Stability of mixed FEMs for non-selfadjoint indefinite second-order linear elliptic PDEs

For a well-posed non-selfadjoint indefinite second-order linear elliptic PDE with general coefficients A,b,γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {A}}, {\mathbf {b}},\gamma $$\end{document} in L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty $$\end{document} and symmetric and uniformly positive definite coefficient matrix A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {A}}$$\end{document}, this paper proves that mixed finite element problems are uniquely solvable and the discrete solutions are uniformly bounded, whenever the underlying shape-regular triangulation is sufficiently fine. This applies to the Raviart-Thomas and Brezzi-Douglas-Marini finite element families of any order and in any space dimension and leads to the best-approximation estimate in H(div)×L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H({{\,\mathrm{div}\,}})\times L^2$$\end{document} as well as in in L2×L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2\times L^2$$\end{document} up to oscillations. This generalises earlier contributions for piecewise Lipschitz continuous coefficients to L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty $$\end{document} coefficients. The compactness argument of Schatz and Wang for the displacement-oriented problem does not apply immediately to the mixed formulation in H(div)×L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H({{\,\mathrm{div}\,}})\times L^2$$\end{document}. But it allows the uniform approximation of some L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} contributions and can be combined with a recent L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} best-approximation result from the medius analysis. This technique circumvents any regularity assumption and the application of a Fortin interpolation operator.

1. Introduction.This section introduces the non-selfadjoint indefinite second-order linear elliptic PDE and its mixed formulations.A brief review of earlier results is followed by the assertion of the stability and the best-approximation results.
1.1.Non-selfadjoint indefinite second-order linear elliptic PDEs.The strong formulations for second-order elliptic problems with coefficients A, b, γ componentwise in L 8 pΩq and f P L 2 pΩq read Lj uj " f a.e. in a polyhedral bounded Lipschitz domain Ω Ă R n with homogeneous Dirichlet boundary condition uj " 0 on BΩ for j " 1, 2 and any dimension n ě 2. For all v P H 1 0 pΩq, the two differential operators (referred to as conservative resp.divergence form throughout this paper) read L1v :" ´∇ ¨pA∇v `v bq `γ v and L2v :" ´∇ ¨pA∇vq `b ¨∇v `γ v. (1) The assumption on ellipticity means that the n ˆn coefficient matrix Apxq is symmetric and positive definite with eigenvalues in one universal compact interval of positive reals for a.e.x P Ω.This makes L1, L2 : H 1 0 pΩq Ñ H ´1pΩq Fredholm operators of index zero and their weak formulations apv, wq :" xL1v, wy H ´1pΩqˆH 1 0 pΩq " xL2w, vy H ´1pΩqˆH 1 0 pΩq , for all v, w P H 1 0 pΩq, are dual to each other in the duality bracket x‚, ‚y H ´1pΩqˆH 1 0 pΩq of H ´1pΩq, the dual of H 1 0 pΩq.Throughout this paper, zero eigenvalues are excluded and the kernel (of one of these operators) Lj is supposed to be trivial, so that L1 and L2 are bijections.It is known from the theory of bilinear forms in reflexive Banach spaces [2,3] that this implies well-posedness and the continuous inf-sup condition (e.g., when H 1 0 pΩq is endowed with the norm }∇ ‚ }) (2) 0 ă α :" inf apv, wq }∇v} }∇w} .
The inf-sup constant is the same for the original and the dual problem; apv, wq could be replaced by apw, vq with the same α.The finite element error analysis is enormously simplified under additional conditions on the coefficients that lead to an ellipticity of ap‚, ‚q and allow an application of the Lax-Milgram lemma [2,3,4].The present situation of a general non-selfadjoint indefinite second-order linear elliptic PDE avoids any of those assumptions and examines coefficients in L 8 , which satisfy the following.
Given the various applications to porous media and ground-water flow with rough and oscillating coefficients merely bounded in a well-posed PDE, this contribution gives an affirmative answer to the fundamental question whether the mixed finite element method be used (and then is stable and provides best-approximation property at least for fine triangulations).
The equivalence to the boundary value problems associated with the linear differential operators in (1) and their well-posedness on the continuous level can be found in [5,Sect. 2].This implies the continuous inf-sup conditions [2,3] (5) 0 ă β :" inf The existence and uniqueness of discrete solutions and optimal L 2 error estimates were introduced in [9] for sufficiently fine triangulations in two and three space dimensions under high regularity assumptions, where the pair pσ h , u h q is approximated in RT k pT q ˆPk pT q with the Raviart-Thomas (RT) for 2D (resp.Raviart-Thomas-Nedelec for 3D) finite elements.Global L 8 and global L 2 and negative norm estimates for the conservation form were discussed in [10,14] for smooth coefficients.Provided the coefficients A and b are Lipschitz continuous, γ is piecewise Lipschitz continuous and H 2 regularity of the adjoint system, an interesting convergence phenomenon for the BDM finite element family is clarified in the fairly general framework of [8].
Let M k pT q be any RT or BDM finite element space of degree k P N0 and define the discrete space V pT q :" M k pT q ˆPk pT q Ă H based on a shape-regular triangulation T with mesh-sizes ď δ, written T P Tpδq.
In case A and b are globally Lipschitz continuous and γ is piecewise Lipschitz continuous, the convergence results in [8] also establish stability in the sense (6) 0 ă β0 ď inf bpx h , y h q }x h }H }y h }H p": β h q for some positive δ and β0.With extra work and refined arguments along the lines of [8], but with reduced elliptic regularity and solution uj P H 1 0 pΩq X H 1`s pΩq to Ljuj " f for some s ą 0. Those arguments are not valid under Assumption (A).
Modern trends in the mathematics of mixed finite element schemes include local stable projections with commuting properties [11,12,13]; those techniques do not seem to allow the proof of discrete stability and best-approximation under assumption (A).
Piecewise Lipschitz continuous coefficients with regularity in H 1`s (for some positive s) lead in [5] to stability for the lowest-order RT FEM.The equivalence to nonconforming Crouzex-Raviart finite elements holds more generally [1] and the combination with the arguments from [17] and [5] might lead to stability results under the assumption (A) for more examples.In comparison, the methodology of this paper provides stability for any degree k and any dimension n (RT and BDM merely serve as popular model examples).
This theorem implies [2,3] that the mixed finite element problems for the RT and the BDM finite element families of any degree k and in any space dimension n are (i) uniquely solvable, (ii) uniformly bounded in H, and (iii) fullfil quasi-optimal error estimates in the norm of H, whenever the underlying shape-regular triangulation is sufficiently fine.
Theorem 1 and the tools of this paper lead to L 2 best-approximation up to oscillations.
Theorem 2 (L 2 best approximation).Suppose δ ą 0 satisfies (6) with bp‚, ‚q defined for general b1, b2 P L 8 pΩ; R n q under Assumption (A).Assume T P Tpδq and that x :" pσ, uq P H (resp. x h " pσ h , u h q P V h :" V pT q :" M k pT q ˆPk pT q) satisfy bpx ´xh , y h q " 0 for all y h P V h .Then the following results (a) and (b) hold.(a) There exists a positive constant C1, which exclusively depends on β0 ą 0, the L 8 norms of (all the components of ) A 1{2 b1, A 1{2 b2, and γ, as well as on the shape-regularity of T, such that the piecewise mesh size hT in T and the L 2 projection Π k onto P k pT q satisfy C ´1 1 p}σ ´σh } A ´1 `}u ´uh }q ď min τ h PM k pT q }σ ´τh } A ´1 `}u ´Πk u} `}hT p1 ´Πk q div σ}.
(b) Suppose b1 " 0 and that the scalar γpxq is Lipschitz continuous in x P intpT q, the interior of T P T , with a Lipschitz constant smaller than or equal to Lippγq.Then there exists a positive constant C2, which depends exclusively depends on β0 ą 0, }A 1{2 b2} L 8 pΩq , Lippγq, and the shape-regularity of T, such that }σ ´τh } A ´1 `}hT pu ´Πk uq} `}hT p1 ´Πk q div σ}.
The additional oscillations }hT pu ´Πk uq} and }hT p1 ´Πk qpdiv σq} can be higher-order contributions and then these terms explain the improved convergence of one variant for the BDM finite element family in [8] under Assumption (A).This article, thus, generalises earlier contributions [5], [8]- [10], [12]- [14] for smooth or piecewise Lipschitz continuous coefficients to L 8 coefficients without any further assumptions.The compactness argument of Schatz and Wang [17] for the displacement-oriented problem does not apply immediately to the mixed formulation in Hpdivq ˆL2 .Remark 12 below explains that no uniform L 2 approximation of the divergence component holds.This paper therefore compensates the lack of compactness by the computation and analysis of an optimal test function (the dual solution y in (7) of Subsection 1.4 below).Recent bestapproximation for the flux in L 2 from the medius analysis [6,15] combines with the compactness for the (dual) PDE.This and a careful shift of the discrete divergence circumvents the aforementioned lack of compactness in the divergence variable.In fact, this new methodology avoids any regularity argument and any Fortin interpolation at all.1.4.Motivation.This subsection outlines the proof of the discrete inf-sup stability (6) in an abstract framework to guide the reader through the arguments.Suppose X h ˆYh is a finite dimensional subspace of H ˆH with dual X h ˆY h and let x h P SpX h q, i.e., x h belongs to X h and has norm }x h }H " 1. Recall ( 5) and the well-posedness of the problem (3).Then, the dual problem is well-posed as well and xx h , ‚yH " bp‚, yq has a unique dual solution y in the Hilbert space pH, x‚, ‚yH q.The continuous inf-sup condition ( 5) shows (7) β }y}H ď }bp‚, yq} H ˚" }x h }H " 1, whence }y}H ď 1{β is bounded.Suppose that y h P Y h is a close approximation to y with }y ´yh }H ď ε for some positive ε ă 1{}b}, where }b} is the operator norm of the bilinear form bp‚, ‚q.Since it remains to bound }y h }H , e.g., with the triangle inequality The combination of the previous two displayed formulas gives a lower bound for }bpx h , ‚q} Y h .Under the assumption that ε is independent of y and so of x h , this estimate reads This proves β0 ď βp1 ´ε}b}q{p1 `εβq provided the approximation error }y ´yh }H is small independently of X h ˆYh and x h P SpX h q.A detailed investigation in Subsection 3.3 below reveals that the above strong form of a uniform approximation appears neither available in the norm of H " Hpdiv, Ωq ˆL2 pΩq (cf.Remark 12) nor necessary for the stability under assumption (A).Recent results from a medius analysis [6,15] and a careful shift of the discrete divergence variable successfully circumvent a uniform approximation in H.
1.5.Structure of the paper.Section 2 starts with the pre-compactness for uniform approximation and the precise assumptions on the set of admissible triangulations T. The other two preliminary subsections concern the L 2 best-approximation of the fluxes and some discrete approximation result for the RT finite element family.The stability analysis in Section 3 is based on the dual solution y in the conservative formulation characterised in Subsection 3.1.One contribution of y involves the PDE L2φ " g and allows for some pre-compacness and uniform approximation in Subsection 3.2.The proof of Theorem 1 concludes Section 3. A combination of the stability result (6) with the approximation arguments leads in Section 4 to Theorem 2, which generalises [6,15] to non-selfadjoint indefinite second-order linear elliptic problems.

Preliminaries.
This section introduces notations used in the paper, fixes the assumptions on the admissible triangulation T, discusses an abstract version of compactness argument in [17], and then recalls some L 2 best-approximation property and concludes with an observation for the RT finite element family.

Notation. Standard notation on Lebesgue and Sobolev spaces
and Hpdiv, Ωq apply throughout this paper.The L 2 scalar product p‚, ‚q L 2 pΩq induces the norm } ‚ } :" } ‚ } L 2 pΩq and the orthogonality relation K.
Whereas } ‚ } denotes the norm in L 2 pΩq with the exception of the abbreviation }b} for the bound of the bilinear form bp‚, ‚q, the vector space L 2 pΩ; R n q is endowed with the weighted scalar product p‚, ‚q A ´1 :" pA ´1‚, ‚q L 2 pΩq and induced norm } ‚ } A ´1 :" }A ´1{2 ‚ } and so, for any τ P L 2 pΩ; R n q, is its distance distpτ, M h q :" minτ h PM h }τ ´τh } A ´1 to any subspace M h of L 2 pΩ; R n q.The norm }pτ, vq}H in the Hilbert space Hpdiv, Ωq is weighted with A ´1 in L 2 pΩ; R n q for the flux variable so the Hilbert space H " Hpdiv, Ωq ˆL2 pΩq has the weighted scalar product x‚, ‚yH with the induced norm }pτ, vq}H, (9) }pτ, vq} 2 H :" }τ } 2 A ´1 `} div τ } 2 `}v} 2 for all pτ, vq P H.
Duality brackets have the dual pairing as an index as in x‚, ‚y H ´1pΩqˆH 1 0 pΩq above.To abbreviate the definition of inf-sup constants throughout this paper, let SpV q :" tv P V : }v}V " 1u for any normed linear space pV, } ‚ }V q.
2.2.Assumptions on the discretization.The finite element spaces are based on admissible triangulations, the set T of all of those has certainly infinite cardinality; the point is that the constants in standard interpolation error estimates become universal through uniform shape regularity.
Definition 3 (admissible triangulations).The set of admissible triangulations T is a set of shape-regular triangulations of the polyhedral bounded Lipschitz domain Ω Ă R n into simplices with uniform shape regularity and arbitrary small mesh sizes.Let hmaxpT q :" max hT for the piecewise constant mesh-size hT for T P T, defined by hT |T :" diampT q in T P T , and abbreviate T pδq :" tT P T : hmaxpT q ď δu.
Given T P T, let P k pT q denote the polynomials of total degree at most k P N0 seen as functions on T P T P T and set P k pT q :" tv k P L 8 pΩq : @T P T , v k |T P P k pT qu.Let Π k : L 2 pΩq Ñ L 2 pΩq be the L 2 projection onto P k pT q with respect to T P T.
Definition 4 (discrete spaces).Any T P T is associated to the finite-dimensional subspace V pT q " M k pT q ˆPk pT q of V :" L 2 pΩ; R n q ˆL2 pΩq with M k pT q :" RT k pT q or M k pT q :" BDM k pT q of order k P N0 from [2].
The best-approximation error reads distpv, V pT qq :" inft}v ´vh } : v h P V pT qu with the weighted L 2 norm, }v} 2 " }τ } 2 A ´1 `}w} 2 for v " pτ, wq P V .The density of smooth functions and standard approximation results for smooth functions proves the well-known pointwise convergence in the sense that each v P V satisfies [2] (10) lim T PTpδq distpv, V pT qq " 0.
The application of the previous lemma to the finite element approximation of the solution of the PDE reads as follows.
2.4.L 2 best-approximation of the fluxes.The medius analysis of mixed finite element methods employs arguments from a priori and a posteriori error analysis [6,15] to prove new L 2 best-approximation results.Recall that Π k is the L 2 projection onto P k pT q and hT is the mesh-size associated to T .
Lemma 7 (flux L 2 best-approximation).There exists a constant C3, which depends on the shape-regularity in T , on Ω and on α, α, such for any p P Hpdiv, Ωq and any T P T, there exists p h P M k pT q such that div p h " Π k div p and This is the L 2 best-approximation result from [15, Lemma 5.1] for mixed finite element approximations for the unit matrix A. Although with a different focus, the paper [6] introduces a general framework with a mesh-dependent norm } ‚ } h in P k pT q; while [11, Eq (3.6)] presents a localized refinement of this lemma.
Proof.Given p P Hpdiv, Ωq, the right-hand sides F pwq :" pw, div pq L 2 pΩq and Gpqq :" pp, qq lead in the elliptic mixed formulation (for the Laplacian) pσ, qq ´pu, div qq L 2 pΩq `pw, div pq L 2 pΩq " Gpqq `F pwq for all pq, wq P H to the unique solution pσ, uq " pp, 0q P H. Its straight-forward mixed finite element discretisation substitutes H by V h :" M k pT q ˆPk pT q and leads to a unique discrete solution pp h , v h q P V h with div p h " Π k div p.This and [6, Thm 2.2] lead to the asserted bestapproximation result (in terms of (non-weighted) L 2 norms) }p ´qh } `}hT p1 ´Πk q div p}.
The constant C4 from [6,15] does not depend on the coefficients A, b, γ but depends on the shape-regularity in T and on Ω.The equivalence of norms concludes the proof and leads to the asserted constant C3, which depends on C4 and α, α.

A discrete approximation result for Raviart-Thomas functions.
In any space-dimension n and degree k, the RT functions satisfy a rather particular approximation estimate with the componentwise L 2 projection Π k onto P k pT q.
Lemma 8. Any τRT P RT k pT q satisfies }τRT ´Πk τRT } ď n pn`1qpn`kq }hT div τRT }.The proof will be postponed to the appendix because of its focus on the RT finite element shape functions.The statement of the above lemma fails for the BDM finite element family.
3. Stability analysis.This section deals with approximation of fluxes and stability result.The design of a test function in the proof of a discrete inf-sup condition is based on the characterisation and approximation of a dual solution.

Dual solution and conservative formulation.
The inner structure of the dual solution y exploits the elliptic PDE and generates some compactness argument in the subsequent subsection.Recall that the operator L2 : H 1 0 pΩq Ñ H ´1pΩq from (1) is bijective.Theorem 9 (dual solution in conservative formulation).Suppose b1 :" A ´1b and b2 " 0 a.e. in Ω in (4).Then x " pσ, uq P H and y " pζ, zq P H satisfy xx, ‚yH " bp‚, yq in H if and only if ζ " σ ´A∇φ and z " div σ ´φ a.e. in Ω for the weak solution φ P H 1 0 pΩq to L2φ " g :" b ¨A´1 σ `pγ ´1q div σ ´u P L 2 pΩq.
The combination of the three preceding identities leads to the PDE ´divpA∇φq `b1 ¨A∇φ `γφ " ´divpzAb2q `pb1 ¨Ab2q z `b1 ¨σ `pγ ´1q div σ ´u in the sense of distributions.Since b2 " 0, the right-hand side g belongs to L 2 .This proves one direction of the assertion; the direct proof of the converse is omitted.l Remark 10 (no divergence formulation).The proof shows the extra term ´divpzAb2q P H ´1pΩq in case (4) is considered for non-zero b2 P L 8 pΩ; R n q.This term does not belong to L 2 pΩq under Assumption (A) and is, therefore, excluded.

Approximation of the fluxes.
The subsequent lemma describes the uniform approximation of the flux variable by a combination of the compactness argument and the L 2 best-approximation of Subsections 2.3 and 2.4.
Lemma 11 (flux approximation).Given any ε ą 0, there exists δ ą 0 such that the following holds for all T P Tpδq and g P L 2 pΩq.There exists some p h P M k pT q that approximates p :" A∇φ P Hpdiv, Ωq for the weak solution φ P H 1 0 pΩq to L2φ " g with div p h " Π k div p and }p ´ph } A ´1 ď ǫ}g}.
Proof.Given any ε ą 0 and the constant C3 from Lemma 7, Lemma 6 leads to a positive δ ď mint1, 2 ´1ǫ{C3u with sup T PTpδq dist ppA∇φ, b ¨∇φ `γφq, V pT qq ď 2 ´3{2 ǫ{C3 }g} (the distance is with respect to the weighted norm } ‚ } A ´1 in L 2 pΩ; R n q and } ‚ } in L 2 pΩq).Lemma 7 applies to p :" A∇φ with div p " b ¨∇φ `γφ ´g P L 2 pΩq and, for any T P Tpδq, leads to some approximation p h P M k pT q with div p h " Π k div p and This concludes the proof.
Remark 12 (no uniform approximation in Hpdivq).Lemma 11 does not state a uniform approximation estimate for the divergence and, in fact, an estimate of the form } divppṕ h q} ď ǫ}g} cannot hold in general.To see this, adopt the notation of the proof of Lemma 11 and a reverse triangle inequality for }g ´Πk g} ´} divpp ´ph q} ď }p1 ´Πk qpb ¨∇φ `γφq} ď ǫ{p2C3q }g}.
Therefore, the approximation error } divpp ´ph q} will not tend to zero uniformly for all g P SpL 2 pΩqq as ǫ and δ tend to zero.l Example 13 (RT approximation in Hpdivq for particular g).The stability analysis in Subsection 1.4 concerns a discrete x h :" pσ h , u h q with norm }pσ h , u h q}H " 1 and leads in Theorem 9 to the particular right-hand side g :" b ¨A´1 σ h `pγ ´1q div σ h ´uh with }g} ď C5 ă 8 for the essential supremum C 2 5 of |A ´1{2 b| 2 `|γ ´1| 2 `1 in Ω.This g allows for a uniform approximation of p by p h in Hpdiv, Ωq for the RT finite element family.
For instance, in the extreme case of piecewise constant coefficients, g´Π k g " b¨A ´1p1Π k qσ h .With C6 :" }A ´1b} L 8 pΩq , Lemma 8 shows }g ´Πk g} ď δC6.The combination with Lemma 11 lead to p h with }p ´ph } Hpdiv,Ωq ď ǫ C5 `δ C6.This and the arguments of Subsection 1.4 lead to the discrete stability (6).
A direct calculation shows b1ppτ, ´vq, pσ, ´uqq " b2ppσ, uq, pτ, vqq for all pσ, uq, pτ, vq P H.This and a duality argument (singular values of a square matrix coincide with those of its transposed) in the last equality show Hence, the divergence formulation has the same discrete inf-sup constant β h .l Remark 14 (δ dependence).The size of δ in (6) is hidden behind a compactness argument of Lemma 11.Besides the norms and parameters mentioned in Assumption (A), the mapping properties of L ´1 2 are of relevance as well.A review of the proofs of this paper shows that there is a finite sub-cover of SpL 2 pΩqq with small balls in H ´1pΩq that leads to a finite number of (without loss of generality) smooth functions k1, . . ., kJ as in the proof of Lemma 5.The size of δ is related to the approximation properties of the weak solutions Φj to L2Φj " kj a.e.The regularity properties of Φj P H 1 0 pΩq are not characterised for Assumption (A): In fact, it is unknown whether Φj belongs to any H 1`s pΩq for any s ą 0. Under Assumption (B) and reduced elliptic regularity, however, the afore mentioned approximation properties could be quantified more and reveal further information on δ.
4. L 2 Best-approximation.The notation of Theorem 2 applies throughout this section with continuous and discrete solutions x " pσ, uq and x h " pσ h , u h q.
4.1.Proof of Theorem 2.a.Given p :" σ P Hpdiv, Ωq and T P Tpδq, choose σ h :" p h P M k pT q as in Lemma 7, and define e h :" pσ h ´σh , u h ´Πk uq P V h .Given β0 ą 0 in (6) there exists some y h " pτ h , v h q P V h with }y h }H " 1 and β0 }e h }H ď bpe h , y h q " bppσ h ´σh , u h ´Πk uq, y h q " bppσ ´σh , u ´Πk uq, y h q.
Since u ´Πk u K div τ h and v h K divpσ ´σh q, the last term is equal to C ´1 3 }σ ´σh } A ´1 ď distpσ, M k pT qq `}hT p1 ´Πk q div σ}.This and the distance distL (measured in the norm } ‚ }L) lead to }e h }H ď C8 maxt1, C3u{β0 pdistLppσ, uq, V h q `}hT p1 ´Πk q div σ}q .This, (16), and a triangle inequality conclude the proof.l 4.2.Proof of Theorem 2.b.Throughout this subsection, let b1 " 0 and b2 :" A ´1b a.e. in Ω in (4) and let f P L 2 pΩq be a fixed right-hand side for the continuous and discrete problem L2u " f .Return to the proof of the previous subsection with e h and follow the first lines until β0 }e h }H ď bpe h , y h q " pσ ´σh , τ h ´vh Ab2q A ´1 `pu ´Πk u, γ v h q L 2 pΩq .
This completes the rest of the proof.l 4.3.Conservative formulation.Theorem 2.a includes an error estimate for the conservative formulation with b1 :" A ´1b and b2 :" 0 in (4) and σ " ´A∇u ´u b with div σ " f ´γu.The refined analog of Theorem 2.b is not expected because of an extra term exemplified in the extreme case of piecewise constant coefficients b1 and γ.The arguments of Subsection 4.1 lead to β0 }e h }H ď bppσ ´σh , u ´Πk uq, y h q " pσ ´σh , τ h q A ´1 `ppu ´Πk uqb1, τ h ´Πk τ h q L 2 pΩq .
The last term is not of higher order for the BDM finite element family as pointed out in [8] through numerical evidence.For the RT finite element family, however, Lemma 8 shows }τ h ´Πk τ h }À}hT div τ h } and then leads to a higher-order contribution in the asserted inequality of Theorem 2.b as the final result.l The arguments could be generalised, but those result are of limited relevance as the convergence order is not generally improved in comparison with Theorem 2.a.An exception is the example of [7, Sect 3.5] (with b " 0 " γ on the unit ball) when Theorem 2.b guarantees Opδ 2 q for the L 2 flux error for k " 0. This proves div τRT " divpgpxq xq `qk´1 " pn `kq gpxq `qk´1 for some q k´1 P P k´1 pT q.The comparison with τRT " gpxq x `pk leads to some polynomial remainder r k P P k pT ; R n q in τRT " pn `kq ´1 pdiv τRT q x `rk for all x P T.