ROBUST A POSTERIORI ERROR CONTROL AND ADAPTIVITY FOR MULTISCALE, MULTINUMERICS, AND MORTAR COUPLING∗

FOR MULTISCALE, MULTINUMERICS, AND MORTAR COUPLING

GERGINA V. PENCHEVA^†, MARTIN VOHRAL´IK^‡, MARY F. WHEELER^†, AND TIM WILDEY^†

Abstract. We consider discretizations of a model elliptic problem by means of different numer- ical methods applied separately in different subdomains of the computational domain and coupled using the mortar technique. The subdomain grids need not match along the interfaces. We are also interested in the multiscale setting, where the subdomains are partitioned by a mesh of sizeh, whereas the interfaces are partitioned by a mesh of much coarser sizeH, and where lower-order poly- nomials are used in the subdomains and higher-order polynomials are used on the mortar interface mesh. We derive several fully computable a posteriori error estimates which deliver a guaranteed upper bound on the error measured in the energy norm. Our estimates are also locally efficient and one of them is robust with respect to the ratioH/hunder an assumption of sufficient regularity. The present approach allows to bound separately and to compare mutually the subdomain and interface errors. A subdomain/interface adaptive refinement strategy is proposed and numerically tested.

Key words. multiscale, multinumerics, mortar coupling, nonmatching grids, a posteriori error estimate, guaranteed upper bound, robustness, balancing error components

AMS subject classifications.65N15, 65N30, 76S05

1. Introduction. We consider in this paper the model problem

−∇·(K∇p) =f in Ω, (1.1a)

p= 0 on ∂Ω, (1.1b)

where Ω ⊂ R^d, d = 2,3, is a polygonal (polyhedral) domain (open, bounded, and connected set), K is a symmetric, bounded, and uniformly positive definite tensor, and f ∈ L²(Ω). Let Ω be divided into several subdomains. We are interested in discretizations by different numerical methods in the different subdomains. The cou- pling of these different methods is achieved by the mortar technique. We allow for the cases where the grids of the individual subdomains do not match along the interfaces and where the subdomain grid elements are a mixture of simplices and of rectangular parallelepipeds. We also investigate the case where the size of the subdomain grids, say h, is much smaller than the size of the interface grid, say H. More precisely, we suppose that H =O(h^β) with β <1; then lower-order polynomials are used in the subdomain grids and higher-order polynomials are used on the mortar interface mesh. Particular examples of such discretizations are the multiscale mortar mixed finite element method proposed in [6] or the multiscale mortar coupled mixed finite

∗A portion of this research was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences. The Center for Frontiers of Subsurface Energy Security (CFSES) is a DOE Energy Frontier Research Center, under Contract No. DE-SC0001114. The authors gratefully acknowledge the financial support provided by the NSF-CDI under contract number DMS 0835745 and King Abdullah University of Science and Technology (KAUST)-AEA-UTA08-687 and DOE grant DE-FGO2-04ER25617. The second author was supported by the GNR MoMaS project “Numerical Simulations and Mathematical Modeling of Underground Nuclear Waste Disposal”, PACEN/CNRS, ANDRA, BRGM, CEA, EdF, IRSN, France.

†Institute for Computational Engineering and Sciences, University of Texas at Austin, USA (gergina@ices.utexas.edu,mfw@ices.utexas.edu,twildey@ices.utexas.edu).

‡UPMC Univ. Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France & CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France (vohralik@ann.jussieu.fr).

1

element–discontinuous Galerkin method of [19]. Note that multiscale mortar tech- niques are especially appealing as the discretization can be reduced to a problem only involving higher-order polynomials on the interface mortar mesh (see [6]). In addi- tion, this immediately leads to a parallel implementation following [20]. A multiscale mortar basis can be constructed following [18] and multiscale preconditioners can be used following [34]. This leads to their extremely high computational efficiency.

The purpose of this paper is to derive a general framework for optimal a posteriori error estimation in the multiscale, multinumerics, and mortar coupling setting. We derive fully and locally computable estimates providing a guaranteed upper bound on the energy error. We propose several estimators, with increasing complexity of evaluation but with increasing precision. All these estimators are also locally efficient;

that is, they also give local lower bounds on the energy error. Importantly, this property holds independently of the use of different discretization schemes in different parts of the domain, of the use of the mortar coupling, and, to a reasonable degree, of the non-alignment of the subdomain meshes at the interfaces. Our estimates are thus robust with respect to the multinumerics and mortar coupling. Moreover, the last of our estimators gives estimates robust with respect to the ratio H/h and is thus robust with respect to the multiscale. We take here up the analysis of [35, 6]

while using tools from [30,33,17]. For recent results on a posteriori error estimation in multiscale discretizations, we refer to [26, 1, 23] and to the references therein.

A posteriori estimates for discretizations with mortar coupling have previously been analyzed in, e.g., [36, 8,7,13,9,10] and a posteriori estimates for multinumerics in, e.g., [10,14].

In Section2, we set up the notation, define the admissible grids, finite-dimensional spaces, and describe the continuous setting. Our a posteriori error estimates are stated in Section3. We do so in a general setting, not mentioning any particular (combination of) numerical methods employed. We only suppose that we are given an approximate fluxuhwhich isH(div,Ωi)-conforming inside each subdomain Ωi, locally conservative inside each subdomain Ωi, and whose normal trace is weakly continuous across the interface; see Assumption 3.1. As mentioned earlier, we derive several estimators.

Some of them only use the given (nonmatching) gridT^h and some of them require a construction of a matching gridTbhand of a globallyH(div,Ω)-conforming fluxt_h. The fluxt_hcan be constructed by directly prescribing its degrees of freedom, by solution of low-order-polynomial-degreeh-size local Neumann problems, or by solution of high- order-polynomial-degreeH-size local Neumann problems by means of the mixed finite element method. All these approaches are described in detail in Section3.

Section 4 investigates the local efficiency of the derived estimates. Once again, this is done generally, without a specification of the underlying numerical scheme(s);

we only need Assumption4.1on the weak continuity of the approximate potential ˜ph. Section 5 then gives examples of multiscale, multinumerics, and mortar discretiza- tions. We therein also verify Assumptions3.1 and4.1 for each example in question.

Sections 6 and 7 respectively collect the proofs of the a posteriori error estimates and of their efficiency. Our estimates allow to distinguish and estimate separately the error coming from the inside of the subdomains and that coming from the mor- tar coupling; an adaptive algorithm keeping the two error contributions in balance and numerical experiments illustrating the theoretical developments are given in Sec- tion8. (The present estimates can also be used to distinguish and estimate separately the discretization and upscaling errors, cf. Remark 3.9below.) Appendix A gives a technical result necessary for the analysis on nonmatching grids.

Ω1 Ω2

Ω3

Ω4

Th

GH G∗

h

Fig. 2.1.Example of a domainΩwith subdomainsΩⁱand nonmatching meshTh(left), interface meshGH (middle), and the corresponding interface meshG_h^∗(right)

2. Preliminaries. We introduce in this section the partitions of Ω, notation, continuous setting, and recall some useful inequalities.

2.1. Partitions ofΩand ofΓ. We suppose that Ω is decomposed into nonover- lapping polygonal (polyhedral) subdomains Ωi,i∈ {1, . . . , n}. This partition can be nonmatching in the sense that neighboring subdomains need not share complete sides (edges if d= 2, faces if d= 3). We denote Γi,j :=∂Ωi∩∂Ωj, Γ := ∪1≤i<j≤nΓi,j, and Γi:=∂Ωi∩Γ. LetTh,i be a matching finite element mesh of Ωi,i∈ {1, . . . , n}, composed of simplices and rectangular parallelepipeds. A mixture of triangles and rectangles is allowed ford= 2 but we only allow for either tetrahedra or rectangular parallelepipeds ford= 3 (we would need to introduce other elements like prisms for d= 3 in order to allow for mixture grids, which we prefer to avoid for the sake of sim- plicity). We then set T^h:=∪ⁿi=1T^h,i and denote by hthe maximal element diameter inT^h; note thatT^h can be nonmatching as neighboring meshesT^h,iandT^h,j need not align on Γi,j. A generic element of the partitionT^h will be denoted byT; hT stands for the diameter ofT. This setting is illustrated in Figure2.1(left).

We use Eh,i^int to denote the interior sides of T^h,i, i ∈ {1, . . . , n}, and set Eh^int :=

∪ⁿi=1Eh,i^int;Eh^intthus contains neither the subdomain interfaces nor the outer boundary of Ω. We denote byEh,i all the sides of Th,i and setEh :=∪ⁿi=1Eh,i. We letEh,i,j^Γ be the partition of Γi,j by the sides of T^h,i and Eh,i^Γ the partition of Γi by the sides of T^h,i. We denote byEh^Γ :=∪1≤i≤nEh,i^Γ all the sides ofT^hlocated at the interface Γ and byEh^ext :=Eh\ Eh^int\ Eh^Γ the faces of Th located at the boundary of Ω. We also set Eh^int,Γ:=Eh^int∪ Eh^Γ. The notationET stands for all the sides of en elementT ∈ Th. A generic side ofEhwill be denoted byeand its diameter byhe.

Next, we let G^H,i,j be the mortar interface finite element mesh of Γi,j. The elements g ∈ G^H,i,j are either line segments (if d= 2) or triangles or rectangles (if d= 3). We do not requireG^H,i,j to be matching in the sense of a (d−1)-dimensional mesh of Γi,j. We setG^H,i:=∪1≤j≤nG^H,i,j and G^H :=∪1≤i<j≤nG^H,i,j, cf. Figure2.1 (middle). Maximal element diameter inG^His denoted byH. In the multiscale setting, h < H ≤1 and the ratio H/hcan be unbounded, H =O(h^β) withβ <1. Note that on an interface Γi,j, G^H,i,j is a unique (d−1)-dimensional surface mesh, whereas there are two (different) meshesEh,i,j^Γ and Eh,j,i^Γ from the two sides of the interface.

Also, the meshesEh,i,j^Γ andEh,j,i^Γ need in general not be refinements ofG^H,i,j; we will, however, need such a requirement at some occasions later. We also assume that the intersection of the meshes Eh,i,j^Γ and Eh,j,i^Γ is a matching mesh of Γi,j consisting of line segments (if d= 2) or triangles or rectangles (ifd= 3), such that for any of its

b

Th TH TH

Fig. 2.2. Example of a matching refinementTbhofTh(left), of a nonmatching and not shape- regular meshTH (middle), and of its modification avoiding nonconvex elements (right)

TH

TH

Fig. 2.3. Example of a shape-regular but nonmatching mesh TH (left) and of a matching and shape-regular meshTH (right)

elementT, the ratio |T|/|T^′| for any elementT^′ ofEh,i,j^Γ or Eh,j,i^Γ is bounded.

We will in the sequel also use the following partition of Γ. Let an interface Γi,j

be given. We define the mesh Gh,i,j^∗ as a set of (d−1)-dimensional sides g, where eachg∈ Gh,i,j^∗ is simultaneously a union of sides fromEh,i,j^Γ and a union of sides from Eh,j,i^Γ . We choose the sidesg so as to be composed of a smallest possible number of sides from Eh,i,j^Γ and Eh,j,i^Γ . There is one meshGh,i,j^∗ for each interface Γi,j. We set Gh,i^∗ := ∪^1≤j≤nG^∗h,i,j and Gh^∗ := ∪^1≤i<j≤nGh,i,j^∗ , cf. Figure 2.1 (right). We denote byHg the diameter of a side g ∈ G^H and by hg the diameter of a side g ∈ Gh^∗. We suppose that there exists a positive constantCG^∗_h such that, for allg∈ Gh^∗,

(2.1) hg

he ≤CG_h^∗ ∀e∈ Eh^Γ, e⊂g.

Assuming (2.1), we avoid the case wherehg/heis only bounded by a function ofH/h.

Finally, two other types of partitions of Ω will be used in the paper. Firstly,Tb^his a matching refinement ofT^h, consisting of simplices and rectangular parallelepipeds (as forT^h, mixture grids are allowed ford= 2). We refer to Figure 2.2(left) for an example of Tbh. We denote by Ebh the sides of Tbh and use the notation Tbh,i for the restriction of Tb^h on the subdomain Ωi. We suppose that Tb^h,i coincides with T^h,i in the interior of each subdomain Ωi and only differs fromTb^hnear the interfaces. More precisely, we assume that for eachT ∈ T^hsuch thatT∩Γ =∅, there exists an element

T^′∈Tb^hsuch thatT =T^′. We also assume that every sidee^′∈Eb^hwhich shares a node with thisT^′either coincides with some sidee∈ Eh^int or belongs to the interior of some T^′′∈ T^h. Finally, we assume thatTb^h adds no new nodes and sides at the interface Γ in comparison withT^h. The last mesh used is denoted by T^H. It will be in general formed by groups of elements from T^h and it will differ in different approaches. It may be matching or nonmatching, shape-regular (cf. Section 4 for our definition of shape-regularity) or not; it’s characteristic feature is that the restriction ofT^H on Γ is the interface mesh G^H. As before, we use the notation T^H,i for the restriction of T^H on Ωi. We refer to Figure2.2(middle and right) and to Figure 2.3for examples of different meshesT^H. For an elementT ∈ T^H, we denote byG^T the set of its sides.

2.2. Finite-dimensional spaces and projection operators. We begin with the mortar space MH. It is the space of discontinuous piecewise polynomials of or- derm on the interface meshG^H; MH thus in particular contains piecewise constant functions on G^H. We next define the spaces onT^h. If T ∈ T^h is a simplex, we let R_r(T) := P_r(T) be the space of polynomials of total degree at most r. If T ∈ T^h is a rectangular parallelepiped, we let R_r(T) :=Q_r(T) be the space of polynomials of degree at mostr in each variable. We then define R_r(Th) as the space such that for eachw ∈ R_r(Th), w|T ∈ R_r(T); we require no continuity at the sides. We also define R_k−1,∗,d(T) by [P_k−1(T)]^d for a simplex and Q_k−1,k(T)×Q_k,k−1(T) if d= 2 andQ_k−1,k,k(T)×Q_k,k−1,k(T)×Q_k,k,k−1(T) ifd= 3 for a rectangular parallelepiped.

LetV_h,i×Wh,i⊂H(div,Ωi)×L²(Ωi) be the Raviart–Thomas–N´ed´elec (RTN) mixed finite element spaces of order k, Vh,i :=RTN^k(T^h,i), Wh,i :=R_k(T^h,i), cf. [11, 28].

The present theory can easily be extended to other mixed finite element spaces. We can also easily take into account spaces with different polynomial degrees in different subdomains and also with different polynomial degrees in different elements; we re- strict ourselves to the given setting for the sakes of brevity and clarity. We then set Vh := Ln

i=1Vh,i, Wh :=Ln

i=1Wh,i. Note that the normal components of vectors in V_h are continuous across the sides between elements in each subdomain Ωi but not across Γ. Based on these spaces, we will also use V_bh, the RTN space on the matching submesh Tb^h ofT^h, and W_bh, the Wh equivalent on Tb^h. We keep the same order k when both T^h and Tb^h are simplicial. When Tb^h contains simplices cutting rectangular parallelepipeds (as in Figure 2.2(left)) and when using the approach of Section3.3.2below, we have to appropriately increase the space order tok^′inV_bhand W_bh, see (3.20a). In the case whereT^H is matching and shape-regular,VH stands for them-th order RTN space on the coarse meshT^H andWH for theWh equivalent on T^H. We use the notation V(S) for the restriction of some spaceV, a priori defined on the whole Ω/meshT^h, to the subdomain/submeshS.

We will also need some orthogonal projections: letPWh be theL²(Ω)-orthogonal projection ontoWh,PW_bh theL²(Ω)-orthogonal projection ontoW_bh,PWH theL²(Ω)- orthogonal projection ontoWH, andPMH theL²(Γ)-orthogonal projection ontoMH,

PWh :L²(Ω)→Wh forw∈L²(Ω), (w−PWh(w), wh) = 0 ∀wh ∈Wh, PW_hb :L²(Ω)→W_bh forw∈L²(Ω), (w−PW_bh(w), wh) = 0 ∀wh ∈W_bh, PWH :L²(Ω)→WH forw∈L²(Ω), (w−PWH(w), wH) = 0 ∀wH ∈WH, PMH :L²(Γ)→MH forµ∈L²(Γ), (µ−PMH(µ), µH)Γ= 0 ∀µH∈MH. We also denote byπlthe orthogonal projection ontoR_l(T) (T is the mesh in question).

2.3. Other notation. LetD⊂Ω. Thenk · k^D stands for theL²(D) norm and (·,·)D for the L²(D) scalar product. Shall D coincide with Ω, the subscript D will be dropped. The L²(D) scalar product forD ⊂R^d−1 will be denoted by h·,·i^D. We also use the notation|D|for the (d−1)-dimensional Lebesgue measure ofD ⊂R^d′, 1≤d^′≤d. For v∈L¹(D), we denote byvD the mean value ofv onD. ForD ⊂Ω, we denote byc^K,D,C^K,D the smallest and largest eigenvalue ofKonD, respectively.

For a sufficiently smooth function v that is double-valued on an interior side e∈ Eh^int, e=T⁻∩T⁺, its jump and average oneare defined as

(2.3) [[v]] :=v|T⁻−v|T⁺, {{v}}:= ¹₂(v|T⁻+v|T⁺).

We set [[v]] :=v|^e and{{v}}:=v|^e fore∈ Eh^ext. We use similar notation for the sides g from GH andGh^∗ and also for the sidese from Ebh. For each sidee ∈ Eh^int,g ∈ GH, or g∈ Gh^∗, we use the notationn_e,n_g for the unit normal vector, pointing from T⁻ towardsT⁺. Similarly,nΓ stands for the unit normal vector to Γ, with arbitrary but fixed orientation. For boundary sides e, ne coincides with the unit normal vector, outward to Ω. Similarly, for D ⊂ Ω, nD is systematically used to denote the unit normal vector, outward toD.

2.4. Bilinear form, weak solution, energy norm. Let the symmetric bilin- ear formAbe given by

(2.4) A(u,v) := (u,K⁻¹v), u,v∈L²(Ω).

The weak solution of (1.1a)–(1.1b) isp∈H₀¹(Ω) such that (2.5) A(K∇p,K∇ϕ) = (f, ϕ) ∀ϕ∈H₀¹(Ω).

Recall that u:= −K∇p is the weak flux satisfying u∈ H(div,Ω). Let H¹(T^h) :=

{ϕ∈L²(Ω); ϕ|^T ∈ H¹(T)∀T ∈ T^h} be the broken Sobolev space. We will use the sign∇to denote the elementwise gradient. We define the energy seminorm onH¹(T^h) (2.6) |||ϕ|||² :=A(K∇ϕ,K∇ϕ) =K¹²∇ϕ², ϕ∈H¹(T^h),

and the energy norm onL²(Ω) by

(2.7) |||v|||²∗:=A(v,v) =K⁻¹²v², v∈L²(Ω).

2.5. Poincar´e and trace inequalities. We recall here two basic inequalities necessary in order to obtain our a posteriori error estimates. LetT be an element of any of the finite element partitions considered, e its side, andϕ ∈ H¹(T). Firstly, Poincar´e’s inequality states that

(2.8) kϕ−ϕTk^T ≤CP,ThTk∇ϕk^T.

The constantCP,Tis equal to 1/πwheneverTis convex. Secondly, the trace inequality states that

(2.9) kϕ−ϕeke≤Ct,T,ehe¹²k∇ϕkT.

It has been shown in [25, Lemma 3.5] that Ct,T,e = (Ct,d)¹²(|e|h²_T/(|T|he))¹², where Ct,d≈0.77708 for a triangle,Ct,d≈3.84519 for a tetrahedron, andCt,d= 1/(πtanhπ) for a rectangle.

3. A posteriori error estimates. We present in this section a general frame- work for a posteriori error estimates in the multiscale, multinumerics, and mortar coupling setting. We propose several guaranteed and fully computable estimates and we discuss their practical evaluation.

We present the results in a general setting, not relying on any particular discretiza- tion method. Recall that the meshesT^h andG^H and the spaceVh are introduced in Section2. We only make the following assumption:

Assumption 3.1 (Properties ofuh). Let 1. uh∈Vh;

2. (∇·u_h,1)T = (f,1)T ∀T ∈ Th; 3.

Xn

i=1

hu_h·nΩi, µHi^Γi = 0 ∀µH ∈MH.

Assumption 3.1 means that the approximate flux uh is H(div,Ωi)-conforming inside each subdomain Ωi, locally conservative inside each subdomain Ωi on the el- ements of T^h,i, and that its normal trace is weakly continuous across the interface sides. We will, in fact, still weaken this assumption in some of the results below.

3.1. Estimates for the flux. We first state estimates for the error in the flux u_h. We need to introduce some more notation. Let TH be a coarse-scale mesh as introduced in Section 2.1. This mesh is only needed in order to state the following estimate, it is not needed for the computation ofuh. For a given subdomain Ωi and a given interface sideg∈ G^H,i, letTi,g denote the element ofT^H,i havingg as a side.

We assume that the elementsTi,g, g ∈ G^H,i, i∈ {1, . . . , n}, do not overlap. Such a meshT^H can always be constructed; remark also that the choice ofT^H is not unique.

For a given i ∈ {1, . . . , n}, g ∈ G^H,i, and the associated Ti,g ∈ T^H,i, recall that we have the trace inequality (2.9) with the constantCt,Ti,g,g. For a givenT ∈ T^h, recall that we have Poincar´e’s inequality (2.8) with the constantCP,T. The first result is:

Theorem 3.2 (A “mortar” estimate for the flux). Let u be the exact flux and letuh satisfy Assumption3.1. Pick an arbitrarysh∈H₀¹(Ω). Then

|||u−uh|||^∗≤ηP+ηR,h+ηM,

where thepotential,residual, andmortar estimatorsare given respectively by ηP:=|||uh+K∇sh|||^∗,

(3.1)

ηR,h:=

(X

T∈Th

C_P,T² h²_Tc⁻¹K,Tkf− ∇·uhk²T

)¹₂ , (3.2)

ηM:=

( _n X

i=1

Xn

j=1

X

g∈GH,i,j

1

2k[[uh·ng]]k^gCt,Ti,g,gHg¹²c⁻K,T¹²i,g

2)¹₂ . (3.3)

Theorem3.2gives a guaranteed upper bound on the error|||u−uh|||^∗. It contains the mortar estimate ηMwhich clearly corresponds to the discontinuity of the normal component ofu_hacross the subdomain interfaces. It is thus possible to distinguish the error component stemming from the subdomain discretization error (ηPandηR,h) and the one due to the mortar coupling at the subdomain interfaces (ηM) and to adaptively decide whether the subdomain or mortar approximation should be improved. The estimate of Theorem3.2may, however, overestimate the mortar error since it is based on the trace inequality (2.9) and features the (known) constantCt,Ti,g,g.

In order to possibly improve on this point, we also present the following theorem (hereT^His either a coarse-scale or a fine-scale partition of Ω, to be determined later):

Theorem 3.3 (An optimal estimate for the flux). Let u be the exact flux and letuh∈L²(Ω) be arbitrary. Pick an arbitrarysh∈H₀¹(Ω) and letth∈H(div,Ω) be arbitrary but such that

(3.4) (∇·th,1)T = (f,1)T ∀T ∈ T^H. Then

(3.5) |||u−uh|||∗≤ηP+ηR,H+ ˜ηM, whereηP is given by (3.1)and

ηR,H :=

(X

T∈TH

C_P,T² H_T²c⁻¹K,Tkf − ∇·thk²T

)¹₂ , (3.6)

˜

ηM:=|||uh−th|||^∗. (3.7)

We remark that the nature of the above estimate is clear. Instead ofηPand ˜ηM, respectively, it can be written with inf_sh∈H0¹_(Ω)|||uh+K∇sh|||^∗ and inf^t_h|||uh−th|||^∗, where t_h are to be taken in H(div,Ω) and such that (∇·t_h,1)T = (f,1)T for all T ∈ TH. Then the first term corresponds to the fact that u_h 6=−K∇sh for some sh ∈H₀¹(Ω), which is which is the case for the exact flux, whereas the second term corresponds to the facts thatuh6∈H(div,Ω) and∇·uh6=f.

3.2. Estimates for the potential. We state here our estimates for the error in the potential. We present them for a general potential approximation ˜ph∈H¹(T^h).

Theorem 3.4 (A “mortar” estimate for the potential). Let pbe the exact po- tential, let p˜h ∈ H¹(T^h) be arbitrary, and let uh satisfy Assumption 3.1. Pick an arbitrarysh∈H₀¹(Ω). Then

|||p−p˜h||| ≤ηNC+ηR,h+ηM+ηDF,

where ηR,h is given by (3.2), ηM by (3.3), and the nonconformity and diffusive flux estimatorsare given respectively by

ηNC:=|||p˜h−sh|||, (3.8)

ηDF:=|||K∇p˜h+u_h|||∗. (3.9)

As in Section3.1, the following improved version of Theorem 3.4holds (hereTH

is as in Theorem3.3):

Theorem 3.5 (An optimal estimate for the potential). Let p be the exact po- tential and let p˜h ∈ H¹(T^h) be arbitrary. Pick an arbitrary sh ∈ H₀¹(Ω) and let th∈H(div,Ω)be arbitrary but such that (3.4)holds. Then

(3.10) |||p−p˜h||| ≤ηNC+ηR,H +ηDFM,

withηNC given by (3.8),ηR,H by (3.6), and thediffusive flux–mortar estimator by ηDFM:=|||K∇p˜h+th|||^∗.

(3.11)

The nature of the estimate is again clear: we can replaceηNCby inf_sh∈H¹

0(Ω)|||p˜h− sh|||which corresponds to the fact that ˜ph 6∈H₀¹(Ω), which is the case for the exact potential. Similarly, one can write the estimate with inft_h|||K∇p˜h+th|||∗ with th ∈ H(div,Ω) and (∇·th,1)T = (f,1)T for allT ∈ T^H, which corresponds to the facts that K∇p˜h 6∈H(div,Ω) and −∇·(K∇p˜h)6=f, which is the case for the exact potential.

Note that, by the triangle inequality, ηDFM ≤ηDF+ ˜ηM, so that we can once again distinguish the subdomain and interface errors.

Remark 3.6 (Residual estimator ηR,H). Recall PW_hb and PWH of Section 2.2.

We will meet particular cases where th of Theorems 3.3and3.5 will be such that

(3.12) ∇·t_h=PW_hb(f).

Then we can set T^H :=Tb^h in Theorems3.3and3.5 and replace ηR,H of (3.6)by

(3.13) η_R,bh:=

(X

T∈Tbh

C_P,T² h²_Tc⁻¹K,Tkf−PW_bh(f)k²T

)¹₂ .

Note thatη_R,bh takes the form of the usual data oscillation estimate on the mesh Tbh. We will also meet cases where th of Theorems3.3 and3.5will be such that

(3.14) ∇·th=PWH(f).

Then the residual estimatorηR,H of (3.6)takes the form

(3.15) ηR,H =

( X

T∈TH

C_P,T² H_T²c⁻¹K,Tkf−PWH(f)k²T

)¹₂ .

Note that suchηR,H takes the form of the data oscillation on the meshT^H.

Remark 3.7 (Theorems 3.2–3.5). In Theorems 3.2 and 3.4, only the meshes T^h and G^H are needed (the mesh Tb^h introduced in Section 2.1 may be employed for the construction of the the potential reconstructionsh whenT^h is nonmatching). For Theorems3.3and3.5, the meshesTbh(whenThis nonmatching) andTH introduced in Section2.1and specified in Section3.3below are essential. Note also that the potential reconstruction sh is needed in all Theorems 3.2–3.5, whereas the flux reconstruction t_h is only needed in Theorems3.3and3.5.

Remark 3.8 (Necessity to practically construct sh and t_h). The a posteriori error estimators above only involve norms featuring the reconstructions sh and t_h. Following, e.g., [4], there may be ways how to compute these estimators (evaluate these norms) without the need to practically constructsh andt_h.

Remark 3.9 (Upscaling ofKand the associated error). In the present setting,K is not related to the fine-scale meshThnor to any coarse-scale meshTH (it is not sup- posed to be, for example, piecewise constant or piecewise polynomial on some mesh).

In practice, some upscaled versionKupsc ofK (for instance, piecewise polynomial on TH) may be used to obtain u_h (andp˜h). We then can estimate

ηP=kK⁻¹²(uh+K∇sh)k ≤ k(K⁻¹² −K⁻

1

upsc2 )uhk+k(K¹²−K

1

upsc2 )∇shk +kK⁻

1

upsc2 (uh+Kupsc∇sh)k,

where the first two terms represent theupscaling errorand the last term thediscretiza- tion error. Similar estimates can be devised for the other estimators.

We will consider below the restrictions of the different estimators to each element T, denoted byη_·,T. Similarly, forg∈ G^H,i,j, we set

(3.16) ηM,g :=k[[uh·ng]]k^gHg¹²

n1

2Ct,Ti,g,gc⁻K,T¹²i,g

2

+

1

2Ct,Tj,g,gc⁻K,T¹²j,g

2o¹₂ .

3.3. Practical construction of H₀¹(Ω)- and H(div,Ω)-conforming recon- structions. We describe here practical constructions of H₀¹(Ω)-conforming recon- structionsh figuring in Theorems3.2–3.5and ofH(div,Ω)-conforming reconstruction t_hfiguring in Theorems3.3and3.5. We only propose one way of constructingshbut we derive several ways of constructingt_h.

Letg∈ G^H,i,j. Then Assumption3.1(3) implies

(3.17) h[[uh·n_g]], µgig= 0 ∀µg∈MH(g).

From (3.17), the precise sense of the weak continuity is obvious: jumps of the normal components ofuh are orthogonal ongto the polynomials contained inMH(g). This in particular allows us to define a function F∈MH such that

(3.18) F|^Γi,j :=PMH((uh|^Ωi·nΓ)|^Γi,j) =PMH((uh|^Ωj·nΓ)|^Γi,j) i, j∈ {1, . . . , n}. Finally, using the weak continuity, we also clearly have

huh|^Ωi·ng,1i^g=huh|^Ωj·ng,1i^g =h{{uh·ng}},1i^g=hF,1i^g

∀g∈ GH,i,j, i, j ∈ {1, . . . , n}. (3.19)

3.3.1. Construction of sh. We propose here a particular construction relying on the meshTb^h; other constructions, only using the meshT^h, are possible.

Recall that Tb^h is the matching submesh of T^h introduced in Section 2.1 and note that R_r(T^h)⊂R_r^′(bT^h), r^′ ≥r, i.e., every piecewise polynomial onT^h is also a piecewise polynomial onTb^h, where possibly the polynomial degreeris increased tor^′. OnR_r^′(Tb^h), we can thus use the averaging interpolateI^av:R_r^′(Tb^h)→R_r^′(Tb^h)∩H₀¹(Ω) defined as follows: for a function ϕh ∈R_r^′(Tb^h), the values ofI^av(ϕh) at a Lagrange nodeV inside Ω are given by the average of the values ofϕh at this node,

I^av(ϕh)(V) = 1

|TbV| X

T∈TbV

ϕh|T(V),

whereTb^V is the set of thoseT ∈Tb^h to whichV belongs and where for any setS,|S| denotes its cardinality. Note that I^av(ϕh)(V) =ϕ(V) at those nodes V lying in the interior of someT ∈Tb^h. At boundary nodes, the value ofI^av(ϕh) is set to zero. We refer to [21,12,31] for more details.

3.3.2. Construction of t_h ∈ V_bh by direct prescription. We define here th∈V_bhwhere Tb^his the mesh introduced in Section2.1, cf. Figure2.2(left).

Remark that{{uh·ne}}is univalued and is a polynomial on alle∈Eb^h. This follows from the assumption thatTb^h is a refinement ofT^h and, consequently, all sidese∈Eb^h either coincide with the sides of E^h, are subsides of the sides ofE^h, or pass through

the interior of someT ∈ T^h. Recall also that by assumption,Tb^h is a matching mesh of Ω. The fluxth is prescribed by the degrees of freedom ofV_bh on the meshTb^hby

th·ne:={{uh·ne}} ∀e∈Eb^h, (3.20a)

(th,rh)T = (uh,rh)T ∀rh∈R_l−1,∗,d(T),∀T ∈Tb^h, (3.20b)

where l = k or k^′, depending on the order k or k^′ of V_bh(T). Remark that th

given by (3.20a)–(3.20b) indeed satisfiest_h∈H(div,Ω), as required in Theorems3.3 and3.5. Recall also that by assumption, the meshTb^h only differs from the mesh T^h near the interface Γ. Consequently, by the above construction, th completely coin- cides with uh on those elements T ofTb^h which do not share a node with Γ and the estimator ˜ηM,T given by (3.7) is equal to 0 on suchT.

To apply Theorems3.3and3.5, we still need to identify a partitionT^H such that the local conservation property (3.4) holds. For a fixed Ωi, we will form two groups of elements ofT^H,i, denoted respectively byTH,i^Γ andTH,i^int. ElementsT ofTH,i^Γ are given as unions of the elements of Tb^h,i sharing a node with Γ, in a way that ∂T∩Γi is a union of sidesg∈ G^H,i. Optimally,∂T∩Γi is just one sideg∈ G^H,ifor eachT ∈ TH,i^Γ . Clearly, by (3.20a), (3.19), Green’s theorem, and Assumption3.1(2), (3.4) on all such T follows. The elements of the second groupTH,i^int are given directly by the elements of T^h,i; consequently, (3.4) follows immediately from Assumption 3.1 (2). Denote T^H :=∪1≤i≤nT^H,i, TH^Γ :=∪1≤i≤nTH,i^Γ , and TH^int :=∪1≤i≤nTH,i^int. The resulting mesh T^H can be nonmatching and needs not be shape-regular, cf. Figure2.2(middle). In order to optimally bound the constantCP,T from Poincar´e’s inequality (2.8), which is used on elementsT ∈ T^H, cf. (3.6), it is however preferable when the elementsT ∈ T^H are convex; modifications ofT^H to a mesh like that in Figure2.2(right) are possible.

Remark 3.10 (th given by (3.20a)–(3.20b)). The definition of th by (3.20a)–

(3.20b)gives a lower-order (k-th or k^′-th order RTN) polynomial t_h and is virtually cost-free as no local linear systems are to be solved. We obtain local conservation on the meshTH in the sense of (3.4). This mesh is, however, rather unconventional and can be nonmatching and not shape-regular. Most importantly, the mortar error is only evaluated in theh-distance from the interfaceΓ, which, as we will see in Theorem4.5 below, leads to its increased overestimation in the multiscale setting whenh≪H.

3.3.3. Construction of th∈V_bh by the solution of h-grid-sizek-th order local Neumann problems. We define hereth∈V_bh such that∇·th=PW_bh(f). We first form a partitionT^H as in Section3.3.2, while however, forming TH^Γ by all those elements ofTbhwhich are located in a band of widthH around the interface Γ. In this way,T^H can be nonmatching but is shape-regular, cf. Figure2.3(left).

On the elementsT fromTH^int, i.e., on those from partitionT^h, we simply set (3.21) th|^T :=uh|^T T ∈ TH^int.

On the elements T ∈ TH^Γ we consider the mesh Tb^h|^T. On this mesh, we will, following [16], solve local Neumann problems by means of ak-th order mixed finite element method. Denote byV_bh,{{u

h·n_T}},Tthe space of suchv_h∈V_bh(T) thatv_h·n_T = {{uh·nT}}on∂T. Denote by V_bh,0,T the space of suchvh∈V_bh(T) thatvh·nT = 0 on

∂T. We look forth∈V_bh,{{u_h·n_T}},T andqh∈W_bh(T) satisfying (qh,1)T = 0 such that (K⁻¹(th−uh),vh)T −(qh,∇·vh)T = 0 ∀vh∈V_bh,0,T,

(3.22a)

(∇·th, wh)T = (f, wh)T ∀wh∈W_bh(T) such that (wh,1)T = 0.

(3.22b)

Recall from [16] that letting t^′_h :=th−uh, (t^′_h, qh) corresponds to the mixed finite element approximation to the local Neumann problem onT

−∇·(K∇q) =f − ∇·uh inT, (3.23a)

−K∇q·nT =−ωg[[uh·ng]] on allg∈ G^T, (3.23b)

(q,1)T = 0, (3.23c)

whereωg:= ¹₂ wheng6⊂∂Ω andωg:= 0 wheng⊂∂Ω. Note that these problems are well-posed as the Neumann boundary conditions are in equilibrium with the load,

X

g∈GT

hu_h·n_T −ωg[[uh·n_g]],1ig =h{{u_h·n_T}},1i∂T = (f,1)T.

This follows by (3.19), Green’s theorem, and Assumption 3.1 (2). Note also that indeedth given by (3.22a)–(3.22b) is inV_bh, as the Neumann boundary condition on T ∈ TH^Γ given by {{uh·nT}} gives the continuity of the normal component of th on

∂T∩Γ (the same condition is imposed from the opposite side), as well as on ∂T\Γ, where{{uh·nT}}=uh·nT, which is the value imposed in (3.21) onT ∈ TH^int. Finally, as the mixed finite element method minimizes the complementary energy, another way to rewrite equivalently (3.22a)–(3.22b) is

th|^T = arg inf

v_h∈V

h,{{b u

h·nT}},T,∇·v_h=PWbh(f)|||uh−vh|||^∗,T. Thus, on eachT ∈ TH,t_his the best choice from the spaceV_bh,{{u

h·n_T}},T to minimize the quantity|||uh−vh|||∗,T (subject to the constraint∇·vh=PW_hb(f)). Remark that this quantity is related to the definition of the ˜ηM,T estimator, see (3.7).

Remark 3.11 (th given by (3.21)–(3.22b)). The definition (3.21)–(3.22b)gives a lower-order (k-th order RTN) polynomial th and is moderately expensive as local linear systems need to be solved. These systems correspond to mixed finite element approximations of orderk posed overH-sized subdomains withh-sized grids. We ob- tain local conservation on the fine mesh Tb^h in the sense of (3.12). Consequently, we can use (3.13)instead of (3.6). The meshTH is here nonmatching but shape-regular and only serves for the computation of the flux t_h and not for the evaluation of the estimates of Sections3.1–3.2. The mortar error is evaluated in the H-distance from the interface Γ. As we will see in Theorem 4.5below, this leads to a smaller overes- timation in the multiscale setting when h≪H, in comparison with the estimator ηM

of (3.3)or with (3.7)evaluated usingth of Section 3.3.2.

3.3.4. Construction of th∈VHby the solution ofH-grid-sizem-th order local Neumann problems. We define here the flux which is of higher-order (m-th order RTN) on the coarse mesh T^H, th ∈ VH. For this purpose, we need T^H to be both matching and shape-regular; the elements of TH^int are no more given by the elements ofT^hbut rather by unions of elements ofT^has forTH^Γ, cf. Figure2.3(right).

Recall that the main requirement on T^H is that the restriction of T^H on Γ is the interface meshG^H.

In this section, we extend the meshG^H from the interface Γ only to Γ∪∂Ω; we define GH on ∂Ω by the sides of TH lying in ∂Ω. We also suppose that the mortar spaceMH is defined over the whole new GH with the samem-th order polynomials.

Let the flux function F be on Γ given by (3.18). We extend it on∂Ω by F|^∂Ω:=PMH((uh·nΩ)|^∂Ω).

Consider a fixed Ωi and the meshT^H,i. We solve local Neumann problems by means of a m-th order mixed finite element method. Denote by VH,F,Ωi the space of such vH ∈VH(Ωi) thatvH·nΓ =F on∂Ωi∩Γ and that vH·nΩi =F on∂Ωi∩∂Ω and by VH,0,Ωi the space of such vH ∈ VH(Ωi) that vH·nΩi = 0 on∂Ωi. We look for th∈VH,F,Ωi andqH ∈WH(Ωi) satisfying (qH,1)Ωi= 0 such that

(K⁻¹(th−u_h),v_H)Ωi−(qH,∇·v_H)Ωi = 0 ∀v_H∈VH,0,Ωi, (3.24a)

(∇·th, wH)Ωi = (f, wH)Ωi ∀wH∈WH(Ωi) such that (wH,1)Ωi= 0.

(3.24b)

Once again, these problems are well-posed as

hFnΩi·nΓ,1i^∂Ωi∩Γ+hF,1i^∂Ωi∩∂Ω= (f,1)Ωi

by the same reasoning as in the previous sections. We also indeed haveth∈VH, as the Neumann boundary condition on Γ given byF gives the continuity of the normal component ofth. Finally, as above, (3.24a)–(3.24b) is equivalent to

(3.25) th|^T = arg inf

v_H∈V_H,F,Ω

i,∇·v_H=P_WH(f)|||uh−vH|||∗,Ωi.

Remark 3.12 (thgiven by (3.24a)–(3.24b)). The definition(3.24a)–(3.24b)gives a higher-order (m-th order RTN) polynomial t_h and is more expensive with semi-local linear systems. These systems correspond to mixed finite element approximations of orderm posed over the subdomainsΩi withH-sized grids. We obtain local conserva- tion on the meshTHin the sense of (3.14). Consequently,(3.6)takes the form (3.15).

The meshT^H is here both matching and shape-regular. The mortar error is evaluated in the Ωi-size-distance from the interface Γ, which, as we will see in Theorem 4.5 below, leads to its optimal estimation in the multiscale setting whenh≪H.

4. Local efficiency of the a posteriori estimates. We state here the local efficiency of the estimators of Theorems3.2–3.5. We suppose for simplicity thatf is a piecewise polynomial of degreeq.

We will need the following assumption:

Assumption 4.1 (Properties of ˜ph). Let 1. p˜h∈R_r(T^h)for somer≥1,

2. h[[˜ph]],1i^e= 0 ∀e∈ Eh^int∪ Eh^ext, 3. h[[˜ph]],1i^g= 0 ∀g∈ G^∗h.

Assumption4.1means that ˜phis a piecewise polynomial, that the means of traces on interior sides in each subdomain are continuous, that the means of traces on bound- ary sides are zero, and that the means of traces on collections of sides inside the interface Γ are continuous. For all the results, we assume Assumption4.1 (1). This assumption is satisfied by all usual numerical methods. We will present some re- sults without Assumptions4.1 (2)–(3), as these assumptions are not satisfied by all numerical methods.

The following theorem gives the efficiency for the estimatorηDF,T and shows that the efficiency of the estimatorsηP,T andηDFM,T is controlled by the efficiency of the estimatorsηDF,T, ηNC,T, and ˜ηM,T.

Theorem 4.2 (Local efficiency of the diffusive flux and potential estimators).

Let p˜h∈H¹(T^h), uh ∈L²(Ω),sh ∈H₀¹(Ω), and th ∈H(div,Ω) be arbitrary. Then, for allT ∈ T^h,

ηDF,T ≤ |||u−uh|||^∗,T +|||p−p˜h|||^T, ηP,T ≤ηDF,T+ηNC,T,

ηDFM,T ≤ηDF,T+ ˜ηM,T.

We will henceforth assume that the mesh families{T^h}^h>0and{Tb^h}^h>0are shape- regular in the sense that there exist constants κ_Th > 0 and κ_Tb

h > 0 such that minT∈ThρT/hT ≥ κTh and min_T∈bT

hρT/hT ≥ κ_Tb

h for all h > 0, where ρT denotes the diameter of the largest ball inscribed inT. For the approach of Section3.3.4, we will also assume that {T^H}^H>0 are shape-regular, i.e., that there exists a constant κTH >0 such that minT∈THρT/HT ≥κTH for all H >0. We will in the sequel use the notation A.B to denote that A≤CB, where the constantC depends on the space dimension d, the polynomial degree r of ˜ph, r^′ of sh, k of u_h, k, k^′, or m of t_h, and qoff, on the shape regularity parametersκTh ofTh,κ_Tbh of Tbh, (and κTH of TH), onK, and on the constantCG_h^∗ from (2.1) but is independent of any mesh size, the domain Ω, and the regularity of the weak solution (p,u). Let, for T ∈ T^h, the notationT_T stand for the union of all elements T^′ ∈ T^h sharing a node withT and T_T,Γ for the union of all elementsT^′∈ T^h sharing a node withT or with thatg∈ Gh^∗

which contains a node ofT. Similarly, for T ∈ T^H, let the notation T_T,Γ stand for the union of all elements T^′∈ T^h sharing a node with thatg∈ G^H which contains a side of T and sethT_{T ,Γ} := minT^′∈T_{T ,Γ}hT^′. We also denote by T_g all the elements of T^h which share a node with a giveng∈ G^H and sethT_g := minT∈T_ghT.

The following theorem is based on the entire Assumption4.1:

Theorem 4.3 (Local efficiency of the nonconformity estimator). Let Assump- tion 4.1 hold and let sh ∈R_r′(Tbh) be given bysh :=I^av(˜ph). Then, for all T ∈ Th,

ηNC,T .|||p−p˜h|||^TT ifT∩Γ =∅, (4.1a)

ηNC,T .|||p−p˜h|||^TT ,Γ ifT∩Γ6=∅. (4.1b)

For the following theorem, Assumption4.1(1) is sufficient:

Theorem 4.4 (Local efficiency of the nonconformity estimator). Let Assump- tion4.1(1) hold and letsh∈R_r′(Tbh)be given bysh:=I^av(˜ph). Then, for allT ∈ Th,

ηNC,T . X

e∈Ebh;e∩T6=∅

h⁻e¹²k[[˜ph]]k^e. (4.2a)

The following theorem hinges on Assumption3.1:

Theorem 4.5 (Local efficiency of the residual and mortar estimators). Let As- sumption3.1be satisfied. Then:

1) Case whereηR,h is given by (3.2)and ηM by (3.3), localized by (3.16).

Let T ∈ T^h andg∈ G^H. Then

ηR,h,T .|||u−uh|||∗,T, (4.3a)

ηM,g . sHg

hT_g|||u−uh|||∗,T_g. (4.3b)

2) Case wheret_h is constructed following Section3.3.2,ηR,H,T is given by (3.6), andη˜M by (3.7).

Let T ∈ Th∩ TH^int. Then

ηR,H,T .|||u−u_h|||∗,T, (4.4a)

˜

ηM,T = 0.

(4.4b)