Efficient, unconditionally stable, and optimally accurate FE algoritms for approximate deconvolution models of fluid flow Leo G. Rebolz Abstract Tis paper addresses an open question of ow to devise numerical scemes for approximate deconvolution fluid flow models tat are efficient, unconditionally stable, and optimally accurate. We propose, analyze and test a sceme for tese models tat as eac of tese properties for te case of omogeneous Diriclet velocity boundary conditions. Tere are several important components to te derivation, bot at te continuous and discrete levels, wic allow for tese properties to old. Te proofs of unconditional stability and optimal convergence are carried out troug te use of a special coice of test function and some tecnical estimates. Numerical tests are provided tat confirm te effectiveness of te sceme. 1 Introduction We consider implicit/explicit finite element timestepping discretizations for approximate deconvolution models ADMs of fluid flow. Tese models, wic are in te class of large eddy simulation LES models, were introduced in 1999 by Stolz and Adams in [1, ] for te purpose of simulating large-scale flow structures at a reduced computational cost compared to direct numerical simulation DNS. Te essential idea beind tese models, and most oter LES models, is to separate large scales from small by filtering, eliminate te small scales, but model teir effect on te large scales. ADMs are unique among LES models because te modeling tey utilize is based on matematics instead of pysical penomenology, wic leads to provable estimates of teir accuracy [1]. Since te resulting models do not solve for fine scales, tey require many less mespoints for full resolution compared to DNS, and tus ADMs can be used for large scale approximations wen DNS is computationally infeasible. Using overbar to denote a spatial average, and D a deconvolution operator tat is an approximate inverse to te averaging defined below, ADMs take te form v t + Dv Dv + q ν v + γv Dv = f 1.1 Dv = 0, 1. were ν is te kinematic viscosity, f is a forcing, γ 0 is a parameter coefficient for te time relaxation stabilization term often used wit ADMs, and v and q are interpreted to be approximations to te large scale velocity and pressure. Since teir development, tese models ave undergone intensive study, and many results ave been found. On te computational front, tey ave been tested on compressible [3, 3] and incompressible flows Department of Matematical Sciences, Clemson University, Clemson, SC 9634 rebolz@clemson.edu, ttp://www.mat.clemson.edu/ rebolz, partially supported by NSF grant DMS111593 1
[1, ] wit great success, and ave also performed well for te quasi-geostropic equations [16]. Many analytical results ave been proven as well, including well-posedness and ig formal accuracy [1], aderence to pysical properties of fluid flow [31, 1, 5, 6], and te size of te ADM microscale wic verifies tat tese models are muc more easily resolved tan a DNS [3]. Despite its success so far, many questions regarding te ADMs remain unanswered, particularly for te design of numerical metods for tem. One main difficulty in computing solutions using 1.1-1. is te need for boundary conditions for te nonlinearity. If C 0 elements are being used, tis presents a significant obstacle for flows wit boundary conditions tat are not periodic or Diriclet. Moreover, te filtering of te nonlinearity is coupled to te system solve in a way tat proibits an unconditionally stable linearization at eac timestep. An alternative formulation of te model tat circumvents tese two problems is to use instead te fourt order ADM formulation, wic is suggested in [18] and is given by assuming te Helmoltz filter: v t α v t + Dv Dv + q ν v + α ν v + γv Dv γα v Dv = f Dv = 0. Unfortunately, fourt order problems ave teir own set of callenges. Here, we need boundary conditions for tird derivatives of v, and tis will affect boundary layers. Also, if C 0 elements are to be used as tey sould be, as C 1 elements are not available in most software packages, and are not trivial to program, ten an extra variable φ = v needs introduced, leading to coupled systems tat need solved at eac timestep. Te purpose of tis paper is to address an important open question for ADMs: In a pysically relevant setting, can an efficient C 0 finite element sceme be devised for te model tat is unconditionally stable wit respect to timestep, and is optimally accurate? We propose a sceme erein tat will sow te answer to tis question is affirmative for te case of omogeneous Diriclet velocity boundary conditions, and we will rigorously prove unconditional stability and optimal accuracy of te sceme. Te main components of te sceme are tat it is built from a reformulation of fourt order version of te model tat reduces te problem back to second order using an additional approximate deconvolution approximation, togeter wit a C 0 finite element spatial discretization and a linearized backward Euler timestepping sceme, an incompressible Helmoltz filter, a linearization at eac timestep via te metod of Baker [5], treating filtered variables explicitly so tat te linear system solve is decoupled, and finally using divergence free finite elements to eliminate te dependence of te error on spatial and temporal derivatives of te Lagrange multipliers arising from te use of an incompressibility constraint in te filter equation. We now derive an ADM based on an incompressible filter, wic will be te starting point for our discretization. In te case of periodic boundary conditions, Helmoltz filtering will preserve incompressibility, but in general it does not, and for tese models incompressibility of te filtered variables is essential for stability and well-posedness. Moreover, discrete incompressibility is essential for well-posedness of te resulting numerical scemes. We begin wit te NS equations, wic are given by u t + u u + p ν u = f, u = 0. We will consider te case of omogeneous Diriclet boundary condition for u. Next, we
define te incompressible Helmoltz filter by α φ + α λ + φ = φ 1.3 φ = 0, 1.4 φ φ Ω = 0. 1.5 For notational convenience, denote by F te solution operator to 1.3-1.5, so tat F φ := φ. We note tere is an ongoing debate about wat te correct boundary condition for φ sould be. We coose 1.5 because we ave found it to work well in practice. Te approximate deconvolution operator we employ erein is van Cittert, and te N t order approximate deconvolution operator is defined by D N = N I F n. Tus, te first few deconvolution operators applied to a function φ are given by D 0 φ = φ D 1 φ = φ φ D φ = 3φ 3φ + φ. 1.6 We note tat te ideas presented erein can easily extend to most oter approximate deconvolution operators, including Tikonov and multiscale deconvolution [11]. Provided tat I D 1, a splitting of te discrete deconvolution operator in te timestepping peraps different tan wat is used erein for van Cittert can be found tat will still provide unconditional stability. Because of te different splittings, te proofs of stability and convergence will differ for eac case, and tus for te most clear presentation, we analyze only a sceme based on van Cittert approximate deconvolution wic is te most common coice in practice and we will note te canges necessary for oter approximate deconvolution operators in section 3. Following te derivation in [1] for te usual witout te incompressibility constraint Helmoltz filter and associated van Cittert approximate deconvolution, a local error estimate can be derived for te accuracy in approximating F 1 by D N. Denoting by A te solution operator to te Stokes problem wit omogeneous Diriclet boundary conditions, we follow te steps of [1] exactly but wit te usual Helmoltz filter replaced wit F and te Laplacian replaced wit A 1, to get φ D N φ = 1 N+1 α N+ A N+1 F N+1 φ, 1.7 and tus φ D N φ H j α N+ φ H k++j, j 0. 1.8 Te estimates 1.7-1.8 justify te use of te approximate deconvolution operator D N as an approximate inverse to te filter, wic is critical in te model derivation tat follows. We wis to derive a system in terms of u instead of u, so tat we solve only for flow averages, in order to reduce computational cost. Because F is invertible, we can write u = F 1 u, and using te filter equation 1.3, we get ut α u t + α λ t + F 1 u F 1 u + p ν F 1 u = f. 3
Altoug te system is now in terms of u, te F 1 implicitly makes te equation in terms of u as well. By approximating te inverse of te filter wit approximate deconvolution, i.e. using te approximation estimate 1.7 to justify F 1 φ D N φ, and writing v = u and q = p + α λ t, we get v t α v t + D N v D N v + q ν D N v = f. 1.9 To our knowledge, tis is te first use of approximate deconvolution in tis way, to reduce fourt order problems back to second order. We also make te approximate deconvolution approximation in te conservation of mass equation, wic yields D N v = 0. Since te incompressible filter already enforces v = v =... = 0, te conservation of mass constraint can be satisfied by simply enforcing v = 0. Tus, we ave arrived at te ADM wit incompressible filter for omogeneous Diriclet velocity boundary condition: v t α v t + D N v D N v + q ν D N v = f 1.10 v = 0 1.11 α v + α λ + v = v 1.1 v = 0 1.13 v Ω = v Ω = 0. 1.14 Except for te viscous term, te system 1.10-1.14 is te analog of te Stolz-Adams ADM witout time relaxation if we apply te filter to eac term in 1.1. It could also be considered as an approximate deconvolution extension of te NS-Voigt model, since wen N=0 te NS-Voigt model is recovered. To our knowledge, te ADM in tis form as not been studied analytically, and so its well-posedness and regularity of its solutions is an open question. However, altoug tis must be cecked in a toroug analysis, it does appear tat bot te results of Euler-Voigt in [0] and te ADM wit Helmoltz filter in periodic case can be extended to prove te well-posedness of 1.10-1.14 and regularity of solutions depending directly on regularity of te data. From a computational point of view, te system 1.10-1.14 appears more attractive tan 1.1-1. for two reasons. First, tere is no filtering of te nonlinearity in 1.10-1.14. For boundary conditions suc as zero-traction, if one computes using a C 0 finite element metod, it is not clear wat an accurate boundary condition for te filtered nonlinearity sould be. More specifically, te nonlinearity will be discontinuous across elements, but te filtered velocity is supposed to belong to H 1, wic is inconsistent wit te trace teorem, and tus some ad-oc approximation must be used to avoid generating oscillations at te boundary. Te second advantage of 1.10-1.14 is tat te additional approximation of te viscous term provides a second order system instead of te fourt order system tat would result oterwise. Tis allows for te use of C 0 elements, eliminates te need for boundary conditions for tird derivatives of v, and once discretized, will allow te system to be decoupled troug a splitting in te timestepping sceme tat treats filtered variables explicitly. In tis work, we will not include te time relaxation stabilization term wit te model. It is a stabilization term wic as been well studied, and is known to add positivity to te system, drive small scales to zero exponentially fast and will only elp numerical accuracy of finite element scemes [10, 4]. All results proven erein will still old if tis stabilization term is used, including if te filtered terms are treated explicitly in te timestepping. Before presenting and analyzing te sceme, wic is done in section 3, we first present some matematical preliminaries and notation in section. In section 4, we present nu- 4
merical experiments to test te effectiveness of te proposed sceme. conclusions in section 5. Finally, we draw Matematical preliminaries and notation We assume a domain Ω R d, d= or 3, to be a convex polygon/polyedra. We will use te notation, and for te L Ω inner product and norm, respectively. All oter norms will be clearly labeled wit subscripts. We will also assume omogeneous Diriclet boundary conditions for velocities, altoug extensions to oter common boundary conditions will be possible wit additional tecnicalities. Te natural function spaces for velocity and pressure are ten X := H0 1 Ω = {v H 1 Ω, v = 0 on Ω}, Q := L 0Ω = {q L Ω, q dx = 0}. Recall tat te Poincare inequality olds in X: For φ X, φ C P φ, were C P is a constant depending on te diameter of Ω. We denote by V te divergence free subspace of X, V := {v X, v = 0}, Define te operator b : X X X R by Ω b u, v, w := u v, w. We state next some important properties of b in te following lemma, wic are proven in []. Lemma.1. If u = 0, ten b u, v, v = 0. Tere exists a constant C dependent on te size of te domain suc tat te following upper bounds old: b u, v, w C u 1/ u 1/ v w, b u, v, w C u v w.1 Discrete setting A mes τ will denote a regular, conforming triangulation/tetraedralization of Ω, wit maximum element diameter. Finite dimensional spaces X X and Q Q are defined wit tis mes to be te Scott-Vogelius SV mixed finite element pair, X, Q := P k τ d, P disc k 1 τ. Tese element ave te special property tat X Q, wic implies tat te space V of discretely divergence free functions, V := {v X, v, q = 0 q Q }, 5
is actually pointwise divergence free. We recall from [30, 34, 33] and references terein tat te inf-sup stability of SV elements may require structure in te mes. For example, if k = d, a barycenter refined triangular mes structure is sufficient. For smaller k, more complex mes macro-structures are necessary, and for larger k, less complex structures are required. We will assume all meses used erein deliver inf-sup stability of te SV pair. We make tis element coice because it will eliminate te dependence of velocity error on te pressure, and of te discrete filtering error on te Lagrange multiplier in te continuous filter equation. Furtermore, recent work wit SV elements ave sown tey provide excellent accuracy [9, 6]. For fixed α, all results erein can be extended to work wit inf-sup stable P k d, P r velocity-pressure elements on general triangular meses, and te same stability result can be found wile te error estimate will contain an extra pressure dependent term and terms tat depend on te Lagrange multipliers tat arise from filtering. Denote by te V -discrete Laplacian operator, defined by: Given φ H 1 Ω, φ V satisfies φ, v = φ, v v V.. Discrete filtering and deconvolution Te discrete filter is defined analogous to te continuous filter, by taking its variational formulation and ten restricting to finite dimensional spaces. Definition.1 Discrete differential filter. Given φ L Ω, define φ to be te solution of: Find φ, λ X, Q satisfying α φ, χ + φ, χ α λ, χ = φ, χ,.1 φ, q = 0. For notational convenience, we will also denote te discrete filter by F : F φ := φ. From [9], we recall tat F is self-adjoint on V in te L inner product, and is positive definite. We note tat an equivalent representation of te discrete filter is: Given φ L Ω, find φ V satisfying α φ, χ + φ, χ = φ, χ χ V..3 Tis second formulation will be more convenient for later analysis. Te next lemma presents upper bounds for filtered variables. Lemma.. For φ L Ω, φ φ. For φ X, φ CΩ φ. For φ V, φ φ. Proof. Te first estimate follows immediately from coosing v = φ in.3. second estimate, we coose v = φ in.3 to get For te α φ + φ = φ, φ. 6
Denoting te L projection into V by P V, we use tat φ V to get for te last term, φ, φ = P V φ, φ = P V φ, φ. Caucy-Scwarz and Young s inequalities now provides te bound α φ + φ PV φ P V φ H 1 CΩ φ, were te last estimate follows from [7] for te stability of te L projection of H 1 functions into V H 1 Ω, and te Poincare inequality. Te tird estimate follows trivially from te proof of te second, since ere φ V by assumption. Lemma.3. We ave te following bound for te difference between filtering and discretely filtering functions φ H k Ω V : φ φ + α φ φ C k+ + α k φ k+1,.4 were C is a constant dependent on te size of Ω and te inf-sup constant, but independent of α,, and φ. Remark.1. Witout te use of divergence free elements, λ k will be part of te te rigt and side of te estimate in Lemma.3. Proof. Multiplying te filter equation 1.3 by arbitrary χ V note χ = 0 and integrating over te domain gives α φ, χ + φ, χ = φ, χ. Subtracting from tis te equation.3, and writing e := φ φ provides us wit α e, χ + e, χ = 0 χ V. From ere, standard finite element analysis and interpolation estimates in V see, e.g. [8] finis te proof..3 Oter coices of deconvolution operators Altoug our numerical analysis will be for te case of van Cittert approximate deconvolution, numerical scemes wit similar desirable properties can be devised oter coices of deconvolution operators. So tat we can remark about tem in te next section, we define ere some oter approximate deconvolution operators tat could also be used to define an ADM model of te form 1.10-1.14..3.1 Multiscale approximate deconvolution Multiscale deconvolution was recently proposed in [11], and is based on using two filters, of different radii. Using te notation of F α for te incompressible filter wit radius α, and F γ for te radius γ incompressible filter, te multiscale deconvolution operator G γ,α is defined by G γ,α φ := α γ φ α γ γ F γ φ 7
It is proven in [11] tat G γ,α F α φ = F γ φ. If γ < α, we expect tat φ F γ φ < φ F α φ, wic justifies te use of G γ,α as a deconvolution operator. Clearly, te smaller γ is, te better te approximation. Furtermore, in [11], it is proven tat for fixed α and in te periodic case, te error in te ADM using multiscale deconvolution as an approximation of te filtered NS equations is Oγ 1/, independent of α. Remarkably, tis result only assumes enoug smootness of te NS solution so tat it is unique. Te discrete operators for multiscale deconvolution are defined analogous, wit F α denoting.1-. wit radius α, F γ te same but wit radius γ, and G γ,αφ := α γ φ α γ γ F γ φ. We note tat a finer mes can be used for te Fγ filter step, altoug in [19] excellent results are found if te finer filtering is done on te same mes wit γ < α..3. Tikonov-Larentiev approximate deconvolution Te Tikonov-Larentiev regularization idea of deconvolution, wic was studied in te context of fluid flow models in [8], is for given φ and µ > 0, φ µ = F + µi 1 φ approximately satisfies F 1 φ. As a discrete problem, tis inverse problem as te form: Given φ, find φ µ, λ X, Q satisfying for all v, q X, Q, It is easy to observe tat α φ µ, v + 1 + µφ µ, v + λ, v = φ, v.5 φ µ, q = 0..6 φ µ 1 + µ 1/ φ and φ µ 1 + µ 1/ φ, and terefore if we define D T,µ to be te solution operator, we ave tat D T,µ < 1 for µ > 0. 3 An efficient and unconditionally stable sceme for te ADM and its analysis Te previous section presented sufficient notation so tat we may now present our numerical sceme in full detail. Te sceme is backward Euler in time, and finite element in space, and we make te assumption of N = 1 van Cittert approximate deconvolution. Algoritm 3.1. Given filtering radius α > 0, initial velocity v 0 V, a forcing f L 0, T ; H 1 Ω, end time T > 0, timestep > 0, set M = T/ and compute for n = 1,,..., M 1, and for all χ X and r Q, α v v n, χ 1 + +b v n vn, v v n v, χ + ν v n, χ p, χ v v n, χ 8 = f, χ,3.1 v, r = 0. 3.
Remark 3.1. Extension to N and a Crank-Nicolson timestepping sceme tat is still unconditionally stable does not appear obvious to us, and so far we ave not been able to derive suc a sceme. It does appear, owever, tat conditional stability results tat are only mildly restrictive on can be proven for tese cases. Our interest erein is for unconditional stability, so we leave tis exploration for future work. We will now analyze te sceme for well-posedness, stability, and convergence. We begin wit existence and uniqueness of solutions to Algoritm 3.1. Lemma 3.1 Existence and uniqueness. Algoritm 3.1 admits unique solutions. Proof. We prove tis teorem by considering a general timestep, and sowing tat given v n V, te solution at te next timestep, v, p X, Q exists uniquely. Since we are given te initial condition v 0 V, tis tecnique will prove tat te algoritm admits unique solutions at every timestep. We also utilize te fact tat at eac timestep, 3.1-3. is a finite dimensional linear problem, so a proof of uniqueness of solutions will also imply teir existence. To begin, given v n V, suppose for all χ, r X, Q, tere are two solutions to 3.1-3., v 1, p 1 and v, p, and let e = v 1 v. Subtracting te respective equations wit tese solutions gives te system, α e, χ + 1 e, χ p 1 p, χ +b v n vn, e, χ + ν e, χ = 0 χ X, 3.3 e, r = 0 r Q. 3.4 Coosing r = p 1 p in 3.3, χ = e in 3.4 and adding te equations vanises bot te nonlinear and te pressure terms, and leaves α e + 1 e + ν e = 0. 3.5 Tis implies e = 0, and tus v 1 = v. Now tat e = 0 is establised, uniqueness of te pressure follows from 3.3 and te discrete inf-sup condition. We now prove tat te unique solutions of Algoritm 3.1 are unconditionally stable wit respect to timestep. Lemma 3. Unconditional stability. Solutions to Algoritm 3.1 satisfy α v M + v M + ν v α + ν v 0 + v 0 + ν 1 f 1 Cν 1, 3.6 were C depends on data but can be considered independent of, ν,, and α. 9
Proof. We begin by coosing χ = v v n and r = p in 3.1-3., wic vanises te pressure and nonlinear terms, and yields α v v n, v v n + 1 + ν v v v n = v n, v v n We bound te forcing term in te usual way see, e.g. [], f, v v n. 3.7 f, v v n 1 f ν 1 + ν v v n. 3.8 For te viscous term, we obtain a lower bound by expanding and using Caucy-Scwarz and Young s inequalities, as ν v v n = 4ν v + ν v n 4ν v, v n 4ν v + ν v n 4 1 v + 1 vn = ν v ν v n. 3.9 Since v n v n, we furter reduce 3.9 to get ν v v n ν v + ν v v n. 3.10 For te second term in 3.7, we first decompose te rigt and side into its individual components to get 1 v v n, v v n v 1 v v n, 1 v v v n, vn v n 1 v, v n 1 + vn, vn. 3.11 Since te discrete filter is self adjoint and positive in te L inner product in V, 3.11 can be furter reduced as 1 v v n, v v n 1 v v n 1 1/ F v, F 1/ v n + 1 1/ F v n 1 v v n + 1 F 1/ v n F 1/ v. 3.1 Similarly for te first term in 3.7, α v v n, v v n α v v n + α F 1/ v n F 1/ v. 3.13 10
Next, we combine 3.7-3.13 to get α v v n + α + 1 F 1/ v v n + 1 + ν v + ν v n F 1/ v F 1/ v n F 1/ v v v n 1 ν f 1. 3.14 Multiplying by, summing from n = 0 to M 1, and reducing gives α v M α 1/ F v M + v M 1 1/ F v M + ν v + ν v M ν v0 + α v 0 α 1/ F v 0 + v 0 1 1/ F v 0 + f ν 1. 3.15 By Lemma., F 1/ φ = φ, φ φ and F 1/ φ = φ, φ φ, we reduce 3.15 and multiply bot sides by to obtain α v M + v M + ν v wic completes te proof. α + ν v 0 + v 0 + ν 1 f 1, 3.16 Remark 3.. We ave now proven tat Algoritm 3.1 admits unique solutions tat are bounded continuously by te data f, ν, v 0, T. Hence te algoritm defines a well-posed problem. We will now prove convergence of te algoritm to te continuous model solution. For continuous in time variables, we will denote φt n := φ n. Teorem 3.1 Convergence estimates. Consider α > 0 to be a fixed filtering radius, and assume v to be te N=1 model solution to 1.10-1.14 equipped wit te given problem data of Algoritm 3.1. If we assume te following smootness conditions on te model solution, v L 0, T ; H k+1 Ω, v tt L 0, T ; H 1 Ω, wit k, ten for any coice of timestep > 0, te error in te numerical solution from Algoritm 3.1 satisfies vt v M + α vt v M + ν vt v Cν 1 expν k 1 + ν 1 + α 4 k + α 4 + + α k+ + ν k, 3.17 11
were C is a constant dependent only on data, and is independent of α,,, and ν. Furtermore, if we assume te following smootness conditions on te velocity solution u of te NS equations equipped wit te problem data of Algoritm 3.1, u L 0, T ; H 5 Ω, u L 0, T ; H k+1 Ω, u tt L 0, T ; H 1 Ω, wit k, ten for any coice of timestep > 0, te difference between te filtered NS solution and te numerical solution from Algoritm 3.1 satisfies ut v M + α ut v M + ν ut v Cν 1 expν k 1 + ν 1 + α 4 k + α 4 + + α k+ + α 8 + ν k + ν α 8, were C is a constant dependent only on data, and is independent of α,,, and ν. Remark 3.3. Since α is fixed, convergence to te model is optimal in space and time. Also, we note tat te dependence on expν results from use of te Gronwall inequality, and is considered a gross overestimate. Tird, because te sceme is linear, te alternative Gronwall estimate of [14] was able to be applied as in [17, 7] so tat no timestep restriction needs assumed. Proof. In tis proof we will use C to represent a generic constant, possibly canging at eac instance, tat is independent of ν,, α, and. We begin wit te NSE, u t + u u + p ν u = f, ten rewrite it by adding and subtracting terms and using te definition of te filter, α u t + u t + D N u D N u + p + α λ t ν D N u = f + D N u D N u u u + ν u D N u. 3.19 Next, coose N=1 and write v := u, and note tat 3.19 represents te model if te last two terms on te rigt and side are removed. Hence we can prove bot estimates of te teorem at te same time, but for te filtered NS case, we will ave tese additional terms. Multiply 3.19 by χ V and integrate over te domain recall V is pointwise divergence free so te pressure term vanises to get α v t, χ + 1 v t, χ + b D 1 v, D 1 v, χ + ν D 1 v, χ = f, χ + b D 1 v, D 1 v, χ b u, u, χ ν u D N v, χ 3.0 and tus for,1,...,m-1, we ave α v v n 1, χ + v v n, χ + b v n v n, v v n, χ + ν v v n, χ = f, χ + Gv, χ, n, 3.1 3.18 1
for every χ V, were v Gv, χ, n = α v n v t t, χ v v n + v t t, χ +ν v v n, χ ν D 1 v, χ b D 1 v, D 1 v, χ b D 1 v n, D 1 v, χ b D 1 v n, D 1 v, χ b D 1 v n, v v n, χ b D 1 v n, v v n, χ b v n v n, v v n, χ + b D 1 v, D 1 v, χ b u, u, χ ν u D N v, χ. 3. Subtracting 3.1, wit χ restricted to V, from 3.1 and writing e n := v n v n gives α e e n 1, χ + e e n, χ + ν e e n, χ + b v n v n, v v n, χ b v n vn, v for all χ V, wit q arbitrary in Q. Next, we decompose e n = v n w n + wn vn =: ηn + φ n, for arbitrary w n V, wic allows 3.3 to be written as v n, χ = Gv, χ, n, 3.3 α φ φ n, χ 1 + φ φ n, χ + ν φ φ n, χ = b e n e n, v v n, χ + b v n vn, e e n, χ α η η n 1, χ η η n, χ + Gv, χ, n. 3.4 Next, we coose χ = φ derivative terms, we obtain φ n, and following analysis in Lemma 3. for te time φ φ n, φ φ n φ φ n + 1 F 1/ φ n F 1/ φ 3.5 and φ φ n, φ φ n φ φ n + 1 F 1/ φ n F 1/ φ. 3.6 13
Using te estimates 3.5-3.6, and te coice of χ = φ φ n in 3.4 yields 1 φ φ n + 1 F 1/ φ n F 1/ φ + α φ φ n + α F 1/ φ n F 1/ φ + ν φ φ n b φ n φn, v v n, φ φ n + b v n vn, φ φ n, φ + b η n η n, v v n, φ φ n + b v n vn + ν η η n, φ φ n α 1, η η n, φ η η n, φ φ n η η n, φ φ n + Gv, φ φ n φ n φ n, n. 3.7 Te fourt and fift terms on te rigt and side of 3.7 can be majorized as in [13], 1 α η η n, φ φ n 4C P F ν t t n η t dt + ν 16 φ φ n, 3.8 η η n, φ φ n t 4α4 η t dt + ν ν t n 16 φ φ n, 3.9 wile te viscous term on te rigt and side is bounded using Caucy-Scwarz and Young s inequalities: ν η η n φ n ν η η n φ ν η φ φ n + ν η n φ φ n ν 16 φ φ n + Cν η + η n. 3.30 For te first nonlinear term in 3.7, we majorize it using Lemma.1, Young s inequality, and Lemma. to get b φ n φn, v v n, φ φ n C φ n 1/ φn φ n φ n 1/ v v n φ φ n ν φ 16 φ n + Cν 1 φ n φ φn n φ n v v n ν φ 16 φ n + Cν 1 φ n φn v + v n ν 16 φ φ n ν + 16 φn + Cν v 4 + v n 4 φ n. 3.31 Tanks to Lemma.1, for te second nonlinear term in 3.7, we get b v n vn, φ φ n, φ φ n = 0. 3.3 Te tird nonlinear term in 3.7 is upper bounded using Lemma.1, ten applying 14
Young s inequality and., wic yields b η n η n, v v n, φ φ n C η n η n v v n φ φ n ν φ 16 φ n + Cν 1 η n v + v n. 3.33 Te fourt nonlinear term gets andled identically to 3.33, and we get b v n vn, η η n, φ ν 16 φ φ n Combining 3.7-3.34 yields 1 φ φ n + 1 φ n + Cν 1 η n v F 1/ φ n F 1/ F 1/ φ n F 1/ φ + α + v n. 3.34 φ φ n + 10ν 16 φ φ n + α φ ν 16 φn +Cν v 4 + v n 4 φ n +Cν 1 η n v + v n + Cν 1 η n v + C P F ν t t n η t dt + 4α4 ν + v n + Cν η + η n t η t dt + Gv, φ t n We turn our attention now to majorizing Gv, φ of G, we use standard estimates based on Taylor series []: v α v n v t t, φ φ n v v n ν 16 φ n, n. 3.35 φ n, n. For te first two terms φ φ n α 4 + v tt L ν t n,t,h 1 Ω 3.36 v t t, φ ν 16 φ n φ φ n C + P α 4 v tt L ν t n,t,h 1 Ω 3.37 For te tird and fourt terms of G, we use Lemma.1 followed by Young s inequality, and Taylor s teorem to find b D 1 v, D 1 v, φ φ n b D 1 v n, D 1 v, φ φ n = b D 1 v v n, D 1 v, φ ν φ 16 ν φ 16 ν 16 φ φ n φ n + Cν 1 D 1 v v n D 1 v φ n + Cν 1 v v n v φ n + Cν 1 v tt L t n,t,h 1 Ω v. 3.38 15
b D 1 v n, D 1 v, φ φ n b D 1 v n, v v n, φ φ n = b D 1 v n, v v n, φ ν φ 16 ν 16 φ n φ n + Cν 1 D 1 v v n D 1 v n φ φ n + Cν 1 v tt L t n,t,h 1 Ω vn. 3.39 Te fift term gets bounded similar to te previous terms using Lemma.1, Young s inequality, Lemma., but also uses Lemma.3 to bound te difference between te continuous and discrete filters, b D 1 v n, v v n, φ φ n b v n v n, v v n, φ φ n = b v n v n, v v n, φ φ n + b D 1 v n, v n v n, φ φ n C v n v n φ φ n D 1 v n + v v n C v n v n φ φ n v n + v ν φ 16 φ n + Cν 1 v n + v v n v n 3.40 ν φ 16 φ n + Cν 1 v n + v α k+ + k v n H k+1 Ω. For te sixt term of G, we Lemma.1 to bound te b terms, ten use tat v = u and te bound 1.8 along wit Young s inequality to get b D 1 v, D 1 v, φ φ n b u, u, φ φ n = b D 1 v u, u, φ φ n + b D 1 v, D 1 v u, φ φ n C u + v D 1 v u φ φ n ν φ 16 φ n + Cν 1 α 8 u + v u 3.41 H 5 Finally, te last term in G is majorized by Caucy-Scwarz and Young s inequalities, and 1.8, wic gives ν u D N v, φ ν φ 16 ν 16 φ n φ n + 4ν u D N v φ φ n + 4να 8 u. 3.4 H 5 16
Combining 3.35-3.4 provides us wit 1 φ φ n + 1 F 1/ φ n F 1/ F 1/ φ n F 1/ φ + α + 3ν 16 φ φ n φ φ n + α φ ν 16 φn +Cν v 4 + v n 4 φ n +Cν 1 η n v + v n + Cν 1 η n v + C P F ν t t n + v n + Cν η + η n t η t dt + 4α4 η t dt ν t n + α 4 v tt L ν t n,t,h 1 Ω + C P α 4 v tt L ν t n,t,h 1 Ω + Cν 1 v tt L t n,t,h 1 Ω v + Cν 1 v tt L t n,t,h 1 Ω vn + Cν 1 v n + v α k+ + k v n H k+1 Ω + Cν 1 α 8 u + v u H 5 + 4να 8 u H 5, 3.43 Multiplying troug by and summing over timesteps from to M-1 gives φ M φ 0 + 1 F 1/ φ 0 F 1/ φ M + α φ M φ 0 + α +Cν 1 F 1/ φ 0 F 1/ ν 16 φ n + Cν φ M + 3ν 16 η n v + v n +Cν 1 + Cν + 4α4 ν t φ φ n v 4 + v n 4 φ n η + η n + C P F ν η t dt + α 4 t n ν η n v + v n t t n η t dt v tt L t n,t,h 1 Ω + C P α 4 v tt L ν t n,t,h 1 Ω + Cν 1 3 v tt L t n,t,h 1 Ω v + Cν 1 + Cν 1 α 8 + Cν 1 3 v tt L t n,t,h 1 Ω vn v n + v α k+ + k v n H k+1 Ω u + v u H 5 + 4να 8 u, 3.44 H 5 17
Next, using te bounds on te NSE, te assumption tat u 0 V so tat φ 0 = 0 and reducing gives us φ M 1 F 1/ φ M + α φ M α ν 16 + Cν 1 + C P F ν T 0 φ n Cν 1/ F φ M + 3ν 16 φ n + Cν 1 η n v + v n + Cν η t dt + 4α4 ν T 0 η t dt + α 4 T ν φ η n φ n η + η n + C P α 4 T ν + Cν 1 T + Cν 1 T α k+ + k + Cν 1 α 8 T + 4να 8 T, 3.45 wic simplifies furter wit regularity assumptions and interpolation estimates to φ M 1 +Cν 1 T F 1/ ν 16 φ M + α φ M α φ n + Cν 1/ F φ M + 3ν 16 φ n + Cν 1 k φ v φ n k + k+ + α 4 k + α 4 + + α k+ + k + α 8 +CνT k +α 8. 3.46 By similar analysis as te viscous term in te stability lemma in 3.10, and using tat φ 0 = 0, we ave tat 3ν 16 φ φ n 3ν 16 φ + φ φ n = 3ν 16 φ + 3ν 16 φm. 3.47 Combining 3.46-3.47, using te stability estimate, subtracting ν 16 φn from bot sides, using te fact tat φ, φ φ, and assuming < 1 gives 1 φm + α φm + ν 8 φ Cν φ n + Cν 1 k 1 + ν 1 + α 4 k + α 4 + + α k+ + α 8 + Cν k + α 8. 3.48 Finally, applying te discrete Gronwall lemma but we note te first sum on te rigt side is M-1, and so te alternate version of [14] given as a remark after Lemma 5.1 can be used, 18
for any > 0 we get φ M + α φ M + ν φ Cν 1 expν k 1 + ν 1 + α 4 k + α 4 + + α k+ + α 8 + ν k + ν α 8. Te triangle inequality completes te proof for te filtered NS case, and for convergence to te model, we need only trace troug and remove te resulting terms from majorizing te last two terms in 3.19. 3.1 Extension to oter types of deconvolution Extension to oter deconvolution operators can be done in a straigt-forward manner, and analogous results to tose below can be obtained wit similar analysis as is done for 3.1-3., provided D is self-adjoint and positive in te L inner product in V. Following similar analysis as above, for multiscale deconvolution, te splitting G γ,αv α γ v α γ γ Fγ v n. can be sown to provide unconditional stability and optimal convergence to te ADM model solution defined wit multiscale deconvolution provided only α γ > 0. For Tikonov-Larentiev, we would use a sligtly different splitting tat exploits tis deconvolution operator as norm less tan 1, following ideas in [4]. Hence we coose D T,µ v v I D T,µ v n, wic wit similar analysis as above will provide an unconditionally stable algoritm tat converges optimally to te ADM model solution defined wit Tikonov-Larentiev deconvolution, for any µ > 0. 4 Numerical test: Cannel flow wit a contraction and two outlets We now test our sceme on a flow problem wit interesting, complex beavior. As wit many nonlinear models and teir associated numerical scemes, it is difficult to numerically verify tat teoretical convergence rates are valid, because it is very difficult to determine analytical expressions for teir solutions. Tis model appears to fit into tat category. Hence, instead, we can only verify tat te sceme works as intended, by giving good results on bencmark problems. Te numerical test we coose is for D cannel flow wit a contraction, one inlet on te left and side, and two outlets, at te top and at te rigt and side, wit Re=1,000. It was first studied by Turek et. al. in [15], and a diagram of te flow domain is given in Figure 1. Te velocity boundary conditions are prescribed no slip on all walls, a parabolic profile for te inlet: u in = 4y1 y, 0 T, and zero traction outflow. We set te kinematic viscosity ν = 0.001, forcing ft = 0, start te flow from rest at T=0, and run te simulation to T=4. Te resolved solution s velocity is sown in Figure as speed contours, for T=1,, 3, and 4. Tis solution was found by computing te NSE directly, using te fully implicit 19
Figure 1: Sown above is a diagram of te domain for test problem 1. Crank-Nicolson finite element sceme see, e.g. [], wit a timestep of = 0.005, and Taylor-Hood elements on a triangular mes tat provides 60,378 total degrees of freedom dof. We observe tat te flow speeds up troug te contraction, ten remains mostly in a single stream until it exits te cannel, but te stream moves up and down on te rigt side of te cannel. By T=4, we observe oter small flow structures form on te rigt side of te cannel as well. Tis is a difficult test problem for a DNS to get correct, as tere are several possible sources of instabilities. We will sow tat te sceme proposed for te ADM will accurately capture te correct flow beavior on muc coarser meses tan a DNS needs. We compute next te NSE directly on a coarse mes sown in Figure 3, again using te fully implicit Crank-Nicolson finite element sceme. We use Taylor-Hood elements, for wic tere was 10,418 total dof, and also SV elements 16,170 total dof. For bot tests we cose = 0.005. Solutions are sown at T=4 in Figure 3, and bot are clearly not good approximations of te true solution. Te Taylor-Hood NSE solution experiences instability at te contraction, wile te SV solution is badly under-resolved, particularly after te contraction. Finally, we compute Algoritm 3.1 wit bot Taylor-Hood and SV elements, using te same coarse mes, timestep = 0.01 and filtering radius α = 0.06. Te solutions are sown at T=4 in Figure 4, and we observe te SV solution for te proposed sceme is able to correctly predict te overall flow beavior of te true solution. Te Taylor-Hood model solution gives an incorrect prediction on te rigt side of te contraction. Te worse performance of te Taylor-Hood solution is consistent wit our analysis, since additional contributions are made to te error by te pressure and by Lagrange multipliers arising from filtering. 5 Conclusions and future directions We proposed a numerical sceme for te ADM wit incompressible filtering, for te case of omogeneous Diriclet boundary conditions, and proved it to be unconditionally stable wit respect to timestep and optimally accurate in bot space and time to te derived ADM model. Te key ideas are to start wit te fourt order version of te ADM, ten 0
DNS, T=1.5 1.5 1 0.5 0 DNS, T=.5 1.5 1 0.5 0 DNS, T=3 3.5 1.5 1 0.5 0 DNS, T=4.5 1.5 1 0.5 0 Te fine mes used for DNS Figure : Sown above is speed contour plots of te resolved Navier-Stokes velocity solution at T=1,,3,4 from top to bottom, wic was resolved wit 60,378 total degrees of freedom. reduce it to second order by using an additional approximate deconvolution approximation in te viscous term, ten coose an appropriate splitting of te deconvolution operator in te timestepping algoritm. Te sceme is quite efficient, since it is linear at eac timestep, and eac filter solve is decoupled from te system solves due to explicit treatment of filtering. It is troug a special coice of test function tat allows unconditional stability 1
Te coarse mes NSE on coarse mes wit TH elements 3 1 0 NSE on coarse mes wit SV elements Figure 3: Sown above is te mes used for coarse mes simulations, and speed contours of te T=4 coarse mes NSE solutions. Model solution on te coarse mes wit TH elements.5 1.5 1 0.5 Model solution on te coarse mes wit SV elements.5 1.5 1 0.5 Figure 4: Sown above are speed contours of te T=4 coarse mes solutions of te proposed sceme for te ADM. and convergence to be proven. Extensions to oter deconvolution operators was discussed, and finally, te sceme was sown to be successful on a flow problem wit complex beavior.
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