the appropriate differential operator, e.g., but not in for the case of interest here.
1.3 Equivalence of local-best and global-best approximations
Finally, in a seemingly rather unconnected recent result, Veeser [Veeser_approx_grads_16] showed that the error in the best approximation of a given scalar-valued function in by continuous piecewise polynomials is equivalent up to a generic constant to that by discontinuous piecewise polynomials. This result is termed equivalence of global- and local-best approximations. A predecessor result in the lowest-order case and up to data oscillation can be easily deduced from Carstensen et al. [Cars_Pet_Sched_comp_FEs_12, Theorem 2.1 and inequalities (3.2), (3.5), and (3.6)], see also the references therein; equivalences between approximations by different numerical methods are studied in [Cars_Pet_Sched_comp_FEs_12]. A similar result is also given in Aurada et al. [Aur_Fei_Kem_Pag_Praet_loc_glob_13, Proposition 3.1], and an improvement of the dependence of the equivalence constant on the polynomial degree in two space dimensions is developed in [Can_Noch_Stev_Ver_hp_AFEM_17, Theorem 4]. This equivalence result might be surprising at a first glance, since the local-best error is clearly smaller than the global-best one. The twist comes from the fact that the function to be approximated is continuous in the sense of traces because of its -regularity, so one does not gain in approximating it by discontinuous piecewise polynomials. For finite element discretizations of coercive problems, this result in particular allows one to obtain a priori error estimates without the passage through the Bramble–Hilbert lemma. Another important application is for approximation classes in the theory of a-posteriori-based convergence and optimality [Veeser_approx_grads_16].
1.4 The present manuscript
Let , , be a Lipschitz polygon or polyhedron with boundary . Let be a (possibly empty) subset and let ; precise definitions are given below. Let be a simplicial and shape-regular (possibly locally refined) mesh of with elements of diameter , with , on which we consider Raviart–Thomas–Nédélec (RTN) piecewise polynomials of order . Variable polynomial degrees can be taken into account by proceeding as in, e.g., [Dol_Ern_Voh_hp_16]. We avoid it here for the sake of clarity of exposition. Our main results can be divided into four groups.
1) Equivalence of local- and global-best approximations in under minimal regularity
Consider the best-approximation error , in a dimensionally consistent weighted -norm defined in (3.1) below, of an arbitrary function by piecewise RTN functions that are -conforming and satisfy a constraint on the divergence. Because of the -conformity (normal trace continuity) constraint, this is a global minimization problem. In our first main result, Theorem 3.1, we show that is, up to a generic constant, equivalent to the local-best approximation errors defined by elementwise minimizations without any constraint on the inter-element continuity of the normal trace or on the divergence. We stress that only depends on on each mesh element ; we simply write to alleviate the notation. The generic constant entering the equivalence result only depends on the shape-regularity of the simplicial mesh , the space dimension , and the polynomial degree . This extends the results of [Aur_Fei_Kem_Pag_Praet_loc_glob_13, Can_Noch_Stev_Ver_hp_AFEM_17, Cars_Pet_Sched_comp_FEs_12, Veeser_approx_grads_16] to the case. A variant of this result, Theorem 3.4, bounds, up to a polynomial-degree robust constant, the global-best error by the local-best approximation errors of one degree lower.
Our main tool to achieve the above results is the equilibrated flux reconstruction. This allows us to transform locally (on patches of elements) a discontinuous piecewise polynomial with a suitable elementwise divergence constraint into a normal-trace-continuous piecewise polynomial with the expected elementwise divergence constraint. This has been traditionally used in a posteriori error analysis of primal finite element methods derived from -formulations, see Destuynder and Métivet [Dest_Met_expl_err_CFE_99], Luce and Wohlmuth [Luce_Wohl_local_a_post_fluxes_04], Braess and Schöberl [Braess_Scho_a_post_edge_08], Ern and Vohralík [Ern_Voh_adpt_IN_13, Ern_Voh_p_rob_15], Becker et al. [Beck_Cap_Luce_flux_rec_16], and the references therein. We now employ it here in the context of a priori error analysis of dual approximations in . In order to prove Theorem 3.1, we combine the equilibrated flux reconstruction with the bubble function technique of Verfürth [Verf_13] (see also Gudi [Gud_a_pr_med_10] for using this tool in a priori error analysis). It is this step that leads to the dependence of the generic constant in Theorem 3.1 on the polynomial degree . As for Theorem 3.4 that leads to a -robust bound, as in several references above, we rather rely on the -robust stability of the right inverse of the divergence operator of Costabel and McIntosh [Cost_McInt_Bog_Poinc_10, Corollary 3.4] on each single element. We then combine this result with the -robust patch stability bounds of Braess et al. [Brae_Pill_Sch_p_rob_09, Theorem 7] in two space dimensions and Ern and Vohralík [Ern_Voh_p_rob_3D_19, Corollaries 3.3 and 3.8] in three space dimensions, which are themselves based on the stable polynomial extension operators of Demkowicz et al. [Demk_Gop_Sch_ext_III_12, Theorem 7.1].
2) A simple stable local commuting projector defined under the minimal regularity
It turns out that the methodology we develop in point 1) above in order to prove the local-best global-best equivalence results stated in Theorem 3.1 immediately leads to the definition of a projector that is defined over the entire space without any additional regularity, enjoys a commuting property with the divergence operator, is locally defined over patches of elements, and is stable in for all functions with piecewise -polynomial divergence. This operator is also stable in up to a data oscillation term for the divergence for all functions , and it is stable in for all functions in that space.
This may be compared to the result of [Falk_Winth_loc_coch_14], where stability in is achieved but not in . Moreover, our projector has a very simple construction, with elementwise local-best approximations combined patch by patch to the final projector via the flux equilibration technique. The essential (no-flux) boundary condition on a part of the computational domain only is here taken into account without any difficulty. All these results are summarized in Definition 3.7 and Theorem 3.8 and improve on [Christ_Wint_sm_proj_08] in that they are local, on [Falk_Winth_loc_coch_14] in that they lead to the above-described -stability up to data oscillation, and on [Ern_Guer_quas_int_best_appr_17] in that they satisfy a commuting property with the divergence operator.
3) Optimal -approximation estimates in
Our third main result is Theorem 3.5 where we derive -approximation estimates. These estimates feature the following four properties: i) they request no global regularity of the approximated function beyond ; ii) only the minimal local regularity is needed, that is, for all , for any ; iii) the convergence rates are fully optimal in both the mesh-size and the polynomial degree , in particular featuring no logarithmic factor of the polynomial degree ; iv) no higher order norms of the divergence of appear in the bound whenever . This improves on [Demk_Buf_q_opt_proj_int_05, Demk_pol_ex_seq_int_06] in removing the suboptimality with respect to the polynomial degree, on [Demk_Buf_q_opt_proj_int_05, Demk_pol_ex_seq_int_06, Mel_Roj_com_p_interp_19] in reducing the regularity requirements, and on approximations using Clément-type operators in removing the need for regularity assumptions over the (overlapping) elemental patches while reducing it instead to (nonoverlapping) elements. The proof of these fully optimal -approximation estimates relies on the elementwise local-best approximation errors described in point 1) and crucially combines Theorems 3.1 and 3.4.
4) Applications to mixed finite element and least-squares mixed finite element methods
We finally showcase how the above results can be turned into fully optimal -a priori error estimates
for two popular classes of numerical methods for second-order elliptic partial differential equations. In mixed finite element methods, c.f. the original contributions of Raviart and Thomas[Ra_Tho_MFE_77] and Nédélec [Ned_mix_R_3_80], or the textbook Boffi et al. [Bof_Brez_For_MFEs_13], the error between the exact flux and its mixed approximation immediately takes the form of the -norm term in the constrained global-best approximation error of Theorems 3.1 and 3.4 (cf. Lemma 5.1), so the application is immediate. For the family of least-squares mixed finite element methods, see Pehlivanov et al. [Peh_Car_Laz_LS_MFEs_94], Cai and Ku [Cai_Ku_LS_MFE_L2_10], Ku [Ku_LS_MFE_min_reg_13], and the references therein, the application is a little less immediate, and for completeness we establish it in Lemmas 5.2 and 5.3. These results allow us in particular to circumvent the typical use of interpolation or quasi-interpolation operators to obtain error estimates that hinge upon increased regularity assumptions. Note also that an immediate application of the above commuting projector in the context of mixed finite elements is the construction of a Fortin operator under the minimal regularity.
The rest of the manuscript is organized as follows. In Section 2, we introduce the discrete setting and the main notation. In Section 3, we state our main results, namely Theorems 3.1 and 3.4 stating the relations between the local- and global-best approximations, Theorem 3.5 stating the optimal -approximation estimates, and Theorem 3.8 about the simple stable local commuting projector. We prove these results in Section 4. Finally, we present an application of our main results to the a priori error analysis of mixed finite element and least-squares mixed finite methods in Section 5.
2 Discrete setting and notation
Let for be an open, bounded, connected polygon or polyhedron with Lipschitz boundary . Let be a given conforming, simplicial, possibly locally refined mesh of , i.e. , where any is a closed simplex and the intersection of two different simplices is either an empty set or their common vertex, edge, or face. Let be a (possibly empty) closed subset of , and let be its (relatively open) complement in , with the assumption that matches and in the sense that every boundary face of the mesh is fully contained either in or in . Let , and . Furthermore, we define the space , where on means that for all functions that have vanishing trace on .
For an open subset , let and . We also denote by and the -inner product and norm for scalar- or vector-valued functions on . In the special case where , we drop the subscript, i.e. and . The diameter of is denoted by , and its outward unit normal as .
For any mesh element , its diameter is denoted by , and we set . Let denote the set of interior vertices of , i.e. the vertices contained in . Let denote the set of vertices of on the boundary , and set . We divide into two disjoint sets and , where contains all vertices in (recalling that is assumed to be closed) and consists of all vertices in . For each vertex , define the patch and the corresponding open subdomain . The piecewise affine Lagrange finite element basis function associated with a vertex is denoted by . Let denote the set of all -dimensional faces of . By convention, we consider faces to be closed sets. Let stand for the set of interior faces of the mesh, i.e. those faces in that are intersections of the boundaries of two mesh elements. Let , respectively , denote the set of all faces in that are contained in , respectively , and let be the set of all boundary faces. The assumption that is a conforming mesh that matches the boundary components and implies that and . For an element , we denote the set of all faces of by , and the set of all vertices of by . For each interior vertex , we let denote the set of all faces that contain the vertex (and thus do not lie on the boundary of ). For boundary vertices , let collect the faces that contain the vertex but do not lie on the Dirichlet boundary . The mesh shape-regularity parameter is defined as , where is the diameter of the largest ball inscribed in .
Let be a nonnegative integer. For , where is an element and is a face, we define as the space of all polynomials of total degree at most on . If denotes a subset of elements of , is the space of piecewise polynomials of degree at most over . Typically, will be either the whole mesh or the patch as defined above. We define the piecewise Raviart–Thomas–Nédélec space , where and denote the space of -valued functions defined on with each component being a polynomial of degree at most in . Note that with this choice of notation, functions in the space do not necessarily belong to ; thus, is a proper subspace of which is classically characterized as those functions in having a continuous normal component across interior faces . Moreover, is a subspace of , the space of piecewise (broken) first-order component-wise differentiable vector-valued fields over . To avoid confusion between piecewise smooth and globally smooth functions, we denote the elementwise gradient and the elementwise divergence by and by , respectively.
If is an interior face, then there exist two mesh elements and in such that . We then let be the unit normal to that points outward and inward . We define for any the jump and mean on by
If is a boundary face, then, for notational convenience, we define as well as .
For each polynomial degree , let denote the -orthogonal projection of order . Similarly, let denote the -orthogonal projection of order on a face , which maps to . Let be the piecewise Raviart–Thomas–Nédélec interpolant. The domain of can be taken (much) larger than , but not as large as piecewise fields; the present choice is sufficient for our purposes. For any , the interpolant is defined separately on each element by the conditions
where denotes the normal trace of , the restriction of to . Note that (2.2) implies that for all faces . A useful property of the operator is the commuting identity:
In the spirit of Braess et al. [Brae_Pill_Sch_p_rob_09] and [Ern_Voh_adpt_IN_13, Ern_Voh_p_rob_15, Ern_Smears_Voh_H-1_lift_17], we define the local mixed finite element spaces by
where contains those boundary faces from that share the vertex . In particular, we observe that when , then on for any . As a result of the above definitions, it follows that the zero extension to all of of any belongs to .
3 Main results
3.1 Equivalence of local- and global-best approximations in
For any function , we consider the global-best approximation error defined as the best approximation, in a weighted norm, from , subject to a constraint on the divergence:
We further consider the local approximation errors defined on each element by
Note that the minimization in (3.2) does not involve a constraint on the divergence nor on the normal component on (whenever relevant). Furthermore, since is the -orthogonal projection onto the broken polynomial space , we have . Thus the local approximation errors involve the local-best approximation errors in plus a weighted best approximation error of the divergence.
Theorem 3.1 (Equivalence of local- and global-best approximations).
There exists a constant depending only on the space dimension , the shape-regularity parameter of , and the polynomial degree , such that, for any ,
Theorem 3.1 shows that the global-best approximation error , is, up to a constant, equivalent to the local-best approximation error, i.e. the sum of the individual best approximation errors over all mesh elements . The first inequality in (3.3) is clearly nontrivial since the local-best approximations do not generally lead to a -conforming object. Since the minimization in (3.2) is done over the space without any normal trace or divergence constraint, the second inequality in (3.3) holds trivially.
Remark 3.2 (Necessity of the divergence error terms).
The terms might seem superfluous at the first sight, since they take an identical form in both and . However, they cannot be removed from the local contributions . Otherwise, it would be possible to choose a sequence of functions in approaching a function but such that the middle term in (3.3) would tend to zero but would remain uniformly bounded away from zero.
Remark 3.3 (Equivalence with constraint on the right-hand side).
Theorem 3.1 also straightforwardly implies that
with the same constant , where the minimization problems in the middle term include a constraint on the divergence to mirror the divergence constraint in .
3.2 Polynomial-degree-robust one-sided bound
Our second result states a bound where the global-best approximation error is bounded in terms of the sums of local-best approximation errors with a constant that is robust with respect to the polynomial degree, but where the polynomial degree in the local approximation errors is instead of . As a result, this is a one-sided inequality and not an equivalence, and it is valid only for .
Theorem 3.4 (Polynomial-degree-robust bound).
There exists a constant , depending only on the space dimension and the shape-regularity parameter of , such that, for any and any ,
3.3 Optimal-order -approximation estimates
We now focus on functions with some additional elementwise regularity. For any and any mesh element , let denote the space of vector fields in with each component in . Our third main result delivers -optimal convergence rates for vector fields in with the minimally necessary additional elementwise regularity. Recall the definition (3.1) of .
Theorem 3.5 (-optimal convergence rates).
Let and let be such that
Let the polynomial degree . Then there exists a constant , depending only on the regularity exponent , the space dimension , and the shape-regularity parameter of , such that
where if and if .
Remark 3.6 (Full -optimality).
3.4 A simple stable local commuting projector in
Our fourth main result is the construction of a local and stable commuting projector defined over the entire that leads to an approximation error that is equivalent to the local-best approximation error. Recall the definition of the broken Raviart–Thomas–Nédélec interpolant from (2.2) and that of the piecewise polynomial patchwise -conforming spaces from (2.4). Recall also that zero extensions of elements of belong to , and that is the piecewise affine Lagrange finite element basis function associated with the vertex .
Definition 3.7 (A simple locally-defined mapping from to ).
Let be arbitrary. Let be defined elementwise by
For each mesh vertex , let be defined by
Extending the functions from the patch domains to the rest of by zero, we define by
The justification that the construction of is well-defined in given in Section 4.1 below. The first step (3.7) in Definition 3.7 considers the elementwise -norm local-best approximation that defines the discontinuous piecewise RTN polynomial closest to under the divergence constraint. The second step in (3.8) can be seen as smoothing over the patch subdomains to obtain an -conforming approximation over each vertex patch with a suitably prescribed divergence. These approximations are then summed into . The overall procedure is motivated by equilibrated flux reconstructions coming from a posteriori error estimation [Dest_Met_expl_err_CFE_99, Braess_Scho_a_post_edge_08, Ern_Voh_adpt_IN_13]. Here we adapt those techniques to the purpose of a priori error analysis. Our main result is the following.
Theorem 3.8 (Commutativity, projection, approximation, and stability of ).
Let a mesh of and a polynomial degree be fixed. Let be arbitrary. Then, from Definition 3.7 satisfies
where are the neighboring elements of . Furthermore, we have
recalling that denotes the diameter of and where the constant only depends on the space dimension , the shape-regularity parameter of , and the polynomial degree .
Property (3.13) readily implies that is globally -stable up to data oscillation of the divergence, since summing over the mesh elements leads to
Similarly, from (3.14), we infer that is -stable, since
We note that, for the divergence term, (3.13) improves the bound (5.2) of [Falk_Winth_loc_coch_14, Theorem 5.2] (in particular, we have in place of ), whereas (3.14) is similar to the combination of the bounds (5.2) and (5.3) of [Falk_Winth_loc_coch_14, Theorem 5.2].
4 Proofs of the main results
This section, whose goal is to prove our main results, namely Theorem 3.1, Theorem 3.4, Theorem 3.5, and Theorem 3.8, is organized as follows. First, in Section 4.1, we provide some justifications on the construction of the mapping from Definition 3.7 and we establish the statement (3.10) from Theorem 3.8 (on commuting with the divergence). Then, in Section 4.2, we prove the statement (3.12) from Theorem 3.8 on the approximation properties of . This is the more technical part of the proofs. Next, in Section 4.3, we finish the proof of Theorem 3.8 by proving the remaining three statements (3.11), (3.13), and (3.14), and we also prove Theorem 3.1. The proof of Theorem 3.4 is then presented in Section 4.4, where we adapt slightly the arguments from the previous sections. Finally the proof of Theorem 3.5, which hinges on Theorem 3.1 and on Theorem 3.4, is contained in Section 4.5.
4.1 Justifications on the construction of
We start by showing that the projector of Definition 3.7 is well-defined on
Lemma 4.1 (Discrete weak divergence of -projection).
For any function , let be defined elementwise in (3.7). Then
First, observe that for any vertex , the hat function belongs to owing to the conformity of with respect to the Dirichlet and Neumann boundary sets. Therefore, , where we use the fact that is the support of . Since is a constant vector on each element , the Euler–Lagrange equations for (3.7) imply that
Consequently, , and (4.1) follows. ∎
We now show that the local minimization problems (3.8) give well-defined local contributions .
Lemma 4.2 (Existence and uniqueness of local problems).
For each vertex , there exists a unique satisfying (3.8).
The minimization problem (3.8) is equivalent to a mixed finite element problem in the patch subdomain . For Dirichlet boundary vertices , this problem is well-posed with a unique minimizer since the space of (2.4) does not impose the normal constraint everywhere on . For interior and Neumann vertices , the source term in the divergence constraint satisfies the compatibility condition
where the second equality follows from Lemma 4.1. Therefore, is also well-defined for interior and Neumann vertices . ∎
It follows from Lemma 4.2 that is well-defined for every