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Advances in Numerical Analysis
Volume 2012 (2012), Article ID 162539, 32 pages
Convergence Analysis of a Fully Discrete Family of Iterated Deconvolution Methods for Turbulence Modeling with Time Relaxation
1Department of Mathematics, University of Pittsburgh, PA 15260, USA
2Departmento de Matemática Pura e Aplicada, Universidade Federal do Rio Grande do Sul, Porto Alegre 91509-900, RS, Brazil
3Department of Mathematics, Wheeling Jesuit University, WV 26003, USA
4Farquhar College of Arts and Sciences, Nova Southeastern University, FL 33314, USA
Received 31 May 2012; Accepted 25 July 2012
Academic Editor: William J. Layton
Copyright © 2012 R. Ingram et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We present a general theory for regularization models of the Navier-Stokes equations based on the Leray deconvolution model with a general deconvolution operator designed to fit a few important key properties. We provide examples of this type of operator, such as the (modified) Tikhonov-Lavrentiev and (modified) Iterated Tikhonov-Lavrentiev operators, and study their mathematical properties. An existence theory is derived for the family of models and a rigorous convergence theory is derived for the resulting algorithms. Our theoretical results are supported by numerical testing with the Taylor-Green vortex problem, presented for the special operator cases mentioned above.
1. Approximate Deconvolution for Turbulence Modeling
Numerical simulations of complex flows present many challenges. The Navier-Stokes (NS) equations (NSE), given by the following is an exact model for the flow of a viscous, incompressible fluid, . For turbulent flows (characterized by Reynold’s number ), it is infeasible to properly resolve all significant scales above the Kolmogorov length scale by direct numerical simulation. Thus, numerical simulations are often based on various regularizations of NSE, rather than NSE themselves. Accordingly, regularization methods provide a (computationally) efficient and (algorithmically) simple family of turbulence models. Several of the most commonly applied regularization methods include: where in each case and is an averaged velocity field , is pressure, and is the Bernoulli pressure. More details about these models can be found for instance in [2–5] and references therein. Although these regularization methods achieve high theoretical accuracy and perform well in select practical tests, those models do not provide a fully-developed numerical solution for decoupling the scales in a turbulent flow. In fact, results show that only time relaxation regularization truncates scales sufficiently for practical computations. Indeed, it is shown that time relaxation term for damps unresolved fluctuations over time [5, 6]. Note that the choice of is an active area of research and that solutions are very sensitive to variations in .
Deconvolution-based regularization is also an active area of research obtained, for example, by replacing by in each (1.2)–(1.5) for some deconvolution operator . In , Dunca proposed the general Leray-deconvolution problem ( instead of ) as a more accurate extension to Leray’s model . Leray used the Gaussian filter as the smoothing (averaging) filter , denoted above by overbar. In , Germano proposed the differential filter (approximate-Gaussian) where is the filter length. The differential filter is easily modeled in the variational framework of the finite element (FE method (FEM)). We provide a brief overview of continuous and discrete operators (Section 3).
The deconvolution-based models have proven themselves to be very promising. However, among the very many known approximate deconvolution operators from image processing, for example, , so far only few have been studied for turbulence modeling, for example, the van Cittert and the modified Tikhonov-Lavrentiev deconvolution operators. Their success suggests that it is time to develop a general theory for regularization models of the NSE as a guide to development of models based on other, possibly better, deconvolution operators and refinement of existing ones.
Herein, we present a general theory for regularization models of the NSE based on the Leray deconvolution model with a general deconvolution operator. We prove energetic stability (and hence existence) and convergence of an FE (in space) and Crank-Nicolson (CN) (in time) discretization of the following family of Leray deconvolution regularization models with time relaxation: find and satisfying the following for some appropriate boundary and initial conditions (the fully discrete model is presented in Problem 1 with energetic stability and well posedness proved in Theorem 4.5).
1.1. Improving Accuracy of Approximate Deconvolution Methods
The fundamental difficulties corresponding to regularization methods applied as a viable turbulence model include ensuring that:(i)scales are appropriately truncated (model microscale = filter radius = mesh width)(ii)smooth parts of the solution are accurately approximated ( for smooth )(iii)physical fidelity of flow is preserved.
Due to the nonlinearity in (1.6), different choices of the filter and deconvolution operator yield significant changes in the solution of the corresponding model. Implementation concerns for deconvolution methods, for example, Tikhonov-Lavrentiev regularization given by , include selection of deconvolution parameter . Iterated deconvolution methods reduce approximation sensitivity relative to -selection and, hence, allow a conservatively large -selection for stability with updates (fixed number of iterations) used to recover higher accuracy (Section 3.3). For example we prove, under usual conditions, (Proposition 5.3), so that iterated modified Tikhonov regularization gives geometric convergence with respect to the update number . In either case, Tikhonov-Lavrentiev or iterated Tikhonov-Lavrentiev regularization, we prove, under usual conditions, (Proposition 5.4) where and represent the discrete deconvolution operator and discrete filter, respectively, and is order of FE polynomial space. We propose minimal properties for a general family of deconvolution operators and filters (Section 3.1) satisfying (Assumptions 3.4, 3.5), for example,(i), forces spectrum of in (ii), controls size of (iii) as , (for smooth ), ensures convergence of method(iv) as (for smooth ), ensures convergence of method.
In fact, the updates satisfying inherit the properties assumed for the base operator in Assumptions 3.4, 3.5 (Propositions 3.10, 3.11). These iterates represent defect correction generalization of iterated Tikhonov regularization operator . We prove that the FE-CN approximation of the general deconvolution turbulence model (Problem 1) satisfies (Theorem 4.6) for smooth enough solutions of the NSE (see variational formulation (2.3)–(2.5)). We show that (for any and ) (Proposition 5.3) and (for ) (Proposition 5.4) for modified iterated Tikhonov-Lavrentiev regularization (see Corollary 5.5 for corresponding error estimate). We conclude with a numerical test that verifies the theoretical convergence rate predicted in Theorem 4.6 (Section 5.2).
1.2. Background and Overview
One of the most interesting approaches to generate turbulence models is via approximate deconvolution or approximate/asymptotic inverse of the filtering operator. Examples of such models include: Approximate Deconvolution Models (ADM) and Leray-Tikhonov Deconvolution Models. Layton and Rebholz compiled a comprehensive overview and detailed analysis of ADM  (see also references therein). Previous analysis of the ADM with and without the time-relaxation term used van Cittert deconvolution operators [5, 6]; although easily programmed, van Cittert schemes can be computationally expensive . Tikhonov-Lavrentiev regularization is another popular regularization scheme . Determining the appropriate value of to ensure stability while preserving accuracy is challenging, see for example, [14–19]. Alternatively, iterated Tikhonov regularization is well known to decouple stability and accuracy from the selection of regularization parameter , see for example, [11, 20–22]. Iterated Tikhonov regularization is one special case of the general deconvolution operator we propose herein.
2. Function Spaces and Approximations
Let the flow domain for be a regular and bounded polyhedral. We use standard notation for Lebesgue and Sobolev spaces and their norms. Let and be the -norm and inner product, respectively. Let represent the -norm. We write and for the corresponding norm. Let the context determine whether denotes a scalar, vector, or tensor function space. For example let . Then, implies that and implies that . Write equipped with the standard norm. For example,
Denote the pressure and velocity spaces by and , respectively. Moreover, the dual space of is denoted and equipped with the norm
Fix and . In this setting, we consider strong NS solutions: find and satisfying Let . Restricting test functions reduces (2.3)–(2.5) to find satisfying and (2.5). For smooth enough solutions, solving the problem associated with (2.6), (2.5) is equivalent to (2.3)–(2.5).
Control of the nonlinear term is essential for establishing a priori estimates and convergence estimates. We state a selection of inequalities here that will be utilized later:
2.1. Discrete Function Setting
Fix . Let be a family of subdivisions (e.g., triangulation) of satisfying so that and any two closed elements , are either disjoint or share exactly one face, side, or vertex. See Chapter II, Appendix in  for more on this subject in context of Stokes problem and  for a more general treatment. For example, consists of triangles for or tetrahedra for that are nondegenerate as .
Let and be a conforming velocity-pressure mixed FE space. For example, let and be continuous, piecewise (on each ) polynomial spaces. The discretely divergence-free space is given by Note that in general (e.g., for Taylor-Hood elements). In order to avoid stability issues arising when FE solutions are not exactly divergence free (i.e., when ), we introduce the explicitly skew-symmetric convective term so that Note that when . Moreover, the trilinear from is continuous and skew-symmetric on .
Lemma 2.1. If ,
Proof. The proof of the first inequality can be found in . The second follows from Hölder’s and Poincaré’s inequalities.
For the time discretization, let be a discretization of the time interval for a constant time step . Write and, if , . Define for any . Write . We say that if the associated norm defined above stays finite as .
The discrete Gronwall inequality is essential to the convergence analysis in Section 4.2.
Lemma 2.2. Let and , , , for any integer and satisfy Suppose that for all , and set . Then,
Proof. The proof follows from .
2.2. Approximation Theory
Let be a generic constant independent of . Preserving an abstract framework for the FE spaces, we assume that inherit several fundamental approximation properties.
Assumption 2.3. The FE spaces satisfy:
Uniform inf-sup (LBB) condition FE-approximation Inverse-estimate The well-known Taylor-Hood mixed FE is one such example satisfying Assumption 2.3.
Estimates in (2.18)–(2.20) stated below are used in proving error estimates for time-dependent problems: for any , where , , and is required, respectively, for some . Each estimate (2.18)–(2.20) is a result of a Taylor expansion with integral remainder.
These higher-order spatial ( or ) and temporal estimates (2.18)–(2.20) require that the nonlocal compatibility condition addressed by Heywood and Rannacher in [26, 27] (and more recently, for example, by He in [28, 29] and He and Li in [30, 31]) is satisfied. Suppose, for example, that is the solution of the (well-posed) Neumann problem In order to avoid the accompanying factor in the error estimates contained herein, the following compatibility condition is necessarily required (e.g., see [27, Corollary 2.1]): Replacing (2.21) with (2.21)(a), (2.22) defines an overdetermined Neumann-type problem. Condition (2.22) is a nonlocal condition relating and . Condition (2.22) is satisfied for several practical applications including start up from rest with zero force, , . In general, however, condition (2.22) cannot be verified. In this case, it is shown that, for example, .
We finish with an approximation property of the -projection. Indeed, Assumption 2.4 holds for smooth enough .
Assumption 2.4. Fix and let be the unique solution satisfying for all . Then for .
3. Filters and Deconvolution
We prescribe the essential properties our filter and deconvolution operator in this section.
Definition 3.1. Let be a Hilbert space and . Write if is self-adjoint and for all and call symmetric nonnegative (snn). Write if is self-adjoint and for all and call symmetric positive definite (spd).
Let be a linear, bounded, compact operator on representing a generic smoothing filter with filter radius : One example of this operator is the continuous differential filter (Definition 3.2), which is used, together with its discrete counterpart (Definition 3.3), for implementation of our numerical scheme (Section 5.2).
Definition 3.2 (continuous differential filter). Fix . Then is the unique solution of with corresponding weak formulation Set so that , defined by , is well defined.
Definition 3.3 (discrete differential filter). Fix . Then is the unique solution of the following
Set so that , defined by , is well defined. Here, is the projection and the discrete Laplace operator satisfying the following
It is well known that and are each linear and bounded, is compact, and the spectrum of and (on and , resp.) is contained in and spectrum of and (on and , resp.) is contained in so that For more detailed exposition on these operators, see .
3.1. A Family of Deconvolution Operators
Assumption 3.4 (continuous deconvolution operator). Suppose that is linear, bounded, spd, and commutes with so that
for some constant . Moreover, suppose that is parametrized by , so that
for smooth enough .
Note that the first estimate in (3.6) is required so that the spectral radius satisfies . The second estimate in (3.6) (which controls the -seminorm of ) is required for the convergence analysis in Section 4.2.
Assumption 3.5 prescribes properties of the discrete analogue corresponding to the continuous deconvolution operator (Assumptions 3.4).
Assumption 3.5 (discrete deconvolution operator). Let satisfy Assumption 3.4. Let be a discrete analogue of that is linear, bounded, spd. Suppose that is linear, bounded, spd, and commutes with such that for some constant . Moreover, suppose that is parametrized by such that and for all for some .
Remark 3.6. If for some continuous map , then commutativity is satisfied . Tikhonov-Lavrentiev (modified) regularization with , given by , is one such example with and , , , see .
Remark 3.7. Letting denote the th (ordered) eigenvalue of a given operator, commutativity of and provides and similarly for the discrete operator .
Proof. Assumptions 3.4, 3.5 guarantee that the spectral radius and . Also, and and commute so that . Similarly, . Therefore, the spectrum of , are each contained in . So, , have spectrum contained in which ensures the non-negativity of both and .
3.2. Iterated Deconvolution
One can show, by eliminating intermediate steps in the definition of the iterated regularization operator in (1.9) with base operator satisfying Assumption 3.4, that Similarly, the discrete iterated regularization operator with discrete base operator satisfying Assumption 3.5, is given by the following We next show that and for inherit several important properties from and , respectively, via Assumption 3.5.
Proposition 3.10. Fix . Then defined by (3.14) satisfies Assumption 3.4. In particular, is linear, bounded, commutes with and satisfies (3.6)(a). Estimate (3.6)(b) is replaced by the following for some constant . Estimate (3.7) is replaced by the following Moreover, and .
Proof. First notice that is linear and bounded since it is a linear combination of linear and bounded operators , for . Moreover, since commutes with , it follows that commutes with and hence with . Next, is a sum of spd and snn operators , . Hence, . Next, notice that Letting denote the th (ordered) eigenvalue of a given operator, we can characterize the spectrum of by summing the resulting finite geometric series (3.18) to get Then under Assumption 3.4, Lemma 3.9 with (3.19) implies that . Hence, satisfies (3.6)(a). Expanding the terms in (3.18) as powers of , we see that (3.18) can be written as a polynomial (with coefficients ) in , so that since can be applied successfully. Therefore (3.16) follows with . Next, start with (3.18) to get Estimate (3.17) follows by noting , , and by Assumption 3.5, .
Proposition 3.11. Fix . Then defined by (3.15) satisfies Assumption 3.5. In particular, is linear, bounded, commutes with and satisfies (3.8)(a). Estimate (3.8)(b) is replaced by the following for some constant . Estimate (3.9) is replaced by the following for any for some . Moreover, for some constant .
Proof. The first two assertions follow similarly as in the previous proof of Proposition 3.10. To prove (3.23), we start by writing and then subtract (3.24) from (3.18) to get where Then taking norms across (3.25), we get Notice that so that . Moreover, via Assumption 3.5. Next, using the binomial theorem and factoring, we get Then, applying , to (3.28) provides Again, via Assumption 3.5. So, we combine these above results to conclude (3.23) with .
3.3. Tikhonov-Lavrentiev Regularization
We provide two examples of discrete deconvolution operators to make the abstract formulation in the previous section more concrete. The Tikhonov-Lavrentiev and modified Tikhonov-Lavrentiev operator (for linear, compact ) is given by the following
Definition 3.12 ((weak) modified Tikhonov-Lavrentiev deconvolution). Fix . Let . For any , let be the unique solution of
Definition 3.13 ((discrete) modified Tikhonov-Lavrentiev deconvolution). Fix . Let and . For any , let be the unique solution of
The iterated modified Tikhonov-Lavrentiev operator (for linear, compact ) is obtained from the Tikhonov-Lavrentiev operator with updates via (1.9):
Definition 3.14 (iterated modified Tikhonov-Lavrentiev deconvolution (weak)). Fix and . Let . Define , then for any and , let be the unique solution of
Definition 3.15 (iterated modified Tikhonov-Lavrentiev deconvolution (discrete)). Fix and . Let , and . Define , then for any and , let be the unique solution of
4. Well Posedness of the Fully Discrete Model
We now state the proposed algorithm.
Problem 1 (CNFE for Leray-deconvolution). Let . Then, for each , find satisfying where .
Notice that when so that the problem of finding satisfying
4.1. Well Posedness
We establish existence of at each time step of (4.3) by Leray-Schauder’s fixed-point theorem.
Lemma 4.1. Let for any and , . Suppose that satisfies Assumption 3.5. Then is a continuous and coercive bilinear form and is a linear, continuous functional.
Proof. Linearity for is obvious, and continuity follows from an application of Hölder’s inequality. Continuity for also follows from Hölder’s inequality and Assumption 3.5. Coercivity is proven by application of (3.13).
Lemma 4.2. Let be such that, for any , solves Then is a well-defined, linear, bounded operator.
Proof. Linearity is clear. The results of Lemma 4.1, and the Lax-Milgram theorem prove the rest.
Lemma 4.3. Fix . Let be a solution of Problem 1 and let satisfy, for any , Then is well-defined, bounded, and continuous.
Proof. For each , the map is a bounded, linear functional (apply Hölder’s inequality and (2.11)). Since is a Hilbert space, we conclude that is well defined, by the Riesz-Representation theorem. Moreover, is bounded on and since the underlying function space is finite dimensional, continuity follows.
Lemma 4.4. Fix . Let be defined such that . Then, is a compact operator.
Proof. is a compact operator (continuous on a finite dimensional function space). Thus, is a continuous composition of a compact operator and hence compact itself.
Theorem 4.5 (well posedness). Fix . There exists satisfying Problem 1. Moreover, for all integers , independent of .
Proof. First, assume that is a solution to (4.1), (4.2). Set in (4.1) so that skew-symmetry of the nonlinear term provides
Duality of with Young’s inequality implies
From (3.13), we have
Then applying (4.9), (4.10) to (4.8), combining-like terms and simplifying provides
Summing from to , we get the desired bound.
Next, let . Showing that has a fixed point will ensure existence of solutions to (4.3). Indeed, if we can show that , then since is given initial data, existence of is immediate. Induction can be applied to prove existence of . To this end, since is compact, it is enough to show (via Leray Schauder) that any solution of the fixed-point problem is uniformly bounded with respect to . Hence, we consider
Test with , use skew-symmetry of the trilinear form and properties of given in Assumption 3.5 and (3.13) to get Duality of followed by Young’s inequality implies Since from the a priori estimate (4.7), we apply Hölder’s and Young’s inequalities to get Applying estimates (4.14), (4.15) to (4.13) we get that independent of . By the Leray-Schauder fixed point theorem, given , there exists a solution to the fixed-point theorem . By the induction argument noted above, there exists a solution for each to (4.3). Existence of an associated discrete pressure follows by a classical argument, since the pair satisfies the discrete inf-sup condition (2.15).
4.2. Convergence Analysis
Under usual regularity assumptions, we summarize the main convergence estimate in Theorem 4.6. Suppose that represents deconvolution with -updates.
Proof of Theorem 4.6, Corollary 4.7. Suppose that , satisfying (2.3), (2.4) also satisfy , , and so that, for each , The consistency error for the time-discretization and regularization/time-relaxation error are given by, for , where . Write . Using (4.20), rewrite (4.19) in a form conducive to analyzing the error between the continuous and discrete models: Let be the -projection of so that . Decompose the velocity error where . Fix . Note that for any . Subtract (4.21) from (4.3), apply (3.13)(b), and test with to get (Spatial discretization error): Fix . First, apply Hölder’s and Young’s inequalities to get Apply (3.12) and duality estimate on to get We bound the convective terms next. First, and estimate (2.11)(b) give and with (2.11)(a) give Next, rewrite the remaining nonlinear term Once again, and (2.11)(a) give Lastly, estimate (2.11)(a) and inverse inequality (2.17) give Apply estimates and from Assumption 3.5 along with estimates (4.26)–(4.30) to get (Time discretization error): First, apply duality estimate on to get