Mesh-Free,Interpolant,Observables,for,Continuous,Data,Assimilation

Animikh Biswas ,Kenneth R.Brown and Vincent R.Martinez

1 Department of Mathematics&Statistics,University of Maryland–Baltimore County,1000 Hilltop Circle,Baltimore,MD 21250,USA

2 Department of Mathematics,University of California–Davis,One Shields Avenue,Davis,CA 95616,USA

3 Department of Mathematics &Statistics,CUNY Hunter College,695 Park Ave,New York,NY 10065,USA

4 Department of Mathematics,CUNY Graduate Center,365 5th Ave,New York,NY 10016,USA

Abstract.This paper is dedicated to the expansion of the framework of general interpolant observables introduced by Azouani,Olson,and Titi for continuous data assimilation of nonlinear partial differential equations.The main feature of this expanded framework is its mesh-free aspect,which allows the observational data itself to dictate the subdivision of the domain via partition of unity in the spirit of the so-called Partition of Unity Method by Babuska and Melenk.As an application of this framework,we consider a nudging-based scheme for data assimilation applied to the context of the two-dimensional Navier-Stokes equations as a paradigmatic example and establish convergence to the reference solution in all higher-order Sobolev topologies in a periodic,mean-free setting.The convergence analysis also makes use of absorbing ball bounds in higherorder Sobolev norms,for which explicit bounds appear to be available in the literature only up to H2;such bounds are additionally proved for all integer levels of Sobolev regularity above H2.

Key words: Continuous data assimilation,nudging,2D Navier-Stokes equations,general interpolant observables,synchronization,higher-order convergence,partition of unity,mesh-free,Azounai-Olson-Titi algorithm.

In recent years,several efforts have been made to develop a first-principles understanding of Data Assimilation(DA),where the underlying model dynamics are given by partial differential equations (PDEs) [2,6,8,11,13,20,39,40,54,56,57],as well to provide rigorous analytical and computational justification for its application and support for common practices therein,especially in the context of numerical weather prediction[1,3–5,24–30,32,36,38,41,42,50–52,55].A common representative model in these studies is the forced,two-dimensional (2D) Navier-Stokes equations (NSE)of an incompressible fluid,which contains the difficulty of high-dimensionality by virtue of being an infinite-dimensional,chaotic dynamical system,but whose longtime dynamics is nevertheless finite-dimensional,manifested,for instance,in the existence of a finite-dimensional global attractor[16,31,59].Given a domain Ω⊂R2,the 2D NSE is given by

supplemented with appropriate boundary conditions,whereurepresents the velocity vector field,νdenotes the kinematic viscosity,fis a time-independent,external driving force,prepresents the scalar pressure field.The underlying ideas in the works above,though originally motivated in large part by the classical problem of DA,that is,of reconstructing the underlying reference signal,has since been extended to the problem of parameter estimation;we refer the readers to the recent works [17,18,53] for this novel application.

Central to the investigations of this paper is a certain algorithm for DA which synchronizes the approximating signal produced by the algorithm with the true signal corresponding to the observations.The algorithm of interest in this paper is a nudging-based scheme in which observational data of the signal is appropriately extended to the phase space of the system representing the truth,(1.1).The interpolated data is then inserted into the system as an exogeneous term and is subsequently balanced through a feedback control term that serves to drive the approximating signal towards the observations.In particular,we consider the approximating signal to be given as a solution to the system whereurepresents a solution of (1.1)whose initialization is unknown,Ihurepresents observed values of the signalu,appropriately interpolated to belong the same phase space of solutions to (1.1),hquantifies the spatial density of the observations,andμis a tuning parameter,often referred to as the “nudging” parameter.The algorithm then consists of initializing (1.2) arbitrarily and integrating it forward.The remarkable property of (1.2) is that although the feedback control-μ(Ihv-Ihu)only enforces synchronization towards the observations,full synchronization of the signalsvanduis nevertheless asymptotically ensured.Indeed,this property is conferred through a nonlinear mechanism,referred to as a Foias-Prodi property of determining values in the context of the 2D NSE,that is inherent to the system itself[19,33,34,44,45];this mechanism asymptotically enslaves the small scale features of the solution to its large scale features in the sense that knowledge of the asymptotic convergence of the large scale features of the difference of two solutions automatically imply asymptotic convergence of its small scale features as well.Morally speaking,any system which possess this property“asymptotic enslavement”of small scales to large scales would guarantee the success of the nudging-based algorithm.

The “nudging algorithm” was originally introduced by Hoke and Anthes in [37]in 1976,for finite-dimensional systems of ordinary differential equations.In a seminal paper of Azouani,Olson,and Titi [2],this nudging scheme was appropriately extended to the case of partial differential equations via the introduction of the“interpolant observable operator,” denoted byIhabove.There,it was shown that forμ,h＞0 chosen appropriately,thatvanduasymptotically synchronize in the topology ofH1(Ω),that is,in the topology of square-integrable functions with squareintegrable spatial derivatives.On the other hand,it was observed in the computational work of Gesho,Olson,and Titi [36] the convergence,in fact,appeared to be occurring in stronger topologies,for instance in the uniform topology ofL∞(Ω).This phenomenon was analytically confirmed in [10] in the setting of periodic boundary conditions,where the observational data was given in the form of Fourier modes.In this setting,it was furthermore shown that synchronization occurs in a far stronger topology,that of theanalyticGevrey topology,which is characterized by a norm in which Fourier modes are exponentially weighted in wave-number,provided that sufficiently many Fourier modes are observed.A distinguished property of this norm is that its finiteness identifies a length scale below which the function experiences an exponential cut-offin wave-number,and thus,can be reasonably ignored by numerical computation.In the context of turbulent flows,this length scale is known as thedissipation length scaleand is directly related to the radius of spatial analyticity of the corresponding flow [7,31,35,49].Hence,the result in [10] rigorously established that the nudging-based algorithm synchronizes the corresponding signals all the way down to this length scale.

The case of other forms of observations,e.g.,volume element,nodal values,etc.,was not,however,treated in [10].One of the central motivations of this paper is to therefore address these remaining cases.In order to do so,we develop a modest,general analytical framework in the spirit of [2] that ultimately allows one to demonstrate higher-order synchronization for the nudging-based algorithm,namely,beyond theH1–topology,and in particular,any L2–based Sobolev topology.This framework accommodates a significantly richer class of interpolant observable operators based on the notion of alocalinterpolant observable operator,which effectively allows one to useanymode of observation withinanylocal region of the domain.These local interpolants are then made global by introducing a smooth partition of unity that allows one to patch the various observations across the domain and interpolate them appropriately into the phase space of the system.Although partitions of unity were already considered in several previous works for the nudging-based algorithm [2,13,41,42],the partitions of unity used there were fixed and explicit,while in this work,we directly introduce the partition of unity as an additional parameter.Indeed,the most attractive feature allowed by the framework developed here is that it liberates the observations from the situation conceived in [2] of being constrained by a given distribution of measurement devices across the domain.Moreover,the possibility of having different spatial densities of measurements across the domain is also accommodated by this framework.This,of course,corresponds to the situation where more spatial measurements are simply available in one region of the domain compared to others.We note that this construction is akin to the“Partition Finite Element Method” introduced by Babuska and Melenk [9],where finite element spaces were generalized to be “mesh-free” in an analogous way via partition of unity,thus imbuing them with a greater flexibility.We also refer the reader to the recent results [5] and [43].In the former work,the efficacy of the nudging-based algorithm in the situation of having observations availableonlyin a fixed subdomain is assessed.The latter work studies higher-order interpolation using finite-element interpolants over bounded domains and the solution produced by the subsequent nudging-based algorithm is compared to solutions obtained by direct-numerical simulation from a semi-discrete scheme.

In Section 2,we introduce the functional setting in which we will work throughout the paper.Note that we will workexclusivelyin the periodic setting;the case of other boundary conditions will be treated in a future work.In Section 3,we introduce the notion of “local interpolant observable operators” and construct a“globalization” of them via partition of unity.Their approximation properties are subsequently developed and several nontrivial examples are provided (see Section 3.1).We point out that due to the amount of flexibility allowed by this construction,a significant portion of this work is dedicated to organizing its salient properties and ultimately identifying the combinations of interpolating operators that ultimately ensure well-posedness of the nudging-based algorithm and the important synchronization property described above.Rigorous statements of the main results of the paper are then provided in Section 4 followed by several remarks.In order to clarify the detailed relation between the structure of the interpolant operators and the system,we introduce hyperdissipation into the system.Of course,all of our results contain the original,non-hyperdissipative case.In fact,akey featureof the results is that synchronization in higher-order Sobolev spaces can be guaranteed under essentially the same assumptions onμ,has were made in[2],i.e.,the assumptions exhibit the same scaling inμ,h.In Section 4,we further identify alternative structural assumptions one can make on the interpolation operators that allow one to consider different families of operators that ultimately lead to the synchronization property(see Section 3.2).The proofs of the main statements are provided in Section 5.We point out that in order to properly quantify the assumptions onμ,hrequired by the analysis to guarantee higher-order convergence,it is crucial to identify absorbing ball estimates with respect to the corresponding higher-order norms.This is captured in Section 4.1,which properly generalizes the bounds obtained in [21] for the radius of the absorbing ball of (1.1) with respect to theH2–topology.Finally,various technical details related to well-posedness (see Appendix 5.2) or regarding the various aforementioned examples introduced in Section 3.1 (see Appendix 5.2 and Appendix 5.2) are relegated to the appendices.

The functional setting throughout this paper will be the space of periodic,meanfree,divergence-free functions over T2[0,2π]2.More precisely,letBper(T2) denote the Borel measureable functions over T2,which are 2π-periodic a.e.in each directionx,y.We define the space of 2π-periodic,square-integrable functions over T2by

For eachk＞0,we define the inhomogeneous Sobolev space,Hk(T2) by

The homogeneous Sobolev space is defined as

By the Poincaré inequality,the topologies induced by(2.2)and(2.3)are equivalent.In particular,we have

for some universal constantc＞0.Also observe that whenk0,we have

Lastly,let us recall the elementary fact that each element in the homogeneous Sobolev space can be identified with a mean-free function belonging to the inhomogeneous Sobolev space (see [12]).We will henceforth assume that each element of(T2) is mean-free over T2.

We additionally incorporate the divergence-free condition by defining,for eachk≥0,the solenoidal Sobolev spaces.Note that due to(2.4),it will suffice to consider only the homogeneous counterpart.In particular,let us define

wheneverZ2andq ≥0.We point out that while this modification of Navier-Stokes is not physical,it is common practice to considerγ,p ＞0 in order to help stabilize numerical simulations.We consider this form of the dissipation in order to highlight the role of the dissipation in establishing the synchronization property of the nudging-based scheme at higher levels of Sobolev regularity.The corresponding nudged system is then given by

Given a solutionuof (2.7) or solutionvof (2.8),the pressure field may then be reconstructed up to an additive constant[16,60].For the remainder of the manuscript,we will consider the study of the coupled system(2.7),(2.8).Note that,as with the Sobolev spaces,we will also abuse notation by writing ()2simply as

The global well-posedness of (2.7) inand the existence of an absorbing ball in the corresponding topology are classical results and can be found,for instance,in[16,31,60].Whenk2,the sharpest estimate for the radius of the absorbing ball is established in [21,Theorem 3.1].To state them,let us also recall the Grashof number,G,corresponding to a given time-independent external forcing,f,which is defined by

whereλ1is the smallest eigenvalue of-Δ.Since the side-length of the spatial domain has been normalized to have length 2π,we see thatλ11.In particular,Gis a non-dimensional quantity.Let us also define the following shape factors of the forcing.Fork≥0,We define thek–th order shape function offby

Observe thatσk ≥1.

for all t≥t0.Moreover,if k≥2,then

for some universal constant c2＞0.

Lastly,let us recall the result proved in [2],where synchronization in the–topology is shown for general interpolant observable operators,Ih,satisfying certain boundedness and approximation properties.For this,let us denote the–absorbing ball for (2.7) by

Moreover,assume thatIhis finite-rank,linear,and satisfies either

or

Although it was only proved for the unperturbed case,γ0,i.e.,without hyperviscosity,we point out that the analysis of[2]still applies to the0 case without any difficulty whatsoever.

for some universal constant c0＞0.Moreover,one has

provided that μ additionally satisfies

for some universal constant ＞0.

In the next section,we expand upon the framework of general interpolation observable operators considered in [2] in order to accommodate approximation in higher-order Sobolev topologies.The specific examples of piecewise constant interpolation,volume element interpolation,and spectral interpolation constitute the original inspiration for the identification of properties(2.14)and(2.15).The framework developed here introduces an additional degree of flexibility for interpolating the data that not only realizes these three examples as special cases,but generates a wealth of new examples that were not treated in [2].

Remark 2.1.Note that we choose to work in the dimensionless domain,T2,rather than [0,L]2,for the sake of convenience.Because of this choice,derivatives and domains are ultimately dimensionless.In particular,throughout the paper velocities and viscosities carry only the physical units of(time)-1.One may,of course,re-scale variables accordingly to introduce a length scale commensurate with the linear size of the spatial domain.In doing so,all physical quantities will then recover their appropriate dimensions.

In [2],a general class of interpolant operators was introduced that could be used to define the nudging-based equation(2.8)and ultimately establish asymptotic convergence of its solution to the corresponding solution of (2.7) in the topology ofL2orH1.One of the main contributions of the present article is to identify a very general class of interpolant operators that allows one to ensure convergence in a stronger topology.In particular,we introduce a class of interpolant operators that generalizes the class introduced in[2]in such a way that accommodates higher-order interpolants by introducing an additional layer of flexibility in their design.When a collection of them are defined locally,subordinate to some open covering of the domain,and they satisfy suitable approximation properties,the family can then be patched together to form a global interpolant;this is one of the main constructions in this paper and is very much akin to the so-called Partition of Unity Method introduced by Babuska and Melenk in [9].

In what follows,we develop basic properties of this more general class of interpolating operators.Firstly,we introduce the notion of a local interpolant operator corresponding to a given subdomain of a givenorderandlevel.We then demonstrate how to“globalize”the construction to the entire domain via partition of unity subordinate to a given covering by subdomains.The main difficulties that arise in doing so are due to the fact that at each subdomain,different interpolant operators can be specified,namely,ones that correspond to different orders and levels.We must therefore systematically develop terminology that distills their salient properties and ultimately allows one to differentiate among the various possibilities of the construction.Then in the local-to-global analysis,the structure of the constants associated to each local interpolation operator must be carefully tracked.

We begin by introducing the notion of a“Q-local interpolation observable operator,”(I.O.O.)whereQrepresents a given subdomain of T2.Note that in the following definition,Hk(Q)or(Q)need not subsume any boundary conditions as it did in the caseQT2that we defined earlier;to distinguish between periodic boundary conditions,we will make use of the notation(Q).WhenQT2,we maintain the convention of dropping the dependence on the domain,e.g.,HkHk(T2).Throughout this section,we will refer to any subset ofQ⊂T2that is bounded,open,and connected as asubdomainof T2.

Definition 3.1.Let m ≥0and k ≥m+1be integers.Let Q ⊂T2be a subdomain and denote hdiam(Q).We say that IQ is a Q–local I.O.O.of order m at level k if IQ is defined on Hk(Q),linear,finite-rank,and whose complement,Id-IQ,for all0≤ℓ≤m,satisfies

for all1≤k′≤k.We say that IQ is a Q–local I.O.O.of order m at all levels if(3.1)holds for all k≥m+1;in this case,we also say at level k∞.

Given a bounded,open,connected set,Q,with finite diameter,hdiam(Q)＞0,we recall[15,Lemma 4.5.3]that sinceQ–local I.O.O.’s have finite rank,the following inverse inequality always holds for all such operators of ordermat levelk:

whenever(Q),for some constantc＞0,depending onℓ,k,but independent ofh.

Remark 3.1.Observe that ifIQis anm–th order local I.O.O.at levelk,then it is also a local I.O.O.of ordermat levelk′,for allk′＞k,as well as a local I.O.O.of orderm′at levelk,for allm′＜m.Indeed,one can simply “de-alias” the matrix induced by the associated constants by setting the additional associated constants to simply be zero.On the other hand,one can also identify a canonical representative for aQ–local I.O.O.by lettingm0be the largest integermsuch that

andk0be the smallest integerksuch that

for allk′＞k.In this case,we may setIQwithout any ambiguity.It will be convenient to exploit the flexibility in the terminology later on (see Lemma 3.4).

Remark 3.2.We will always associate an I.O.O.to a subdomainQ.It will thus be more convenient to denote the associated constants ofIQsimply by(Q),rather than(IQ).This convention will be enforced after Definition 3.3 below.

A key object in this paper is the patching together of a family of local interpolant operators to form a global one.This is done via partition of unity.Givenk≥2 and a coveringQ{Qq}by subdomainsQq ⊂T2,let us fix any family of functions Ψ{ψq}q ⊂Cksatisfying one hasδ-1hq ≤hq′≤δhq.

We refer to (P5) as theδ–adic condition.Indeed,this condition implies that all“neighbors,”Qq′,ofQqhave diameters equivalent toQqup to the fixed multiplicative factorsδ,δ-1.We will refer to Ψ as aδ–adic,Ck–partition of unity subordinate toQ.If Ψ additionally satisfies Ψ⊂C∞(Ω)and(P4)holding for allk,then Ψ is aδ–adic,C∞–partition of unity.For the majority of the manuscript,it will be assumed that Ψ satisfies (P1)–(P5),so we will simply refer to Ψ as a partition of unity.Lastly,it will also be useful to have additional control on the diameters in the covering.For this,we say thatQis a uniform cover at scalehif there existsh＞0 such thatδh≤hq ≤δ-1hfor allq.

Before proceeding to define a global interpolant operator,let us establish two useful facts which are consequences of the various partition of unity assumptions.In particular,for the moment,we do not necessarily assume that Ψ satisfies every property (P1)–(P5).

Lemma 3.1.Let {fq}q ⊂L2(Ω).Suppose that {ψq}q ⊂L∞(Ω)satisfies(P2)and(P5),and ‖ψq‖L∞≤λ(hq)for all q,for some monotonic,homogeneous function λ:[0,∞)→[0,∞)of degree ρ.Then

Proof.By the Cauchy-Schwarz inequality,it follows that

where we applied(P2),(P5),and the boundedness hypothesis of theψqin obtaining the final two inequalities.

Lemma 3.2.1(Ω)such that φ≥0.Suppose thatΨsatisfies(P1)–(P3).Then

where π0is the constant from(P2).

Remark 3.3.Partitions of unity satisfying (P1)–(P5) were constructed in [2,13,41].There,a collection of augmented squares overlapped in a regular manner to cover the domain multiple times;one may refer to this property as having “finite partition multiplicity.”In general,the collection of open sets to which a partition of unity is subordinate,need not satisfy this property.Indeed,let us formally introduce this notion as follows:

Definition 3.2.Let Q{Qq}q be a covering ofΩby bounded,open,connected subsets.We say that Q has partition multiplicity,M ＞0,if there exists an integer,M ＞0,and subcollections Q1,···,QM ⊂Q such that

for all Q,Q′Qj,for all j1,···,M,where¯Q denotes the closure of Q.

Lemma 3.3.Suppose Q is a covering ofΩwith partition multiplicity M.Then

1(Ω)such that φ≥0,and for all j1,···M.

Proof.LetQ1,···,QMdenote theMsubcollections ofQthat each cover Ω and each of whose respective elements can overlap only on sets of zero Lebesgue measure.Observe that for allj1,···,M,we have

which implies the upper bound.Now,upon averaging injand applying Fubini’s theorem,we arrive at

which produces the lower bound,as desired.

We therefore see that the first inequality of Lemma 3.2 already follows from the assumptions (P1)–(P5) (see Lemma 3.2).Indeed,property (P2) basically asserts a type of “local multiplicity,” whereas a cover with finite partition multiplicity is a form of “global multiplicity.” In contrast,the assumptions on Ψ allow for the possibility of having an infinite open covering in the case of a general bounded domain,i.e.,bounded,open,connected subset of the plane.Indeed,if Ω is a disk centered at the origin,then the open covering given by a small disk centered at the origin followed by consecutively overlapping concentric open annuli with geometrically decreasing length,appropriately proportional to the radius of the disk,provides such an example.

Let us now define a global interpolant operator.For convenience,whenever we refer to a partition of unity,we will specifically consider ones of the type described above,that is,satisfying (P1)–(P5).

Definition 3.3.Given a covering,Q{Qq}q,ofΩby bounded,open,connected subsets with hqdiam(Qq),we say that the family,I{I(q)}q,of local I.O.O.’s is subordinate to Q if for each q,I(q)is an mq–th order Qq–local I.O.O.at level kq,for some integers mq ≥0and kq ≥mq+1.We furthermore say that the family is Q–uniform if the associated constants of each I(q)

We say that I is an(m,k)–generic family if there exist m≥0and ∞≥k≥m+1such that mmq and kkq,for all q.

Given m≥0and m+1≤k≤∞,we say an operator Im,k is a(I,Ψ)Q–subordinate global I.O.O.of order m at level k if there exists a Q–subordinate family,I,of I.O.O.’s,and Q–subordinate Ck–partition of unity,Ψ,such that m ≤infqmq and k≥supqkq,and

whenever

Note that when the associated partition of unity is clear,we will simply say thatIm,kis theI–subordinate global I.O.O.with associated coveringQ.On the other hand,in light of the “dealiasing” procedure described in Remark 3.1,we see that ifIm,kis an (I,Ψ)Q-subordinate global I.O.O.,then eachI(q)is anm–th orderQq–local I.O.O.at levelksuch thatm≤infqmqandk ≥supqkq.In particular,we immediately deduce the following fact.

where(mq,kq)denotes the order and level associated to the canonical representative of I(q).

Without loss of generality,we may therefore always assume that any global I.O.O.,Im,k,derives from an (m,k)–generic familyIof local I.O.O.’s.Now,as a consequence of Definition 3.1,the properties of the partition of unity,and (3.7),we have the following.

Proposition 3.1.Let m,k≥0be integers such that k≥m+1.Let Im,k be an(I,Ψ)Q–subordinate global I.O.O,where I is(m,k)–generic,and Q{Qq}q denotes the associated covering.Then there exist constants{(Qq)}q such that for all0≤ℓ≤m

for some constant c＞0,independent of h,and where ε,j can be specified as

(Qq)are the constants associated to I(q)(q)interpolates optimally over Qq (at level k),for all q,then for all1≤k′≤k and0≤ℓ≤k′-1

where

Proof.Since Ψ is a partition of unity,observe that

Letαbe a multi-index such that|α|ℓ,where 0≤ℓ≤m.It then follows from the Leibniz rule that

Upon taking absolute values,squaring both sides,integrating over Ω,summing over|α|≤ℓ,then applying (3.1),(P4),and Lemma 3.1 (withφq∂α-βψq),we obtain

where we shifted indices to obtain the last inequality.Finally,changing the order of summation yields

which is (3.8).

On the other hand,ifIm,kIkinterpolates optimally,then for 1≤k′≤kand 0≤ℓ ≤k′-1,we apply (3.2) in (3.12),then (P4) and Lemma 3.1,as before,to obtain

which is (3.10),as desired.

In light of Proposition 3.1,we may define the following terminology.

Definition 3.4.Let Im,k be an(I,Ψ)Q–subordinate global I.O.O.such that I is(m,k)–generic.We say that Im,k is Q–uniform if I is a Q–uniform family.If Im,k is Q–uniform and Q is a uniform cover at scale h,then we say that Im,k interpolates uniformly at scale h.If I(q).

From Definition 3.1 and(3.7),one also easily obtains as a corollary to Proposition 3.1 and Lemma 3.2,the following interpolation error estimates for various special cases.

Corollary 3.1.Let m,k≥0be integers such that k≥m+1.Let Im,k be an(I,Ψ)Q–subordinate global I.O.O,where I is(m,k)–generic,and Q{Qq}q denotes the associated covering.If Im,k is Q–uniform,then for all0≤ℓ≤m

If Q is a uniform cover at scale h,then for all0≤ℓ≤m

In particular,if Im,k interpolates uniformly at scale h,then for all0≤ℓ≤m

if,additionally,Im,kIk interpolates optimally,then for all0≤ℓ≤k′-1and1≤k′≤k

Lastly,from Proposition 3.1,we also immediately deduce the following boundedness property of global I.O.O.’s.

Corollary 3.2.Let m,k≥0such that k≥m+1and Im,k be an(I,Ψ)Q–subordinate global I.O.O.If Im,k is Q–uniform,then there exists a constant c＞0such that

If,moreover,Im,k interpolates uniformly at scale h,then

where c is independent of Q.In particular,if Im,k interpolates uniformly at scale h,then Im,k:,is a bounded operator for all k≥m+1,where m≥0.

Proof.Suppose 0≤ℓ≤m.By the triangle inequality and (3.14),we have

In the particular casem≥ℓandℓ0,1,we may apply the fact that 0＜hq ≤2πfor allq,Lemma 3.2,and Poincaré’s inequality to deduce

This establishes (3.18).

Now,ifIm,kinterpolates uniformly at scaleh,then (3.20),Lemma 3.2,and Poincaré’s inequality imply

This establishes boundedness fromfor all 0≤ℓ≤m.

Now suppose thatm＜ℓ≤kandIm,kinterpolates uniformly at scaleh.By the product rule,Lemma 3.1 (withφq∂γψq,γα-β,|α|ℓ,|β|i),and (3.1),we deduce

We then applyQ–uniformity,sum overq,and apply Lemma 3.2 to deduce

Upon returning to the estimate ofIm,kφ,combining the above considerations,we apply Lemma 3.2 to complete the estimate of the first sum in (3.21),the fact thath≤2π,and Poincaré’s inequality to finally arrive at

as desired.

Remark 3.4.The universal constants appearing in each of the above estimates in Proposition 3.1,Corollary 3.1,and Corollary 3.2 depend additionally onℓ,k,m,and Ψ through properties (P1),(P2),(P4),and (P5).In particular,they are always independent of the diameters associated to the covering.

3.1 Examples of globalizable local interpolant observable operators

In this section we provide examples of local I.O.O.’s in the sense of Definition 3.1,as well as their corresponding globalized counterparts in the sense of (3.7).We only present a small sample of examples of immediate relevence to the context of Data Assimilation,e.g.,nodal values or local averages of velocity,but remark that several other examples exist which accommodate other forms of data,e.g.,nodal values or local averages of derivatives of the velocity,boundary flux data,etc.We refer the reader to [14,15,23] for these additional examples.

3.1.1 Spectral observables

LetQ[a,b]2,where 0≤a＜b≤2πsuch that 2πhb-a,where 0＜h≤2π.Then,givenN ＞0,let

whereκh2πh-1,is the indicator function of the ball,BN,of radiusNin Z2,centered at the origin,and

3.1.2 Piecewise constant interpolation

In light of (3.1),we see that we may take

for some constantc＞0,independent ofh;observe that(Q)ch.

3.1.3 Taylor polynomials

LetQ ⊂Ω be a star-shaped bounded,open,connected subset of diameterh ＞0.Given3(Q)andxQQsuch that|x-xQ|≤h,for all,letT1φ(·;xQ)denote the first-order Taylor polynomial ofφcentered atxQ.In particular

Then we have

This is an elementary extension of the corresponding fact for constant interpolation proved in [2,45] in dimensiond2;the details are provided in Appendix 5.2,where it is established in the greater generality of dimensiond≥2.Moreover,observe that

which implies

3.1.4 Sobolev polynomials

There are obvious shortcomings to using the Taylor polynomial as a means to interpolate nodal observations in the context of data assimilation,specifically since it requires one to make observations on derivatives ofφat given nodes.One may slightly relax this requirement by replacing nodal values of the derivatives with their spatial averages.This was done in the zeroth order case in Section 3.1.2,above,by replacingφ(xQ) byThe study of such polynomials of any order is classical and was introduced by Sobolev in [58].We recall their properties here following the treatment in [15].The reader is also referred to [22].

LetQ⊂Ω be a ball of radiushwith centerxQΩ.Fork≥1,denote the Taylor polynomial of orderkcentered atxQof(Ω) byTkφ(·;xQ),so that

Then (see [15,Lemma 4.3.8]) for all 0≤ℓ≤k

3.1.5 Lagrange polynomial

In the context of data assimilation for the 2D NSE where it is preferable and more reasonable that velocity measurements at nodal points are collected rather than(spatial) derivatives of velocity.A class of interpolants that leverage nodal values of a function to reconstruct higher-order features of the function are Lagrange polynomials.We define them here in a configuration that fits our setting suitably,but point out that more flexibility is allowed in general,for instance,in the arrangement of the prescribed nodes.We refer the reader to [15] for additional details.

Note that Σkrepresents the dual basis ofPkand that

Let Θkdenote the basis ofPkand represent its elements,θγ,by tensor products of one-dimensional polynomials as

3.1.6 Volume element polynomials

Spatial averages of the velocity field constitute another class of physical observations.This type of data is used in the finite volume method to approximate true solutions with piecewise constant functions in theL2-topology.They may also be used to construct higher-order polynomial approximations with similar error bounds to the Lagrange polynomial interpolants in higher-order Sobolev topologies.

LetQ ⊂Ω,Γk,NQ,Pkbe as in Section 3.1.5.We define functionals given by integration on square patches withinQas follows.Let

The unisolvence of the polynomial space with respect to the functionals,along with a similar argument to that for (3.33) (see,for instance,[15,Theorem 4.4.4]) gives the following bound: for all 0≤ℓ≤k′-1 and 1≤k′≤k,there exists a constant＞0,independent ofφandh,such that

3.1.7 Hybrid interpolation

LetQ{Qq}qbe a covering of Ω by bounded,open,connected subsets such thathqdiam(Qq).Given any (I,Ψ)Q–subordinate family,Proposition 3.1 ensures that an estimate of the form (3.8) holds.In particular,Imay be any family comprising of any combination of the operators from above.Four possible categories of such combinations are given by the following.

· Repeated-type,Uniform.TheI(q)are all of the same type of interpolating operator,e.g.,all Taylor,all Sobolev,all Lagrange,etc.,and there existmmqandkkqfor allq.The examples of operators considered in[2]belong strictly to this class;

· Repeated-type,Non-uniform.TheI(q)are all of the same type,butmq,kqare allowed to vary.In this case,the induced global I.O.O.would be given byIm,k,whereminfqmqandksupqkq(see Remark 3.1);

· Hybrid-type,Uniform.TheI(q)consists of different types,butmq,kqare constant inq;

· Hybrid-type,Non-uniform.TheI(q)consist of different types,butmq,kqare allowed to vary inq.

We formally state our main results here.The first of the three main theorems provides estimates for the radius of the absorbing ball inHkfork≥2.In particular,we properly generalize the bounds in Proposition 2.1 to all higher orders of Sobolev regularity.Indeed,to establish the desired higher-order convergence between the nudged solution and true solution,we will make crucial use of the a priori bounds available for the true solution when initialized in the absorbing ball with respect to a Sobolev topology of arbitrary positive degree.

holds for some universal constant ck ＞0for all t≥t0for some t0sufficiently large,depending only on the diameter of the bounded set.

Remark 4.1.Note that by interpolation,one may immediately obtain absorbing ball bounds inHsfor alls＞1.

For the remainder of the manuscript for eachk ≥0,we will denote theHk–absorbing ball of (2.7) byBkso that

where c＞0is a universal constant,π0is the constant from(P2),εi,j(Qq)are the constants associated to Im,k,and where[p]denotes the greatest integer ≤p.Moreover,if Im,k interpolates uniformly at scale h,then it suffices to instead assume

in place of(4.3).Note that we use the convention that

when γ0.

Since the analysis performed in [2,Theorem 6] in the periodic setting can be easily extended to prove Theorem 4.2,we relegate the proof of this theorem to Appendix 5.2.

Remark 4.2.We note that since ΩT2is a compact manifold without boundary,property (P2) implies thatQis finite.For general bounded domains,however,Qmay be infinite.We refer the reader to Remark 3.3 and Remark 4.4 for further comments.

The third main result provides sufficient conditions on the nudging parameter,μand the density of data,determined byh,in terms of the system parameters,ν,f,alone that ensure convergence of the approximating signal,as determined by the nudging-based system,to the true signal,as represented by a solution to (2.7),in higher-order Sobolev topologies,provided that the observables are interpolated in a suitable manner.In particular,we assume that the observables are interpolated using a sufficiently nice global I.O.O.in the sense of (3.7).

where[p]denotes the greatest integer ≤p,provided that μ,h additionally satisfy

where π0is the constant from(P2).Moreover,if Im,k interpolates uniformly at scale h,then it suffices for μ,h to instead satisfy

in place of(4.6).Note that we use the convention that

when γ0.

Under stronger assumptions onIm,k,one can obtain convergence up to the regularity level of the solution from which the data derives.

provided that μ,h additionally satisfy

where π0is the constant from(P2).Moreover,if Ik+1is uniformly interpolating at scale h,then it suffices for μ,h to instead satisfy

in place of(4.9).

Remark 4.3.In [10],it was proved that convergence with respect to the analytic Gevrey norm was ensured under slightly more stringent conditions than (2.16) in the particular case when only spectral observations are used.

Remark 4.4.In [2],the case of Dirichlet boundary conditions was also treated,where convergence inL2was obtained for a wide class of observable quantities,including nodal value observations.In light of these results and Theorem 4.3,it remains an interesting issue to investigate whether one can show higher-order convergence in the setting of Dirichlet boundary conditions or others,when data is particularly given by nodal values.Moreover,in light of Remark 4.2,the framework we establish here may accommodate the case where infinitely many I.O.O.’s are used across the domain in the Dirichlet setting.We leave the study of this case to a future work.

Remark 4.5.From Corollary 3.1,we may immediately deduce from Theorem 4.3 that convergence inH2can be ensured in the case of nodal observables by using either Taylor polynomials of degree 1 or quadratic Lagrange polynomial as the method of interpolation.SinceH2embeds intoL∞,this provides rigorous confirmation of the observation from the numerical simulations carried out in [36] that the approximating solution was in fact converging uniformly in space to the reference solution.In particular,this also supplements the result in [10],where the synchronization property with respect to the uniform topology,L∞,was established in the particular case of the spectral observables.In the absence of hyperdissipation,i.e.,γ0,we note that the same assumption onμ,his imposed in either case of nodal observables or spectral observables,up to an absolute constant.

Remark 4.6.The case of hyperviscosity is included here in order to illustrate the interplay between the order of dissipation and the order of interpolation.It is a well-known fact that forp≥1/4,the corresponding system (2.7) in dimensiond3 has global unique strong solutions.We point out that the analysis developed here applies in a straightforward manner to that setting as well.We refer the reader to the work of [61] for the relevant details.

We will first prove Theorem 4.1 in Section 5.1.We will then prove Theorem 4.3 in Section 5.2.Recall that the proof of Theorem 4.2 will be supplied in Appendix 5.2.To prove these results,it will first be useful to collect various estimates for the trilinear term that appears in the estimates.The proof of these estimates are elementary,but we supply them here for the sake of completeness.

Lemma 5.1.Let u be a smooth,divergence-free vector field inT2,and v be any smooth function.Then given ℓ≥2for any |α|ℓ,we have

For all β ≤α,we have

for some constants c＞0,depending on ℓ.

Proof.By the Leibniz rule and the fact thatuis divergence-free,we have

where we interpretβ＜αasβi＜αifori1,2.By H¨older’s inequality and interpolation we have

On the other hand,by H¨older’s inequality and interpolation we have

Combining the estimates forIaandIb,then summing over|α|ℓ,yields (5.1).

Now consider

We treatIIaas

We treatIIbas

Finally,we treatIIcas

Upon combiningIIa,IIb,IIcand applying the Poincaré inequality,we obtain(5.2a).

Lastly,we consider

We estimateIIIawith H¨older’s inequality,interpolation,and Young’s inequality to obtain

We treatIIIband consider the caseℓ2 separately.Whenℓ2,we apply H¨older’s inequality and interpolation to obtain

Whenℓ≥3,we use the divergence-free condition to writeIIIbas commutator.In particular

By H¨older’s inequality,a classical commutator estimate [46] and interpolation we obtain

Upon combiningIIIa,IIIb,and applying the Poincaré inequality,we arrive at(5.2b).

5.1 Higher-order absorbing ball estimates: proof of Theorem 4.1

Denotingu(t)u(t;u0,f)for allR.Then,owing to the divergence-free condition,the basic energy balance inis given by

In particular,by choice ofu(0)u0,observe that

so that

By (5.1) of Lemma 5.1,we have

ForII,we integrate by parts and apply H¨older’s inequality to obtain

Upon combining the estimates forI,II,we arrive at

Observe that by interpolation

With this and Young’s inequality,we estimate

Hence

Returning to (5.4),it follows that

which proves (4.1),as desired.

5.2 Synchronization in higher-order Sobolev topologies

Letp≥0 andu0,v01∩Bk,where 1+m≤k ≤2+panduandvdenote the corresponding unique strong solutions of the following initial value problem

whereJm,kIm,k-〈Im,k〉,whereIm,kis an (I,Ψ)Q–subordinate global I.O.O.with associated coveringQ,and〈Im,k〉denotes the operator such that

Letw:v-uandw0v0-u0,so thatwsatisfies

We will ultimately show that

Our approach will be to bootstrap convergence in higher-order Sobolev topologies,starting fromH1.Note that we adopt the following convention for applications of the Cauchy-Schwarz inequality or Young’s inequality in the analysis below,which we will invoke repeatedly:

for somec ＞0 depending onp,p′.There is nothing essential about the constant 1/100,except that we never add more than 50 of such terms in a given argument.In particular,we make no attempt whatsoever to optimize such constants.This can certainly be done by the interested reader,but in order for this to be a meaningful exercise,one must also carefully track the constants from Lemma 5.1,which we also neglect to do.The important feature that we care to emphasize is the manner in which the constants from the I.O.O.’s appear in the analysis,as the development of these operators is the main novelty of this work.

Lemma 5.2.Let m ≥0and1+m ≤k ≤2+p.Let Im,k be an(I,Ψ)Q–subordinate global I.O.O.that is(m,k)–generic.Suppose that μ,h satisfies

where[p]denotes the greatest integer ≤p.Then for c,c′＞0sufficiently large and μ additionally satisfying

one has

If,additionally,Im,kIk+1interpolates optimally,then it suffices for μ,h to satisfy

where π0is the constant from(P2).

On the other hand,if Im,k is uniformly interpolating at scale h,then we may instead suppose that μ,h satisfies

in place of(5.8).If,additionally,Im,kIk+1interpolates optimally,then it suffices for μ,h to satisfy

Proof.Upon taking theL2–inner product of (5.7) with-PσΔw,integrating by parts,then using the fact that〈(w·∇)w,Δw〉0,we obtain

To estimateI,we again invoke the property that〈w·∇w,Δw〉0,then apply H¨older’s inequality,the Brézis-Gallouet inequality,and Proposition 2.1 to obtain

To estimateII,we appeal to Proposition 3.1 and invoke (5.8) and Poincaré’s inequality,so that

IfIm,kIk+1is further assumed to interpolate optimally,we then proceed to apply Young’s inequality,(3.10) of Proposition 3.1,and invoke (5.11),so that

Now let us consider the case whenIm,kinterpolates uniformly at scaleh.To estimateII,we apply the Cauchy-Schwarz inequality,Corollary 3.1,Young’s inequality,and (5.12) to obtain

IfIm,kIk+1is further assumed to interpolate optimally,then we integrate by parts first to write

where we have used the fact that∇Jk+1∇Ik+1.IfIk+1also interpolates uniformly at scaleh,then by (3.17) of Corollary 3.1,Young’s inequality,and (5.13),it follows that

Sinceμadditionally satisfies (5.9) (see [2,Lemma 2]),it follows that

so that Gronwall’s inequality yields

as desired.

We are now ready to prove Theorem 4.3.

Proof of Theorem4.3. Observe that Lemma 5.2 covers the casem0,1.It suffices then to consider 2≤ℓ ≤m.Let∂αdenote any partial with respect toxof order|α|ℓ.Upon applying∂αto (5.7),taking theL2–inner product of the result with∂αw,integrating by parts,then summing over all|α|ℓ,we obtain the following energy balance

We estimate each the termsIandII.Observe thatImay be expanded as

We treatIawith (5.1) of Lemma 5.1,then apply Young’s inequality to obtain

We treatIbby first integrating by parts,then applying (5.2a) of Lemma 5.1 and Young’s inequality to obtain

ForIc,we apply (5.2b) of Lemma 5.1 and Young’s inequality to obtain

Upon combining the estimatesIa,Ib,Ic,then applying Theorem 4.1,we obtain

Now we treatII.First observe that upon integrating by parts,we get

Note that we used the fact thatPσcommute with derivatives and that∂βJm,k∂βIm,kfor any|β|＞0.then by the Cauchy-Schwarz inequality,Proposition 3.1,Young’s inequality,and the assumptions that 1≤k≤2+p,we have

On the other hand,ifIm,kis uniformly interpolating at scaleh,we estimate as above and apply Corollary 3.1 in place of Proposition 3.1 to obtain

Finally,upon returning to (5.14) and combining the estimates forIand eitherIIorII′,then applying the Poincaré inequality and (4.3) or (4.4),respectively,we see that

Lastly,we invoke the fact thatu0the estimate (5.10) of Lemma 5.2,and ultimately Gronwall’s inequality to deduce

for all 2≤ℓ≤m.The remainder of the proof can be completed by a basic induction argument,where the assumption of the induction step is

Thus,we complete the proof.

Proof of Theorem4.4. We letℓkand proceed exactly as in proof of Theorem 4.3,except that we instead treatIIwithout integrating by parts in(5.15)and use(3.10)of Proposition 3.1 to obtain

Now observe that from Lemma 3.2,we have

Hence,arguing as we did in Theorem 4.3,we arrive at

Upon invoking (4.9),then applying of Gronwall’s inequality and Lemma 5.2,we obtain (4.8).

On the other hand,ifIk+1is also uniformly interpolating at scaleh,then

Arguing we did before,we obtain

We then apply (4.10) and deduce (4.8) once again.

Appendix A: Well-posedness of nudging-based equation in higher-order Sobolev spaces

We will now supply the proof of Theorem 4.2 in,wherek≥2.Recall that we will consider the system (2.7).Recall that forp＞0,we denote by (-Δ)pthe operator defined by

wheneverZ2{0}.Givenγ ≥0,consider

whereJm,k:Im,k-〈Im,k〉,〈Im,k〉denotes the operator such that

We prove the existence of solutions to(2.8)via Galerkin approximation.LetPNdenote the Galerkin projection at levelN ＞0 and letvNdenote the unique solution to the following system of ODEs

wherep≥0 andγ ≥0.We will develop uniform bounds forvNin the appropriate topology over the maximal interval of existence[0,TN),independent ofN.This will imply global existence for the projected system,and therefore global existence for(A.3).Uniqueness and continuity with respect to initial data will follow along the same lines as in [2].

To prove Theorem 4.2,we will make use of the following lemma,which controls the growth ofJm,ku,whereuis a strong solution to(2.7)evolving within an absorbing ball.

Lemma A.1.Let m≥0and k≥2such that k≥m+1k-1,let Bk denote the Hk–absorbing ball of(2.7).Suppose that Im,k interpolates uniformly at scale h.Then given u01≤ℓ≤m,there exists a universal constant c＞0such that

where u denotes the corresponding unique strong solution of(2.7).

Proof.Sinceℓ ≥1,by the boundedness of the Leray-projection and the triangle inequality,observe that

From Corollary 3.1,Proposition 2.1(forℓ1),and Theorem 4.1(forℓ≥2),it follows that

Combining these estimates,then taking the supremum overt≥0 yields (A.5).

Lemma A.2.Let μ＞0and k≥m+1such that k≥2.Let

There exists a universal constant c＞0such that

where π0is the constant from(P2).Moreover,if Im,k interpolates uniformly at scale h,then

holds for all μ＞0,0＜h≤2π,and N ＞0.

Proof.Upon taking theL2–inner product of (A.4) with-ΔvN,using the identity〈Pσ(vN·∇)vN,ΔvN〉0,and integrating by parts,we obtain

We estimateI1by applying the Cauchy-Schwarz inequality,Young’s inequality,so that

On the other hand,we treatII1by applying the Cauchy-Schwarz inequality,Corollary 3.1,and Young’s inequality,to obtain

Upon combiningI1,(A.9),and Lemma 3.2,we obtain

On the other hand,ifIm,kinterpolates uniformly at scaleh,then

Hence,upon combiningI1and (A.10),we obtain

This completes the proof.

Lemma A.3.For all ℓ≥2,let

Let m≥0,k≥2be given such that k≥m+1.There exists a universal constant c＞0such that

holds for all μ＞0,0＜h≤2π,and N ＞0.

Proof.Now,we estimate in.By taking theL2–inner product of (A.4) with(-1)|α|∂2αvN,where|α|k,integrating by parts,then summing over all|α|k,we obtain

We treatby integrating by parts,then applying the Cauchy-Schwarz inequality,Young’s inequality,(2.9),and (2.10),to obtain

Next,we treatIIk.We observe that∂αJm,kφ∂αIm,kφ,then apply the Cauchy-Schwarz inequality,(3.3),Corollary 3.1,Poincaré’s inequality,and Young’s inequality to obtain

Lastly,we estimateIIIk.Indeed,observe that due to the divergence free condition,we have

Thus,by H¨older’s inequality,a classical commutator estimate (see for instance [47,48]),and interpolation we have

Upon returning to (A.13) and combiningIk,IIk,andIIIk,we have that

Hence,by Gronwall’s inequality,it follows that

as desired.

Under certain assumptions onIm,k,one may then identify conditions onμso that the sequence{vN}N＞0is bounded uniformly in time in,independent ofN.Provided that the initial data belongs to an absorbing ball for the dynamics,these bounds can then be expressed explicitly in terms of its radius.

Corollary A.1.Suppose that1+m≤k≤2+p.Then there exists a universal constant c＞0,independent of N ＞0,such that if μ＞0and {hq}q satisfy

then

On the other hand,if Im,k interpolates uniformly at scale h,then there exists a universal constant c＞0,independent of N ＞0,such that if μ＞0and0＜h≤2π satisfy

where[p]denotes the greatest integer ≤p,then

In particular,if v01and u01∩Bk,then

for all N ＞0.

Proof.By (A.18) and the Poincaré inequality we have

Since (A.17) holds andk ≤2+p,it then follows upon shifting the index and upon applying Lemma 3.2 that

We deduce from Gronwall’s inequality that

as desired.

On the other hand,ifIm,kinterpolates uniformly at scaleh,we apply Lemma A.2,so that by (A.8) we have

Since (A.19) holds,it follows that

An application of Gronwall’s inequality,then yields

which implies (A.20).We deduce (A.21) by applying Lemma A.1 and Proposition 2.1.

Upon combining Lemma A.3 and Corollary A.1,we obtain the following corollary.

Corollary A.2.Suppose that1+m≤k≤2+p.If μ and {hq}q satisfy(A.17),then

holds for all T ＞0.Moreover,if Im,k interpolates uniformly at scale h and μ,h satisfy(A.19)in place of(A.17),then(A.22)still holds.

Lemma A.4.Under the assumptions of CorollaryA.2,we have

for all T ＞0.

Proof.Observe that

We treatIby Poincaré’s inequality and Corollary A.2.ForII,we apply Cauchy-Schwarz inequality and interpolation to obtain

Thus,we may ultimately controlIIwith Poincaré’s inequality and Corollary A.2.We treatIIIwith the Cauchy-Schwarz inequality and Lemma A.1.Lastly,we treatIVwith the Poincaré inequality,Corollary 3.2,and the fact that 1+m≤k ≤2+p,so that we have

Hence,by Corollary A.2,we conclude that

for allT ＞0.

Finally,we are ready to prove Theorem 4.2.

Appendix B: Taylor interpolant

We prove a preliminary lemma,which is a generalization of that found in [45].

Lemma B.1.Let h＞0and d≥2.Let Q[0,h]d and φCk(Q),where0≤k ≤d.For each1≤k ≤d-1,there exist universal constants bα ＞0,for each multi-index0≤|α|≤k,depending only on d,such that

Proof.Letkd-1.Foryd ≤xd,we have

By applying the Cauchy-Schwarz inequality,then integrating with respect todx1···dxdover Ω,we deduce

for allyd[0,h].Dividing byhd-1establishes (B.1) fork1.Observe that it now suffices to assumed≥3.

It follows that

Applying Cauchy-Schwarz yields

Integrating with respect todx1···dxdover Ω then gives

It follows from the induction hypothesis that

Therefore,upon substituting the bounds in (B.4) into (B.3),then combining like terms we arrive at

The proof is complete upon dividing byhd-k+1.

Proposition B.1.Suppose d≥2.Let Q[0,h]d and φC1(Q)1φ(·;y)denote first-order Taylor polynomial of φ centered at y.There exists an absolute constant C ＞0,independent of y,such that

Proof.Let (x1,x2,(y1,y2).Then observe that

It follows from H¨older’s inequality that

The Cauchy-Schwarz inequaliy then implies that

By Lemma B.1,we have

as desired,which establishes the cased2.

Now supposed ≥3 and letx(x1,···,xd) andy2(y1,···,yd),where.For convenience,in addition to the notationintroduced in the proof of Lemma B.1,we defineso thatzBy the fundamental theorem of calculus,we have

Similarly,forjd-1,d,we have

Upon returning to (B.7),applying (B.8),(B.9),taking the square of the result,integrating over[0,h]dwith respect todx1···dxd,then applying the Cauchy-Schwarz inequality and (B.10),we have

Finally,we apply Lemma B.1 to obtain

Switching the order of summation completes the proof.

Appendix C: Volume elements

We describe an approximation operator based on data given by integration over subsets of each cell.In particular we construct the operator on the unit cube [0,1]d,from which its definition on affine images in the domain follows.For this particular section,we refer the interested reader to [14] for additional details.

First we define an index set and collection of subsets of the cube

The degrees of freedom are then given by integration on the subsets.Define the set of functionals Σwhere([0,1]d)→R are given by

Now let us recall that unisolvence of a function spaceXwith respect to a collection of functionals Σ is equivalent to Σ forming a basis for the dual space ofX.Unisolvence will ensure that the approximation operator constructed from the functionals will act as identity onX.Prior to proving unisolvence of the tensor product volume element in general,we begin in one dimension.

ˆMis the difference of two simpler matrices

In particular,if we define

We therefore conclude that

We proceed to the case of general dimension.

Letn≥1,and define for eachα(α1,···,αd)a polynomialθα Pm-1,n

We have

Thus,givenN,there exists a collection of polynomials Θsuch that

As dimPm-1,n|Σ|,we conclude that Σ forms a basis forbi-orthogonal to Θ as constructed.

By Proposition C.1

Observe thatImis indeed a projection ontoPm-1,n;for any polynomial1,nandwe haveσα(p)σα(Imp),and therefore by Proposition C.1 we havep(x)Imp(x),as desired.

Acknowledgements

The work of Vincent R.Martinez was partially supported by the award PSC-CUNY 64335-00 52,jointly funded by The Professional StaffCongress and The City University of New York.The authors would like to thank Michael S.Jolly and Ali Pakzad for insightful discussions in the course of this work,as well as the referees for their careful reading of the manuscript and the generous comments they shared to improve it.

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