PARTITIONED METHODS FOR COUPLED FLUID FLOW PROBLEMS

Transcription

1 PARTITIONED METHODS FOR COUPLED FLUID FLOW PROBLEMS by Hoang A. Tran B.S. in Matematics and Computer Sciences, University o Natural Sciences, Ho Ci Min City, Vietnam, 006 France-Vietnam M.S. in Applied Matematics, Université d Orléans, Orléans, France, 008 Submitted to te Graduate Faculty o te Department o Matematics in partial ulillment o te requirements or te degree o Doctor o Pilosopy University o Pittsburg 03

2 UNIVERSITY OF PITTSBURGH MATHEMATICS DEPARTMENT Tis dissertation was presented by Hoang A. Tran It was deended on April 4t 03 and approved by Pro. Catalin Trencea, Dept. o Matematics, University o Pittsburg Pro. William Layton, Dept. o Matematics, University o Pittsburg Pro. Ivan Yotov, Dept. o Matematics, University o Pittsburg Pro. Noel Walkington, Dept. o Matematical Sciences, Carnegie Mellon University Dissertation Advisors: Pro. Catalin Trencea, Dept. o Matematics, University o Pittsburg, Pro. William Layton, Dept. o Matematics, University o Pittsburg ii

3 PARTITIONED METHODS FOR COUPLED FLUID FLOW PROBLEMS Hoang A. Tran, PD University o Pittsburg, 03 Many low problems in engineering and tecnology are coupled in teir nature. Plenty o turbulent lows are solved by legacy codes or by ones written by a team o programmers wit great complexity. As knowledge o turbulent lows expands and new models are introduced, implementation o modern approaces in legacy codes and increasing teir accuracy are o great concern. On te oter and, industrial low models normally involve multi-pysical process or multi-domains. Given te dierent nature o te pysical processes o eac subproblem, tey may require dierent meses, time steps and metods. Tere is a natural desire to uncouple and solve suc systems by solving eac subpysics problem, to reduce te tecnical complexity and allow te use o optimized legacy sub-problems codes. Te objective o tis work is te development, analysis and validation o new modular, uncoupling algoritms or some coupled low models, addressing bot o te above problems. Particularly, tis tesis studies: i explicitly uncoupling algoritm or implementation o variational multiscale approac in legacy turbulence codes, ii partitioned time stepping metods or magnetoydrodynamics lows, and iii partitioned time stepping metods or groundwater-surace water lows. For eac direction, we give compreensive analysis o stability and derive optimal error estimates o our proposed metods. We discuss te advantages and limitations o uncoupling metods compared wit monolitic metods, were te globally coupled problems are assembled and solved in one step. Numerical experiments are perormed to veriy te teoretical results. iii

4 TABLE OF CONTENTS PREFACE x.0 INTRODUCTION Explicitly uncoupled variational multiscale stabilization o luid lows..... Partitioned time stepping metods or magnetoydrodynamic lows Partitioned time stepping metods or te evolutionary Stokes-Darcy problems 7.4 Analytical tools Tesis outline EXPLICITLY UNCOUPLED VARIATIONAL MULTISCALE STABI- LIZATION OF FLUID FLOWS Metod descriptions Notation and Preliminaries Te Postprocessed VMS Metod Growt o Perturbations in te discrete sceme Algoritms or Computing te Projection A Computationally Attractive Variant Numerical experiments Green-Taylor vortex Test wit an L-saped discontinuity advected skew to mes Decaying Homogeneous Isotropic Turbulence Conclusions PARTITIONED TIME STEPPING METHODS FOR MAGNETOHY- DRODYNAMICS FLOWS iv

5 3. Problem setting Notation and preliminaries Te partitioned time stepping scemes Stability Error analysis Numerical experiments Test : Convergence rates Test : Stability Conclusion IMPLICIT-EXPLICIT BASED PARTITIONED METHODS FOR THE EVOLUTIONARY STOKES-DARCY PROBLEMS Introduction Te continuous problem and semi-discrete approximation Discrete ormulation Long time stability BEFE Stability BELF Stability Error analysis Numerical experiments Test : Convergence rates Test : Stability Conclusion SPLITTING BASED PARTITIONED METHODS FOR THE EVO- LUTIONARY STOKES-DARCY PROBLEMS Notations and preliminaries Discrete ormulation Analysis o stability o SDsplit, BEsplit/ and CNsplit SDsplit Stability BEsplit Stability BEsplit Stability v

6 5.3.4 CNsplit Stability Numerical experiments Test : Convergence rates Test : Stability in case o small parameters Conclusion CONCLUDING REMARKS AND FUTURE RESEARCH Fast-slow wave splitting or atmosperic and ocean codes BIBLIOGRAPHY vi

7 LIST OF TABLES. Error and convergence rate data or Uncoupled VMS were te projection term is solved iteratively Error and convergence rate data or Uncoupled VMS were te projection term is lagged Corresponding C* or dierent inite element large scale spaces Te convergence perormance or PARA Te convergence perormance or SEQU Te convergence perormance or Algoritm PARA: more oscillatory true solution Te convergence perormance or Algoritm SEQU: more oscillatory true solution Comparison o error o PARA, SEQU and corresponding monolitic metods Convergence rate or BEFE wit t = Convergence rate or BELF wit t = Convergence rate or BEFE wit t = / Convergence rate or BELF wit t = / Comparison o error o te ully coupled implicit metod, BEFE and BELF Te convergence perormance or SDsplit metod Te convergence perormance or BEsplit metod Te convergence perormance or BEsplit metod Te convergence perormance or CNsplit metod vii

8 LIST OF FIGURES. Flow diagram o a numerical simulation wit Uncoupled VMS algoritms Advection o an L-saped ront: Problem description Advection o an L-saped ront: te solutions produced by using Backward Euler and Crank-Nicolson scemes in Step Te decay o energy wit time or dierent scemes Energy spectra observed wit te Smagorinsky model in comparison to a postprocessed Smagorinsky step Energy spectra observed wit 3 3 degrees o reedom or te velocity and te usual VMS model in comparison to a postprocessed VMS step, and a zoom into te plot on te rigt Energy spectra observed wit 64 3 degrees o reedom or te velocity and te usual VMS model in comparison to a postprocessed VMS step Number o iterations in Uncoupled VMS were te projection term is solved iteratively and te decay o te L-norm o te approximated solution Te decay o system energy computed by PARA (let and SEQU (rigt wit several dierent time steps cosen Te decay o system energy computed by PARA (let and SEQU (rigt wit several dierent time steps cosen and larger parameters Te decay o kinetic energy or dierent numerical metods Variation o kinetic energy wit nu= and kmin=.0e Variation o kinetic energy wit nu=.0e- and kmin=.0e viii

9 5. E N using dierent time step sizes and splitting metods wit kmin = and S0 =.0e E N using dierent time step sizes and splitting metods wit kmin =.0e- and S0 = E N using dierent time step sizes and splitting metods wit kmin =.0e-3 and S0 =.0e E N using dierent time step sizes and splitting metods wit kmin =.0e-4 and S0 =.0e E N using dierent time step sizes and splitting metods wit kmin =.0e-4 and S0 =.0e E N using dierent time step sizes and splitting metods wit kmin =.0e- and S0 =.0e Stability o CNsplit at dierent small values o kmin and S ix

10 PREFACE I would like to express my deep appreciation and gratitude to my advisor Dr. Catalin Trencea. Trougout my years at University o Pittsburg, e provided constant encouragement, sound advice, great patience and excellent teacing due to tose tis project was possible. Wit me, e is not only a great advisor but also a great riend, wose caring extends beyond researc progress, and I ave torougly enjoyed te time we spent to discuss bot mat and non-mat topics. I wis to tank my co-advisor Dr. William Layton or all is support, caring and inspiration. Te conidence e as ad in me and is invaluable guidance enabled me to carry out tis tesis. He as done an excellent job in aiding me in understanding te deep concepts o luid dynamics and turbulence models. I must also tank im or teacing me or many additional neccessary tools to build an academic career. I am very grateul to Dr. Ivan Yotov or is stimulating and entusiastic lectures, wic gave me an insigtul understanding o various topics in numerical analysis. My researc signiicantly beneited rom is class. I also wis to tank Dr. Noel Walkington or te time e spent reading my tesis and or is valuable comments and suggestions. I would like to tank Dr. Volker Jon at WIAS and Dr. Clayton Webster at ORNL or generously inviting me to visit teir institutions, and involving me in te impressive works tere. A special tank to all o my passionate teacers, especially Nguyen Tan Dung, Tran Nam Dung, Le Ba Kan Trin. Teir classes nurtured my entusiasm o matematics and greatly inluenced my decision to pursue a career in mat. Many tanks to my great riends and colleagues, in order o appearance: Tran Vin x

11 Hung, Nguyen Kac Din, Nguyen Tien Kai, Lars Röee, Nick Hurl, Micaela Kubacki, Marina Moraiti, Xin Xiong, Nan Jiang, Martina Bukač or te wonderul time we studied and discussed matematics togeter. I particularly tank Lars Röee and Xin Xiong, rom collaboration wit wom I acieved te results appearing in Capter and 5 o tis tesis. I wis to tank my parents, Tran Van Loan and Dang Ti Viet or teir unconditional love and or supporting me in watever direction I coose. Te completion o tis work would not ave been possible witout te constant love, encouragement and understanding o my loving wie, Ha. Words are not enoug to express ow muc grateul I am to er or saring te past years wit me. To er I dedicate tis dissertation. xi

12 .0 INTRODUCTION Te low o liquids and gases occurs in many processes in nature and plays an important role in science and industry. Obtaining accurate, eicient and reliable prediction o quantities in luid lows is crucial to understand and predict te related real-world penomena. Many luid lows in engineering and tecnology are solved by complex codes or coupled to oter pysical eects. Te ability o ast reining tese models wen understanding is improved and using te legacy and best codes or subprocesses poses an important modeling problem. Tis tesis involves te development and testing o new numerical metods wic elp address te above diiculty in te modeling and simulation o some complex lows. In particular, we ave studied new variational multiscale algoritms or implementation o modern turbulence models in legacy codes, partitioned metods to uncouple lows wit electricity and magnetism in magnetoydrodynamics, partitioned metods or groundwater - surace water models. Te ollowing sections will describe eac o te topics in details.. EXPLICITLY UNCOUPLED VARIATIONAL MULTISCALE STABILIZATION OF FLUID FLOWS Numerical simulation o turbulent lows generally requires extremely ine mes size due to small caracteristic size o tese structures. Tis results in solving large systems o equations

13 and large amounts o computing time and power. Clearly, a direct numerical simulation o many interesting and important lows is not practically possible at te present time. One alternative promising approac is large eddy simulation (LES. LES is motivated by te pysical idea tat te dynamics o te large and small structures o turbulent lows are quite dierent. Te large structures (large eddies evolve deterministically; in te meantime, te small eddies are sensitive on perturbations o te problem data. Teir random caracter does, owever, ave universal eatures so tat tere is ope tat teir mean eects on te large eddies can be modelled. Te goal o LES tus consists o accurately resolving te large scale eatures and modeling te eect o missing, unresolved scales on resolved scales. Te dominant LES model over te years is Smagorinsky eddy viscosity model. Tis approac owever is unable to successully separate between large and small scales. Introduced by Huges in [6] and used irst in turbulence modeling in [6], Variational Multiscale (VMS metod as proven to be a remarkably successul interpretation o te LES concept witin a variational ormulation o te Navier-Stokes equations. Tis metod uses variational projection to dierentiate scales and conines modeling to te small scale equation rater tan all scales, overcoming many sortcomings o classical LES. Given te success o te VMS approac, tere is a natural desire to introduce a VMS treatment o turbulence witin legacy codes, in complex multi-pysics applications and in oter settings were reprogramming a new metod rom scratc is daunting. We propose, analyze and test erein (and also in [78] a metod to induce a VMS treatment o turbulence in an existing NSE discretization troug an additional, modular and uncoupled projection step. An uncoupled metod generally cannot be as accurate as a ully coupled one. We ind in our analysis and irst tests owever tat te uncoupled VMS metod studied is consistent to a ig level wit te ully coupled implementation o te same VMS realization and not appreciably less accurate. To introduce ideas, suppressing spatial discretization and te pressure, write te NSE as u t + NSE(u =. Te algoritm we developed adds a modular, uncoupled, projection-like Step to a standard metod, Step :

14 Step : Call or NSE solver to compute w. Given u n u(t n, or te Crank-Nicolson metod: w u n t ( w + u n + NSE = + n. Step : Postprocess w to obtain u : u = Πw. Eliminating Step gives u u n t ( w + u n + NSE + t (w Πw = + n. (. We deined te operator Π in Step so tat te extra, bold term in (. is exactly a VMS eddy viscosity term acting on marginally resolved scales: u = Πw satisies, or all v in te discrete velocity space, (Extra Term, v = t (w u, v = (ν T (x, [I P ] w + u, [I P ] v, were ν T is te eddy viscosity coeicient and is assumed to be a known, positive, constant elementwise unction. Also P is an L projection tat deines te VMS velocity gradient averages (so [I P ] deines te luctuations. Te exact ormulation o Step does not depend on te time discretization used in Step. Tereore, te projection based stabilization in Step is an uncoupled, independent second step and amenable to implementation in legacy codes. Te explicitly uncoupled variant o te VMS metod presented in tis tesis is based on ideas rom ilter based postprocessing in [44] and [77]. Hig Re lows ave many diiculties including convection dominance and vortex stretcing. Te ormer can be studied in linear problems and is muc better understood: many stabilized metods are available or convection dominance. In te context erein, tese would be included in Step. Turbulence models, suc as VMS, are directed at nonlinear beavior suc as error stretcing, backscatter (see [3], equipartition o energy due to truncation o te energy cascade and so on. Projection based VMS metods give some stabilization or convection dominance, see [7]; tey are known to reduce but do not eliminate oscillations near layers, [7, 73]. Properly tuned to mimic energy loss due to breakdown o eddies rom resolved to unresolved 3

15 scales, teir added dissipation eliminates equipartition. It also elps control error growt rom smaller to larger scales due to vortex stretcing. Backscatter is a more complex issue. Witin te VMS ramework, E u in (.5 below is te approximation to u and u E u is te model or u. Tus, te VMS metod we study allows or backscatter rom u to u but not rom completely unresolved scale to u. Tere is a wide range o metods adding numerical dissipation on all scales o a low, like e.g. te residual based stabilization tecniques [7]. One can ind an overview in [06]. In LES modeling te aim is to simulate te large scales o a low accurately, e.g., [4, 67]. In [6, 5] and [6], te VMS approac to LES was developed. In VMS modeling te velocity is decomposed into means, deined to be tat wic is resolvable on te given mes, and te luctuations being everyting else. Only te equation or luctuations is approximated and te resulting luctuation model inserted exactly into te mean equation. Several luctuation models ave been used suc as deining luctuations using eiter a pair o FEM spaces o increasing local polynomial degree or wit te same polynomial degree on a iner mes (te cases considered erein; see [75], and [49, 68] or oter work wit tese coices. Te most common coice (tat is not considered erein is to deine luctuations using bubble unctions wic connects tis development o te VMS idea wit SUPG and oter stabilized metods; see Bensow and Larson [3] and Hsu, Bazilev, Calo, Tezduyar and Huges [58] or recent and interesting development o tis VMS realization.. PARTITIONED TIME STEPPING METHODS FOR MAGNETOHYDRODYNAMIC FLOWS Te MHD equations describe te motion o electrically conducting, incompressible lows in te presence o a magnetic ield. I an electrically conducting luid moves in a magnetic ield, te magnetic ield exerts orces wic may substantially modiy te low. Conversely, te low itsel gives rise to a second, induced ield and tus modiies te magnetic ield. Initiated by Alven in 94 [], MHD models occur in astropysics, geopysics as well as engineering. Understanding tese lows is central to many important applications, e.g., liquid 4

16 metal cooling o nuclear reactors [9, 55, ], sea water propulsion [87], process metallurgy [34]. Te magnetic Reynolds number R m is an important parameter in MHD, being indicative o te relative strengt o induced magnetic ield and imposed magnetic ield: R m = Induced ield Applied ield = µσul. Here µ is te permeability o ree space, σ is te electrical conductivity, u and L are te caracteristic velocity and lengt scale correspondingly. Large values o velocity and lengt scale are unreacable in most industrial and laboratory lows. Consequently, MHD lows in terrestrial applications typically occur at small magnetic Reynolds number. Wile te magnetic ield considerably alters te luid motion, te induced ield is usually ound to be negligible by comparison wit te imposed ield. Neglecting te induced magnetic ield reduces MHD models to te reduced MHD system: Given body orce and external imposed magnetic ield B, ind te luid velocity u, pressure p and electric potential φ suc tat: N ( u t + u u u + p = + (B φ + B (B u, M φ = (u B, and u = 0. Here M, N are te Hartman number and interaction parameter given by σ M = BL ρν, N = σb L ρu were B is te caracteristic magnetic ield, ρ is te density, and ν is te kinematic viscosity, all assumed constant. Justiication o using simpliied MHD equations to model MHD lows in terrestrial applications can be ound in [35, 74, 04]. Tere are several dierent, almost equivalent ormulations or RMHD. Te one we study in tis tesis was also considered in [03, 53, 7]. MHD lows involve dierent pysical processes: te motion o luid is governed by ydrodynamics equations and te electric potential is governed by electrodynamics equations. Despite te importance o MHD in science and engineering and te large computational experience, numerical metods or MHD ave not been well developed. Te results on existence, 5

17 uniqueness and inite element approximation o te steady-state MHD problems were developed troug work in [5] (or two dimensional case, [03] (or small magnetic Reynolds number case and [53] (or ull MHD lows wit perectly conducting wall condition. In [9, 93, 84, 94], Meir et. al. studied variational metods and numerical approximation or solving stationary MHD equations under more pysically realistic boundary conditions tat account or te electromagnetic interaction o te luid wit te outside world. For urter discussions on matematical and numerical analysis o steady-state MHD lows, we reer to [5, 37]. Tere are muc less works on time-dependent MHD. Scmidt [08] developed a ormulation or evolutionary MHD and establised te existence o global-in-time weak solutions via te Galerkin metod. To te best o our knowledge, te irst paper dealing wit time discretization scemes o MHD problems was o Yuksel and Ingram [7], in wic te autors studied te stability and error analysis o te ully coupled, monolitic Crank-Nicolson metod or reduced MHD equations. In [7], Trencea proposed a metod tat decouples o te evolutionary ull MHD system in te Elsässer variables. Wilson, Labovsky and Trencea developed tat result by introducing a ig-order accurate deerred correction metod, wic also decouples te MHD system, [4]. In tis tesis (and also in [79], [80], we propose two new implicit-explicit partitioned metods (based on uncoupling pysical variables or solving te evolutionary MHD equations at small magnetic Reynolds number. Te metods we study include a irst order, one step sceme consisting o implicit discretization o te subproblem terms and explicit discretization o coupling terms. Algoritm. Given u n, p n, φ n, ind u, p, φ satisying ( u u n + u u N t M u + p = + ( B φ n + B (B u, φ = (u n B and u = 0. Te second sceme we consider employs second order, tree level backward dierentiation ormulas (BDF discretization or te subproblem terms. Te coupling terms are treated 6

18 by two step extrapolation in Navier-Stokes equation and by implicit metod in Maxwell equation. Since one needs te updated value o u at current time level to compute φ, tis metod is uncoupled but sequential: φ n u φ. Algoritm. Given u n, u n, p n, p n, φ n, φ n, ind u, p, φ satisying ( 3u 4u n + u n + u u N t M u + p = + ( B (φ n φ n + B (B u, φ = (u B and u = 0. We prove tat tese metods are stable over 0 t < and convergent at irst and second order respectively. In particular, we sow tat our irst order partitioned metod is long time stable witout needing a restriction on time step size t, altoug we treat te coupling terms explicitly. Te perormance o our metods is compared wit tat o monolitic metod by numerical tests..3 PARTITIONED TIME STEPPING METHODS FOR THE EVOLUTIONARY STOKES-DARCY PROBLEMS Groundwater, orming two-tirds o te world s res water, is vital to uman activities. One serious global problem nowadays is groundwater contamination, wic occurs wen man-made pollutants are dissolved in lakes and rivers and get into te groundwater, making it unsae and unit or uman use. To predict and control te spread o suc contamination requires te accurate solution o coupling o groundwater lows wit surace water lows (te Stokes-Darcy problem. Te essential problems o estimation o te propagation o pollutants into groundwater are tat (i te dierent pysical processes suggest tat codes optimized or eac sub-process need to be used or solution o te coupled problem, (ii te large domains plus te need to compute or several turn-over times or reliable statistics require calculations over long time intervals and (iii values o some system parameters, e.g., ydraulic conductivity and speciic storage, are requently very small. To address tese 7

19 issues, we study te stability and errors over long time intervals o uncoupled metods or te ully time dependent Stokes-Darcy problem. We are particularly interested in analyzing and comparing te perormance o te studied metods or small parameters. In tis work (also in [8, 8], we propose several implicit-explicit based and splitting based partitioned metods or uncoupling te evolutionary Stokes-Darcy problem. Suppressing tecnical complexities, write te Stokes-Darcy equation as u t + A u + Cφ =, φ t + A φ C T u = g. were A, A are symmetric positive deinite (sel-adjoint, maximal monotone and coercive operators. Te two equation are linked by exactly skew symmetric coupling. Te metods we study erein include te irst order Backward Euler - Forward Euler (BEFE, wic was also studied in [98]: Given u n, φ n, ind u, φ satisying u u n + A u + Cφ n =, t φ φ n + A φ C T u n = g, t and Backward Euler - Leap Frog (BELF, a new two step partitioned sceme motivated by te orm o te coupling: Given u n, u n, φ n, φ n, ind u, φ suc tat u u n + A u + Cφ n =, t φ φ n + A φ C T u n = g. t We prove te long time (over 0 t < stability o bot metods and derive an optimal error estimate tat is uniorm in time over 0 t <. For uncoupling a coupled problem, general experience wit partitioned metods suggests tat some price is inevitably paid. Our proposed metods wit explicit coupling terms inerit restrictions on time step size t t C 0 min {k, S 0 } k (. were S 0 is speciic storage, k is ydraulic conductivity and C 0 is a generic positive constant independent o mes size, time step and inal time. Te values o S 0 and k are requently 8

20 very small, see [], [4], and in tose cases, te dependence indicated in (. becomes too pessimistic. To overcome tis problem, we propose and analyze our novel uncoupling metods or Stokes-Darcy equations, wic ave stronger stability properties, using ideas rom splitting metods. Tese metods include ones stable uniormly in S 0 or moderate k and uniormly in k or moderate S 0. Tey are tus good options wen one o te parameters is small. Te literature on numerical analysis o metods or te Stokes-Darcy coupled problem as grown extensively since [38], [85]. See [4] or a recent survey and [], [4], [0], [0], [07], [3] and [85] or teory o te continuum model. Tere is less work on te ully evolutionary Stokes-Darcy problem. One approac is monolitic discretization by an implicit metod ollowed by iterative solution o te non-symmetric system were subregion uncoupling is attained by using a domain decomposition preconditioner; see, e.g., [4], [3], [], [97], [0], [36], [40], [39], [60], [66], [95], [0]. Partitioned metods allow parallel, non-iterative uncoupling into one (SPD Stokes and one (SPD Darcy system per time step. Te irst suc partitioned metod was studied in 00 by Mu and Zu [98]. Tis as been ollowed by an asyncronous (allow dierent time steps in te two subregions partitioned metod in [0] and iger order partitioned metods in [], [83]. In most o tese works, stability and convergence were studied over bounded time intervals 0 t T < and te estimates included e αt multipliers. Alternate approaces or coupling surace water lows wit groundwater lows include Brinkman one-domain models, Angot [], Ingram [64], wic are a more accurate description o te pysical processes. One-domain Brinkman models are also more computationally expensive. Monolitic quasi-static models (one domain evolutionary and te oter assumed to instantly adjust back to equilibrium ave also been studied, e.g., [6]. Wile tey are not considered erein in detail, te metods considered also give non-iterative, domain decomposition scemes or quasi-static models (e.g., set S 0 0 in (4.7, (4.8 below. Partitioned metods employ implicit discretizations o te sub-pysics/ subdomain problems and explicit time discretizations o te coupling terms, e.g., [8], [98], [8], [9], [5], [3], [3], [30]. Tus tere is a very strong connection between application-speciic partitioned metods and more general IMEX (IMplicit - EXplicit metods; te latter developed 9

21 in, e.g., [], [9], [5], [33], [45], [63], [45], [3], [3] and []. On te oter and, applicationspeciic partitioned metods are oten motivated by available codes or subproblems, [8]. Examples o partitioned metods include ones designed or luid-structure interaction [8], [9], [5], Maxwell s equations, [] and atmospere-ocean coupling, [3], [3]. Te idea used in our CNsplit sceme to compute in parallel two approximations and ten average occurs in te Dyakunov splitting metod, e.g., [9], [90], [6], [59]. Long time stable numerical scemes ave also been introduced or related problems, especially or D Navier-Stokes equations. For suc works, we reer to [6], [48], [5] and [7]..4 ANALYTICAL TOOLS In tis section, we state some well-known results and assumptions wic will be utilized in te analysis trougout tis tesis. Let Ω be an open, regular domain in R d (d = or 3. We denote te L (Ω norm and inner product by and (,. Likewise, te L p (Ω norms and te Sobolev Wp k (Ω norms are denoted by L p and W k p, respectively. For te semi-norm in Wp k (Ω we use W k p. H k is used to represent te Sobolev space W k (Ω, and k denotes te norm in H k. Te space H k denotes te dual space o H0 k. Teorem.4.. (te trace teorem Let Ω be a grap o a Lipscitz continuous unction. I u L (Ω and u L (Ω, ten u Ω L ( Ω and u L ( Ω C u ( / u + u /4. Teorem.4.. (te Poincaré inequality Tere is a constant C = C(Ω suc tat u C u or every u H0(Ω. Teorem.4.3. For any u, v, w H0(Ω, tere is C = C(Ω suc tat u v wdx C u u v w. (.3 Ω 0

22 For te proo, see [76]. Lemma.4.4. (discrete Grönwall inequality Let D 0 and κ n, A n, B n, C n integer n 0 and satisy 0 or any N N N A N + t B n t κ n A n + t C n + D or N 0. n=0 n=0 n=0 Suppose tat or all n, tκ n <, and set g n = ( tκ n. Ten, ( ] N N N A N + t B n exp t g n κ n [ t C n + D n=0 n=0 n=0 or N 0. For te details, see, e.g., [57]..5 THESIS OUTLINE Tis tesis begins in Capter wit a study o an algoritm or implementation o VMS approac in a legacy turbulence code. A complete stability and convergence analysis o tis metod is given in Section.3. In Section.4 we turn to te problem o actually computing te operator in Step eiciently. A related VMS metod (adding ideas rom [3] wic is sligtly less accurate but more eicient is given in Section.5. Numerical experiments are given in Section.6. In Capter 3, we discuss partitioned metods to uncouple conducting luid lows wit electricity and magnetism in magnetoydrodynamics. Our scemes are introduced in Section 3.. We sow in Section 3.3 tat tese ormulations ave a stable solution or long time periods. Te main convergence results are presented in Teorem Te numerical experiments in Section 3.5 support tese teoretical results. Capter 4 will be devoted or uncoupling algoritms or solving groundwater - surace water system. Section 4.3 presents two implicit-explicit based partitioned metods or Stokes- Darcy problem: BEFE and BELF. In Teorem 4.4. and 4.4.5, we prove tat tese metods are long time and uniormly in time stable. Section 4.5 (particularly, Teorem 4.5. gives

23 a compreensive error analysis and Section 4.6 ollows wit numerical tests wic conirm te teory. Te study on partitioning te groundwater - surace water low is extended in Capter 5. We introduce our uncoupling scemes wic are more stable in motivating applications involving small pysical parameters. Tese algoritms are presented in Section 5.. We analyze long time stability and derive te associated timestep restrictions in Section 5.3. In Section 5.4, we give computational experiments to veriy te accuracy and stability o our metods.

24 .0 EXPLICITLY UNCOUPLED VARIATIONAL MULTISCALE STABILIZATION OF FLUID FLOWS. METHOD DESCRIPTIONS In tis capter, we develop a modular, postprocessing metod to implement a variational multiscale metod in complex (possibly legacy and possibly laminar low codes. Suppressing te pressure and spatial discretization, suppose te Navier-Stokes equations are written as u t + N(u + νau = (t. Add one uncoupled, modular, projection-like step (Step to te standard Crank-Nicolson Finite Element Metod (Step : given u n u(t n, compute u by ( Step : Compute w w u n w + u n via: + N + νa w + u n t Step : Postprocess w to obtain u : u = Πw, = /, were / = ( n + /. We will develop teoretical results or te unstabilized Crank- Nicolson FEM in Step, but te setting o Step is independent o te time discretization, see Remark.3.4. Te deviations rom previous work considered erein are tat ( te projection based stabilization is an uncoupled, independent second step and tus amenable to implementation in legacy codes, and ( te projection in Step is not a ilter but constructed to recover te VMS eddy viscosity term. Eliminating Step gives u u n t ( w + u n + N + νa w + u n + t (w Πw = /, (. 3

25 wic is a time relaxation discretization o te original problem wit time relaxation coeicient / t. We deine te operator in Step so tat (see Section.4 or details te extra, bold term in (. is exactly a VMS eddy viscosity term acting on marginally resolved scales: u = Πw satisies, or all v in te discrete velocity space, (Extra Term, v = t (w u, v = (ν T (x, [I P ] w + u, [I P ] v. Te subscript = (x denotes te local meswidt o a FEM mes and P, deined precisely in (.5, is an L projection tat deines te VMS velocity gradient averages (so [I P ] deines te luctuations. Full details are given in Section.3. Also ν T (x, is te cosen eddy viscosity coeicient. We sall assume (motivated by te nonlinear case in wic its value is oten extrapolated rom previous time levels in tis report tat: Condition... ν T elementwise. = ν T (x, is a known, positive, bounded unction wic is constant. NOTATION AND PRELIMINARIES We deine te norms ( m < v,k := EssSup [0,T ] v(t, k, and v m,k := ( T /m v(t, m k dt. 0 or unctions v(x, t deined on te entire time interval (0, T. Te Navier-Stokes equation wit boundary and initial condition are: Given time T > 0, body orce, ind velocity u : [0, T ] Ω R d, pressure p : [0, T ] Ω R satisying u t + u u ν u + p = (x and u = 0 in Ω, or 0 < t T u(x, 0 = u 0 (x in Ω, u = 0 on Ω, or 0 < t T. (. Te velocity and pressure spaces are X := (H 0(Ω d, Q := L 0(Ω, wit v X := v. 4

26 Te space o divergence ree unctions is given by V := {v X : ( v, q = 0 q Q}. A weak ormulation o (. is: Find u : [0, T ] X, p : [0, T ] Q or a.e. t (0, T ] satisying (u t, v + (u u, v (p, v + ν( u, v = (, v v X (.3 u(x, 0 = u 0 in X and ( u, q = 0 q Q. (.4 We consider our analysis on te inite element metod (FEM or te spatial discretization (te results extend to many oter variational metods. Te inite element velocity and pressure spaces considered are built on a conorming, edge to edge triangulation wit maximum triangle parameter denoted by a subscript. We assume tat tey satisy te usual discrete in-sup condition or div-stability. For a given selection o velocity and pressure elements, tis can require enricment by bubbles or impose implicitly a condition on te mes. In our tests we ave used te common Taylor-Hood pair o conorming quadratics or velocity and conorming linears or pressure. Tere are many cases were low order elements and very ine meses are needed. Tese oten require additional stabilizations or te incompressibility-pressure coupling. Extension o te metods erein to include suc stabilizations is naturally an important problem tat must be analyzed and tested on a case by case approac. We sall denote conorming velocity, pressure inite element spaces by X X, Q Q. We also must select a space o well resolved velocities and pressures, denoted by X H X, Q H Q. Tree commonly seen examples o te deinition o te well resolved spaces are: Te ine space (containing means and luctuations X arises rom augmentation o a given FEM space X H wit element bubble unctions; see Hsu, Bazilev, Calo, Tezduyar and Huges [58]. 5

27 A coarse mes velocity and pressure space X H, Q H (wit meswidt denoted by subscript H is constructed. I te meses are nested and te space uses te same elements as te ine mes space ten X H X X, Q H Q Q; see [50, 89] or examples. Te space o well reined velocities and pressures are deined on te same mes but using inite element spaces o lower polynomial degree. In tis case also X H X X, Q H Q Q; see [68, 05] or examples. Te irst approac is most commonly seen and not considered erein. Te second requires a code wit only one element but pointers between te two meses (as are commonly ound wit adaptive codes wile te tird works only on one mes but requires at least two velocity elements (suc as in p adaptive codes. We sall assume tat X H/, Q H/ satisy te usual in-sup condition necessary or te stability o te pressure, e.g. [54]. Te discretely divergence ree subspace o X H/ is V H/ = {v H/ X H/ : ( v H/, q H/ = 0 q H/ Q H/ }. Note tat V H V in general. Taylor-Hood elements (see [6, 54] are one common example o suc a coice or (X, Q, and are also te elements we use in our numerical experiments. Furter, we denote te space o (typically discontinuous coarse mes velocity gradient tensors by L H := X H = { v H : or all v H X H }, and analogously or L. Te weigted L and elliptic projections are deined as usual (in general and in tis speciic case ollowing [4], Section.6 by P H u = G H L H satisies (ν T (x, [G H u], l H = 0, l H L H, E H u = ũ X H satisies (ν T (x, [ ũ u], v H = 0, v H X H. (.5 Te motivation or te deinition in (.5 is tat means (and tus luctuations deined by elliptic projection are equivalent to means o deormations deined by L projection (see [4], Lemma.0 u := E H u, P H u = E H u. 6

28 Furter, wile computation o velocity means is global, wen te means o deormation are deined by L projection into a C 0 inite element space, P H u can be computed in parallel element by element. Deine te usual, explicitly skew symmetrized trilinear orm b(u, v, w := (u v, w (u w, v. Let v(t / = v((t + t n / or a continuous unction in time and v / = (v + v n / or unctions o time tat are bot continuous and discrete..3 THE POSTPROCESSED VMS METHOD In tis section we will give a precise ormulation o te metod and prove stability and an a priori error estimate. Deinition.3.. Given w, u = Πw V is te (unique solution o ( w u, v = (ν T [I P H ] w + u, [I P H ] v t ( u, q = 0 or all q Q + ( λ, v or all v X (.6 Step requires taking a velocity on a mes and rom tat solving Stokes like problems eiter on a dierent mes ollowed by interpolation back to te mes used or Step or on te same mes. Our analysis assumes Step will be perormed on te mes used or Step. Extending te metod and numerical analysis to te case o using Step treating Step as a black box solver requires tis extra error (and oter subtleties as well to be understood. Te orm o te uncoupled projection step o Step does not depend on te time discretization sceme used in Step. 7

29 Algoritm.3.. Given u n Step : Compute w ( w u n, v + b t ( w compute u by V satisying: or all v V + u n, w + u n, v + ν ( w + u n, v = ( /, v Step : Apply projection Π on w to obtain u u = Πw. Eliminating Step gives ( u u n t ( w, v + b ( w + + u n, w + u n, v + ν Πw, v = ( /, v. t ( w + u n, v Te last term on te let and side is te additional term rom Step. By deinition (.6, tis term recovers te VMS eddy viscosity term and te projected velocity is discretely divergence ree. Te ollowing lemma quantiies te eddy viscosity induced by Step. Lemma.3.3. [Numerical Dissipation induced by Step ] Let ν T... Ten, tere olds ulill Condition w = u + t νt [I P H ] w + u. Proo. Set v = w and q = λ + u in (.6 and obtain t ( w u = νt [I P H ] w + u, were we used tat w V. Tis already proves te claim ater rearranging. 8

30 Remark.3.4. Te special coice in (.6 used in Step wit te argument o te orm w + u (and not w + u n does not depend on te time discretization sceme in Step. Wit a dierent discretization used in Step we would get te same induced eddy dissipation terms in Step witin te proo o Lemma.3.3. Tis is wy te explicitly uncoupled Step o Algoritm.3. does not depend on te time discretization sceme in Step and wy Step can be used wit an arbitrary CFD code. Lemma.3.3 is one key to prove stability o Algoritm.3.. Teorem.3.5. Let ν T satisy Condition.., ten [ u N N ν + t w + u n + νt [I P H ] w + u n=0 + u n u 0 + t ν N n=0 / Proo. Set v = w in Step and obtain ( w u n t + ν w + u n Application o Lemma.3.3 to tis equation gives [ ( u u n t + ν w + u n. ( = /, w + u n ]. + νt [I P H ] w + u ( = / w + u n,. Summing tis up rom n = 0 to n = N results in [ N u N + t ν w + u n + νt [I P H ] w + u ] n=0 = N ( u 0 + t / w + u n,, (.7 were we can apply Young s inequality to te rigt and side inside te sum to see ( t / w + u n, t / + ν t ν w + u n. Hiding te last term on te let and side o (.7 proves te claim. n=0 ] 9

31 Teorem.3.5 also gives a stability estimate or w N. In particular, Corollary.3.6 sows tat w N is also not te usual CN approximation. Corollary.3.6. Let ν T ulill Condition.., ten N w N + t n=0 ν w + u n + t N n=0 νt [I P H ] w + u u 0 + t ν N n=0 /. Proo. Apply Lemma.3.3 or w N to Teorem.3.5. As a next step we will give an a priori error estimate or te approximation sceme, Algoritm.3.. Let t n = n t, n = 0,,,..., N T, and T := N T t. Also introduce te ollowing discrete norms: v,k := max v n k, v /,k := max v n / k, 0 n N T n N T v m,k := ( NT /m v n m k t, v / m,k := ( NT n=0 n= /m v n / m k t. In order to establis te optimal asymptotic error estimates or te approximation we need to assume te ollowing regularity o te true solution: u L (0, T ; H k+ (Ω H (0, T ; H k+ (Ω H 3 (0, T ; L (Ω W 4 (0, T ; H (Ω, p L (0, T ; H s+ (Ω, and H (0, T ; L (Ω. (.8 For te error between u n u n we ave te ollowing teorem and corollary. 0

32 Teorem.3.7. For u, p, and satisying (.8, (.3 and (.4, and u n, wn Algoritm.3. we ave tat, or t suiciently small, given by un u N + t 4 N n=0 ( ν (u(t / (w + u n / + ν T [I P H ] (u (w + u / C k+ u,k+ + Cν k u,k+ + Cν T k u,k+ + Cν T H k u,k+ + C k ν u,k+ + C k+ ν ( u 4 4,k+ + u 4,0 4 s+ + C p /,s+ ν ( + C k+ u t,k+ + C( t 4 ν u 4 4,0 + ν u / 4 4,0 + u ttt,0 + ν u tt,0 + ν u tt 4 4,0 + tt,0. For k =, s = Taylor-Hood elements, i.e. C 0 piecewise quadratic velocity space X and C 0 piecewise linear pressure space Q, we ave te ollowing asymptotic estimate. Corollary.3.8. Under te assumptions o Teorem.3.7, wit t = C, ν T =, H = and (X, Q given by te Taylor-Hood approximation elements, we ave u N u N + t N n=0 ( ν (u(t / (w + u n / + ν T [I P H ] (u (w + u / C ( ( t Te rest o tis section is devoted or proving Teorem.3.7. Tis proo is tecnical and exibits te usual limitations in te inal result tat arise rom employing te discrete Grönwall inequality o exponential error growt and te assumption tat t is suiciently small. Proo. (o Teorem.3.7 Denote w / := w. To begin te analysis we rewrite Algoritm.3.. As te spaces X and Q satisy te usual in-sup condition, Algoritm +u n

33 .3. is equivalent to: For n = 0,,..., N T ind w, u V suc tat (w, v + t b( w /, w /, v + t ν( w /, v = (u n, v + t ( /, v, v V, (.9 t (w u, v = (ν T [I P H ] w + u, [I P H ] v, v V. (.0 To establis te optimal asymptotic error estimates or te approximation we assume true solution satisies te regularity condition (.8 rom Section.3: At time t / = (n + / t te true solution u o (.3, (.4 satisies (u u n, v + t ν( u /, v + t b(u /, u /, v t (p(t /, v = t ( /, v + t Intp(u ; v, (. or all v V, were Intp(u ; v, representing te consistency error, denotes Intp(u ; v = ( (u u n / t u t (t /, v + ν( u / u(t /, v + b(u /, u /, v b(u(t /, u(t /, v + ((t / /, v. (. We split te error into a Step error ε according to (.9, a Step error e according to (.0, and an approximation error Λ u w = (u I u + (I u w =: Λ + ε, u u = (u I u + (I u u =: Λ + e, (.3 were I u V will be an interpolation o u in V later in te proo but is an arbitrary element in V at tis point. Now we subtract (.9 rom (. and use (ε test unction v to obtain ( ε e n + t ν (ε + e n = (Λ Λ n, (ε t b(u /, u /, (ε + e n t ν( Λ /, (ε + e n + t b( w / + e n, w /, (ε + e n + e n V as + t(p(t / q, (ε + e n + t Intp(u ; (ε + e n. (.4

34 Te key to tis equation is tat (ε +e n is discretely divergence ree and ence a possible test unction v. Next we estimate te terms on te RHS o (.4 and get (Λ Λ n, (ε = ( t t t t Ω Ω t ( t t n t t n + e n Λ n t Λ t Λ t dt dω + t (ε + e n Λ t dt dω + t (ε + e n t n Λ t dt + 4 t ( ε + e n, + t (ε + e n (.5 ν( Λ /, (ε + e n ν 0 (ε + e n + Cν Λ /. (.6 We rewrite b(u /, u /, (ε b(u /, u /, (ε = b(u /, u /, (ε + b( w /, u /, (ε + e n + e n b( w / b( w/, w /, (ε, w /, (ε + e n + e n b( w /, u /, (ε + e n + e n b( w / = b( ((u w + (u n u n, u /, (ε + e n, w /, (ε + e n + e n as + b( w /, ((u w + (u n u n, (ε + e n (.7 = b(λ / + (ε + e n, u /, (ε + e n + b( w /, Λ / + (ε + e n, (ε + e n = b(λ /, u /, (ε + e n + b( (ε + b( w /, Λ /, (ε + e n, + e n, u /, (ε + e n were we used te skew symmetry o b. Using (.3 and Young s inequality, we bound te terms on te RHS o (.7 as ollows. b(λ /, u /, (ε + e n C Λ / Λ / u / (ε + e n ν 0 (ε + e n + C ν Λ / Λ / u / (.8 3

35 b( (ε + e n, u /, (ε + e n C (ε + e n / (ε + e n 3/ u / ν 0 (ε + e n + C ν 3 u / 4 (ε + e n ν 0 (ε + e n + C ν 3 u / 4 ( ε + e n (.9 b( w /, Λ /, (ε + e n C w / Λ / (ε + e n ν 0 (ε + e n + C ν w / Λ / (.0 (p(t / q, (ε + e n p(t / q (ε + e n ν 0 (ε + e n + C ν p(t / q. (. ollows. Te consistency error term t Intp(u ; (ε + e n in (.4 can be bounded as Lemma.3.9. Under te regularity assumption (.8 rom Section.3 tere olds t Intp(u ; (ε + e n t ( ε + e n + ν t 4 (ε + e n + C( t5 ( u / 4 + u(t / 4 ν t + C( t 4 t n ( u ttt + ν u tt + ν u tt 4 + tt dt. Proo. We want to estimate every term in te deinition o Intp(u ; v rom (. and obtain ((u u n / t u t (t /, (ε + e n (ε + e n + (u u n / t u t (t / 4 ε + 4 en + ( t 3 80 t t n u ttt dt, 4

36 ((t / /, (ε 4 ε + 4 en + ( t e n (ε + e n + (t/ / t t n tt dt, ν( u / u(t /, (ε + e n ν 8 (ε + e n + ν u / u(t / ν 8 (ε + e n + ν ( t3 48 t t n u tt dt, were we used classical Caucy-Scwarz inequality and Taylor expansion. Also wit tese inequalities we get an estimate o te terms o te nonlinearity b(u /, u /, (ε = b(u / u(t /, u /, (ε + e n b(u(t /, u(t /, (ε + e n + e n + b(u(t /, u / u(t /, (ε +e n C (u / u(t / (ε + e n ( u / + u(t / ν 8 (ε + e n + Cν (u / u(t / ( u / + u(t / ν 8 (ε ν 8 (ε ν 8 (ε + e n ν 8 (ε + e n ( t3 ( + Cν u / + u(t / t 48 + e n ( t3 + Cν 48 ( t3 + Cν 48 ( t + e n + C ( t3 48ν t t n u tt 4 dt + t n ( t n u tt dt u tt ( u / + u(t / dt t t n ( u / 4 + u(t / 4 dt t ( u / 4 + u(t / 4 + t t n u tt 4 dt. Combining all estimates yields te lemma. 5

37 Te application o Lemma.3.9 to (.4 togeter wit te estimates (.5 (. gives ( ε e n + t ν 4 (ε + e n C t ( + ν 3 u / 4 ( ε + e n + Cν t Λ / + C t ν + C w / Λ / + C t u / Λ / Λ / ν ν p(t/ q + C( t5 ( u / 4 + u(t / 4 ν t ( u ttt + ν u tt + ν u tt 4 + tt dt. + C t t t n Λ t dt + C( t 4 t n (. As u and w are connected troug te variational multiscale projection in Step, we next use tat equation to obtain a relationsip between ε n and en. Lemma.3.0. Tere olds ε = e + t ν T [I P H ] (ε + e + t(ν T [I P H ] (Λ u, [I P H ] (ε + e. Proo. From (.0 we ave ( w u, v = t (ν T [I P H ] w + u, [I P H ] v and set v = (w I u + (u I u = (ε + e. We obtain ( (ε e Hence t, (ε + e = (ν T [I P H ] (ε + e + I u t ( ε e = ν T [I P H ] (ε + e, [I P H ] ( (ε + e. (ν T [I P H ] I u, [I P H ] (ε + e and wit I u = u Λ rom (.3 we conclude te proo. 6

38 Substituting Lemma.3.0 into (., we obtain ( e e n + t ( ν 4 (ε + e n + ν T [I P H ] (ε + e C t( + ν 3 u / 4 ( e + e n + Cν t Λ / ( + C( t ( + ν 3 u / 4 ν T [I P H ] (ε + e + (ν T [I P H ] (Λ u, [I P H ] (ε + e + t (ν T [I P H ] (u Λ, [I P H ] (ε + e (.3 + C t ν + C t + C w / Λ / + C t ν u / Λ / Λ / ν p(t/ q + C( t5 ( u / 4 + u(t / 4 ν t t n Λ t dt + C( t 4 Since we can estimate t t n ( u ttt + ν u tt + ν u tt 4 + tt dt. (ν T [I P H ] (Λ u, [I P H ] (ε + e 8 ν T [I P H ] (ε + e + C ν T [I P H ] (Λ u, it is possible to coose t suiciently small, i.e., C t < 6 ( + ν 3 u / 4 suc tat te terms stemming rom te VMS metod are idden and ater summing tis up rom n = 0 to n = N equation (.3 results in en + t 4 { N n=0 N n=0 ( ν (ε + e n + ν T [I P H ] (ε + e C t( + ν 3 u / 4 ( e + e n + Cν t Λ / + C t ( ν T [I P H ] Λ + ν T [I P H ] u + C t ν + C t + C w / Λ / + C t ν u / Λ / Λ / ν p(t/ q + C( t5 ( u / 4 + u(t / 4 ν t t n Λ t dt + C( t 4 t t n ( u ttt + ν u tt + ν } u tt 4 + tt dt. 7

39 Now we coose te interpolation operator in V, constructed in [47, 4], and a usual interpolation operator or te pressure, wic leads us to u I u r C k+ r u k+, were r k and k is te polynomial degree o te corresponding FE space. Since P H also ulills te interpolation property, due to te regularity assumptions and Teorem.3.5 tis gives en + t 4 N n=0 N C t ( + ν 3 u 4,0 e n n=0 ( ν (ε + e n + ν T [I P H ] (ε + e + Cν k u,k+ + Cν T k u,k+ + Cν T H k u,k+ + C k ν u,k+ + C k+ ν ( u 4 4,k+ + u 4,0 4 s+ + C p /,s+ ν ( + C k+ u t,k+ + C( t 4 ν u 4 4,0 + ν u / 4 4,0 + u ttt,0 + ν u tt,0 + ν u tt 4 4,0 + tt,0. Te next step will be te application o Lemma.4.4, te discrete Grönwall inequality. Let t be suiciently small, i.e., C t < ( + ν 3 u 4,0, it is allowed to apply te lemma and we obtain en + t 4 N n=0 ( ν (ε + e n + ν T [I P H ] (ε + e Cν k u,k+ + Cν T k u,k+ + Cν T H k u,k+ + C k ν u,k+ + C k+ ν ( u 4 4,k+ + u 4,0 4 s+ + C p /,s+ ν ( + C k+ u t,k+ + C( t 4 ν u 4 4,0 + ν u / 4 4,0 + u ttt,0 + ν u tt,0 + ν u tt 4 4,0 + tt,0. 8

40 Now we ave an estimate or te model error e and it is let to ind an error estimate or te wole error. We obtain un u N + t N (ν (u(t / (w + u n 4 / n=0 + ν T [I P H ] (u(t (w + u / N Λ N + e N + Cν t ( (u / u(t / + Λ / + (ε + e n n=0 N + C t ( ν T [I P H ] Λ + ν T [I P H ] (ε + e, n=0 were te upcoming new terms are eiter already contained in te RHS o te model error, or easy to andle like e.g. wit Lemma.3.9. Combining all estimates rom above we get Teorem.3.7 and (in te particular case Corollary Growt o Perturbations in te discrete sceme. Te question naturally arises o dependence o te constant C in Teorem.3.7 upon te inal time T. Tis dependence is exponential (relecting exponential stretcing in te continuous NSE and inevitably arising rom te discrete Grönwall inequality. It is related to te maximal Lyapunov exponent in te discrete model given by Algoritm.3.. In tis subsection we derive an estimate or te Lyapunov exponent o tis model and tus its error growt. To simpliy te notation we will suppress te index, altoug we only consider discrete solutions ere. Let (u, w, and (u, w, be two solutions wit dierent problem data rom Algoritm.3.. By subtracting te two corresponding equations in Step, we obtain ( (w w t + b ( w + u n (u n u n, v + ν, w + u n, v b ( (w w ( w + u n + (u n u n, w + u n, v =, v ( / /, v 9

41 or all unctions v V. Setting v = [(w w + (u n u n ] gives ( w w t u n u n + ν ( w + u n b b ( w + u n + (w w + (u n u n = + (u n u n + (u n u n + (u n u n, w + u n, (w w, w + u n, (w w ( / /, (w w As a next step we estimate all terms on te RHS and start wit te easy one ( / /, (w w + (u n u n ν (w w + (u n u n 8 To bound te nonlinear term we use (.3 = = ( w + u n b b b ( w + u n ( w + u n + b b b ( w + u n ( w + u n ( (w w, w + u n, (w w, w + u n, (w w, (w w + (u n u n, w + u n, (w w, w + u n, (w w + (u n u n, w + u n + (u n u n + ν + (u n u n ( / /, (w w + (u n u n + (u n u n + (u n u n, (w w + (u n u n (w w + (u n u n C (w w + (u n u n 3ν (w w + (u n u n 4 + C4 w + u n 4 (w w + (u n u n 4ν 3 3/ w,. (.4 + u n. 30

42 were te actor w +u n 4 w can also be replaced by min i i=, +u n i 4 wen we apply te same steps or w +u n again and use bot estimates. Wit tis in mind (.4 becomes ( w w u n u n t + ν 8 8ν min 3 i=, C4 w i + u n 4 ( i w w (w w + (u n u n + u n u n + ( / / ν. To get a connection between u and w, we use a variant o Lemma.3.3 or te dierence o te solutions and get ( u u u n u n t + ν (w w + (u n u n 8 + νt [I P H ] (w w + (u u C4 8ν min w i + u n 4 ( i u 3 i=, u + u n u n + ( / / ν + tc4 min w i + u n 4 i 4ν 3 i=, νt [I P H ] (w w + (u u. At tis point let us assume tat ( C 4 t 3ν min 3 i=, w i + u n i 4 to get ( u u u n u n t + ν (w w νt [I P H ] (w w + (u C4 8ν min w i + u n 4 ( i 3 i=, u u + (u n u n u + u n u n + ν ( / / 3

43 and sum up te inequalities rom n = 0 to n = N. It olds N ( ν t un u N + (w w + (u n u n 8 n=0 + 4 νt [I P H ] (w w + (u u t u0 u 0 + C4 N min w i + u n 4 ( i u 8ν 3 i=, u + u n u n were + ν N n=0 κ n = C4 4ν 3 ( / n=0 / = N n=0 κ n un u n + ν N n=0 ( / 4ν 3 C t + min 4 i=, w i +u0 i 4 or n = 0 ( wi min n+un i i=, 4 + w i +u n i wi min N +un i i=, 4 or n = N. / 4 or n =,..., N Wen we now apply te discrete Grönwall inequality rom Lemma.4.4, we get u N N ( ν u N + t (w w + (u n u n 4 n=0 + νt [I P H ] (w w + (u u ( { N exp t g n κ n tκ 0 (u 0 u t N ( / / ν n= n=0 were g n = ( tκ n under te assumption tat tκ n <. Now, we will look at te exponential multiplier. For clarity, let us deine Given in addition tat t exp ( t w + u 4, : = max ( C 4 ( N g n κ n exp n= min n=0,...,n i=, w min ν 3 i i=, wn i + u n i +u n i 4 we can estimate t C4 ν 3 w + u 4, ( t C4 ν 3 w + u 4, ( exp N t C4 ν w + 3 u 4, exp 4. N n= ( T C4 ν 3 w + u 4, }. (.5,, 3

44 Remark.3.. Te result in (.5 is wat one can expect rom te discrete Grönwall inequality. Neverteless it would be better to ave an improvement o te actor C4 ν 3 to C 4 (ν+ν T 3. Te analysis erein ailed to produce tis because o te mismatc in te arguments o te usual Galerkin terms in comparison to te term stemming rom te VMS projection step. Te Galerkin terms ad an argument o te Crank-Nicolson time discretization sceme, i.e. wi n + u n i, were te terms rom te VMS projection step ad an argument wi n + u n i. Recall tat te projection step does not depend on te time discretization, Remark ALGORITHMS FOR COMPUTING THE PROJECTION In Step te action o Π must be computed. We consider two approaces to solving te linear system to compute te projection Πu in tis section and one approac in Section.5 were te diicult term in (.6 involving te operator P H is simply lagged to te previous time level, reducing complexity to one Stokes solve and circumventing tis possible diiculty. Te simplest metod is a ixed point iteration in wic te terms involving P H are in te RHS residual calculation. We prove convergence in Teorem.4.4. Tis metod was used in our computable experiments in wic convergence was seen in 5 steps or less. Te proo o Teorem.4.4 can be adapted to give an estimate o te number o steps tat is not in accord wit te rapid convergence observed in our experiments. Step involves solving a linear system wit a mixed structure. Let RHS denote a rigt and side known rom previous values and let { φ, φ N} denote a basis or te velocity space X. Ten we ave te system M + ta C u = RHS, were (.6 C t 0 λ 0 (M + t A ij = B(φ i, φ j := (φ i, φ j + t (ν T [I P H ] φ i, [I P H ] φ j. Te, block M + t A is SPD. However, te diiculty in tis system is tat (or some common coices o P H i it is assembled it as a large bandwidt. For example, i P H is te (weigted L projection onto a coarse mes space, ten it is very easy to compute it in a 33

45 residual term but it couples ine mes basis unctions across te coarse mes macro element. Our standard approac to mixed type systems is to solve te Scur complement system C T (M + t A Cλ = C T (RHS by an iterative metod in wic te inner action o (M + t A is evaluated by anoter iterative metod. We sow in Proposition.4. tat cond(m + t A = O( so tis inner iteration is not callenging (and te action o P H is computed in te residual calculation at eac step. Tis suggests tat alternate approaces (wose delineation is still an open question are easible. To study te condition number o te, block o (.6, we make te ollowing two assumptions on te velocity space wic old or many spaces on sape-regular meses. Condition.4. (Inverse Estimate and Norm Equivalence. (i Tere is a C INV or every v X we ave v C INV v. suc tat ave (ii Tere are positive constants C, C suc tat or every v X, v = N i= α iφ i, we C d v N αi C d v. i= Proposition.4.. Suppose te velocity space satisies Condition.4. and ν T = ν T (x,. Ten cond (M + t A C [ + C t ( INV max ν C T (x, ]. x Proo. First note tat M + t A is clearly SPD. Let α = (α,, α N t be an eigenvector o M + t A and deine v := N i= α iφ i. We ave λ α = α t (M + t A α = B(v, v = = v + t (ν T [I P H ] v, [I P H ] v. 34

46 I λ = λ min ten dropping t (ν T [I P H ] v, [I P H ] v and using norm equivalence we ave λ min = v + t (ν T [I P H ] v, [I P H ] v α C d. I λ = λ max ten majorizing t (ν T [I P H ] v, [I P H ] v and using te inverse estimate and norm equivalence we ave λ max = v + t (ν T [I P H ] v, [I P H ] v α v + t(ν T v, v C α d v + t (max x ν T (x, v v C d v + t (max x ν T (x, CINV v v C d [ + C INV t ( max ν T (x, ]. x Te result ollows by dividing tese two estimates. In many cases te dependence o ν T (x, upon scales like O(, implying (in tese cases tat cond (M + t A = O(. Consider next te ixed point iteration or solving (.6. Algoritm.4.3. Until convergence criteria are satisied, given u j satisying V ind u j+ V (u j+, v + t (ν T u j+, v = t (ν T P H u j, v + (w, v t (ν T [I P H ] w, v or all v V. Teorem.4.4. Let {u j } j N be determined by Algoritm.4.3. Suppose tat tere exists a constant C suc tat 0 < ν T C <. Ten u j Πw in X as j. Proo. Subtracting te above equalities deining u j and u j+ yields (u j+ u j, v + t (ν T (u j+ u j, v = t (ν T P H (u j u j, v. 35

47 Set v = u j+ u j and applying Young s inequality to te RHS give u j+ u j + t ν T (u j+ u j t 4 ν T P H (u j u j + t 4 ν T (u j+ u j t 4 ν T (u j u j + t 4 ν T (u j+ u j. Applying te inverse estimate u j+ u j C INV (u j+ u j, we obtain C INV (u j+ u j + t 4 ν T (u j+ u j t 4 ν T (u j u j. Since ν T is bounded rom above by C we ave ( Tereore, C INV C ν T (u j+ u j C INV (u j+ u j. + t CINV C 4 ν T (u j+ u j t 4 ν T (u j u j. Tis implies (as a consequence o Contraction Mapping Teorem bot existence and uniqueness o a solution u to (.6 and convergence..5 A COMPUTATIONALLY ATTRACTIVE VARIANT Te projector Π in Algoritm.3. is te solution o t (w u, v = (ν T [I P H ] w + u, [I P H ] v. Te diiculty wit tis system or u is coupling across many ine mes elements caused by te projection P H. First note tat te above is equivalent to t (w u, v = (ν T w + u, v (ν T P H w + u, v. Tus te diiculty is given by te second term alone. We consider te modiication o Step in Algoritm.3. o just lagging tis term to te previous time level wit no iteration. Te complexity o Step is ten one Stokes solve. 36

48 Step : Given w V, ind u V satisying t (w u, v = (ν T w + u, v (ν T P H wn + u n, v, v V. (.7 In (.7 te action o P H is calculated or a known unction and goes into te RHS o te linear system (.7. Surprisingly, we sow tis to be unconditionally stable and second order accurate. We tus consider te modiication o Algoritm.3. below. Algoritm.5.. Step : Given u n ind w X, p Q satisying ( w u n t, v + b( w + u n, w + u n, v + ν( w + u n, v (p /, v = ( /, v, or all v X, (.8 ( w, q = 0, or all q Q. Step : u := Πw were (u, λ X Q is te unique solution o t (w u, v (λ, v = (ν T w + u, v (ν T P H wn + un, v, ( u, q = 0, v X, q Q. Figure. sows a low diagram o algoritms proposed erein - Algoritm.3.,.4.3 and.5.. Teorem.5.. Assume ν T is constant in space at eac time level. Consider Algoritm.5.. It satisies, or any N > 0, te ollowing energy equality, implying stability, [ u N + t ν T P H wn N ( ν T [I P H ] w + t = [ n=0 + un ] N + t n=0 ν w + u + t 8 ν T P H u 0 + t ν T P H w0 + ] u0 + t N n=0 + u n [ w w n t ( /, w + u n. + u u n t ] 37

49 Figure.: Flow diagram o a numerical simulation wit Algoritm.3. and.4.3. Te number crit is te stopping criterion o Algoritm.4.3. In Algoritm.5. we simply omit te iteration or Step in te diagram. 38

50 Proo. Take te L inner product o (.8 wit (w + u n /. Rearranging te result gives t [ u u n ] + ν = ( /, w + u n. w + u n + [ w u ] t Now consider Step. Set v = ( w + u /. Tis gives [ w u ] t = νt w + u ( ν T P H wn + un, w + u = νt w + u ( ν T P H wn + un, P H w + u = { νt w + u νt P H wn + un + νt P H w + u [ w + u νt P H = νt (I P H w + u + [ w + u νt P H { + νt P H w + u νt P H wn + un }. wn + ] } un ] wn + un [ Now insert te above RHS in te energy estimate or te term t w u ]. Tis gives { [ u u n t ] + νt P H w + u + ν + u n + νt (I P H w + u w + νt P H [ w + u wn + un ] = ( /, w νt P H wn + un + u n. } Summing rom n = 0 to N yields te result. 39

51 Remark.5.3. Te orm o te kinetic energy and numerical diusion induced by Algoritm.5. is [ Kinetic Energy = u N + t νt P H wn V iscous Diusion = ν w + u n, V MS Diusion = νt [I P H ] w + u [ w w n νt P H t Additional Algoritmic Diusion = t 8 + un, + u ], u n t ]..6 NUMERICAL EXPERIMENTS We present numerical experiments to test te algoritms presented erein. Using te Green- Taylor vortex problem, we conirm te predicted convergence rates rom te teory. Next, te eects o te metods as stabilization tecnique are tested wit an advected L-saped ront. Furter testing is ten perormed using te well-known bencmark o te decaying omogeneous isotropic turbulence to compare te algoritms presented erein to te usual approac were everyting is applied in one step. We used FreeFEM++ [56] or te Green- Taylor vortex and advection o L-saped discontinuity and deal.ii [6, 7] or te decaying omogeneous isotropic turbulence..6. Green-Taylor vortex. For te irst test we select te velocity ield given by te Green-Taylor vortex, [4], [3], wic is used as a numerical test in many papers, e.g., Corin [8], Tati [], Jon and Layton [69], Barbato, Berselli and Grisanti [8] and Berselli [5]. Te exact velocity ield is given by u (x, y, t = cos(ωπx sin(ωπye ω π t/τ, u (x, y, t = sin(ωπx cos(ωπye ω π t/τ, p(x, y, t = 4 (cos(ωπx + cos(ωπye 4ω π t/τ. (.9 40

52 We take ω =, t =, τ = Re = 500, Ω = (0,, = /m, t = /0, H =, were m is te number o subdivisions o te interval (0,. We utilize Taylor-Hood inite elements or te discretization. Newton iterations are applied to solve te nonlinear system wit a w (j+ w (j H (Ω < 0 0 as a stopping criterion. For te ixed point iteration in Algoritm.4.3, te convergence criterion is u (j+ u (j H (Ω < 0 0. Convergence rates are calculated rom te error at two successive values o in te usual manner by postulating e( = C β and solving or β via β = ln(e( /e( / ln( /. Te boundary conditions could be taken to be periodic (te easier case. Instead we take te boundary condition on te problem to be inomogeneous Diriclet: u = u exact, on Ω. Te errors and rates o convergence are presented in Table. and.. From te tables, we see tat te rates o convergence o bot algoritms conirm te predicted convergence rates rom teory. Algoritm.5. (in wic te projected term in Step is simply lagged to te previous time level proves itsel to be eective. Wile it does not utilize any iterative metod in Step, te quality o its errors is as good as ull solve VMS algoritm. t u u,0 rate u u,0 rate /6 / e e- /5 /50.306e e-.84 /36 / e e-.3 /49 / e e-.5 /64 / e e-.70 /8 / e e-.75 /00 /000.64e e-3.6 Table.: Error and convergence rate data or Algoritm Test wit an L-saped discontinuity advected skew to mes. We also test and compare te perormance o our uncoupled, partitioned approac to VMS wit te usual, centered FEM witout any stabilization or advection dominance on te 4

53 t u u,0 rate u u,0 rate /6 / e e- /5 /50.303e e-.83 /36 / e e-.3 /49 / e e-.5 /64 / e e-.70 /8 / e e-.75 /00 / e e-3.6 Table.: Error and convergence rate data or Algoritm.5. bencmark problem o advection o an L-saped discontinuity rom [0]; see [75], [7] and [73] or an analysis o projection based VMS or stabilization o advection. We solve φ t + a φ (κ φ = on Ω, φ = g on Ω, were is te known source term, g is te prescribed boundary data, a is a solenoidal velocity ield and κ is te diusivity. Te problem setup is given in Figure.. Te domain is a unit square subdivided into triangle elements. Te number o subdivisions in eac side o te domain is 5. At te initial time, te value o φ is set to in te interior o te L-saped block located in te lower letand corner o te domain and 0 elsewere. We coose te angle o advection to be 45 and te diusivity κ = 0 6. Te solution is marced to t = 0.5 wit time step t = We utilize C -piecewise quadratic inite element in tis test o tree dierent approaces. Te irst approac is simply using a standard discretization sceme. Te result is ten compared wit tat o our proposed modular, postprocessing VMS approac. In te second test, te solution is computed by te Algoritm.4.3 and in te tird test we use Algoritm.5.. Te results in [0] using bubble unction based VMS and quintic smootest splines were signiicantly better tan all te results using projection based VMS and quadratic splines. 4

54 Figure.: Advection o an L-saped ront. Problem description, rom [0]. Te perormance o te mentioned metods is sown using two dierent approximation scemes in Step. In te irst sceme, te numerical solutions in Step are generated by Backward Euler discretization wile in second sceme, we use Crank-Nicolson FEM. Results are sown in Figure.3 erein. We observe tat in bot case, te unstabilized solutions sow te expected wiggles. Te stabilized solutions are consistent wit te results in [7] and [73]. Tere are less oscillations in solutions produced by te two later metods. In tis test, te uncoupled VMS approac were te projected term is lagged again sow its ig consistence and competitiveness wit te ull solve VMS algoritm s solution..6.3 Decaying Homogeneous Isotropic Turbulence. Our next numerical illustration is or te tree dimensional low o te decaying, omogeneous, isotropic turbulence. Te setting is a domain Ω = [0, π] 3 wit periodic boundary conditions on all sides o Ω and rigt and side = 0. For comparison, we consider te experimental results o [9] wic provide energy spectra 43

55 Figure.3: Advection o an L-saped ront. In te let is te solutions produced by using Backward Euler time discretization sceme in Step : (a beore stabilization, (b ater stabilized by Algoritm.4.3, (c ater stabilized by Algoritm.5.. In te rigt is te solutions produced by using Crank-Nicolson time discretization sceme in Step : (d beore stabilization, (e ater stabilized by Algoritm.4.3, ( ater stabilized by Algoritm.5.. Te plotted solutions is at time t = 0.5 wit time step t =