ROBUST MULTISCALE ITERATIVE SOLVERS FOR NONLINEAR FLOWS IN HIGHLY HETEROGENEOUS MEDIA

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1 ROBUST MULTISCALE ITERATIVE SOLVERS FOR NONLINEAR FLOWS IN HIGHLY HETEROGENEOUS MEDIA Y. EFENDIEV, J. GALVIS, S. KI KANG, AND R.D. LAZAROV Abstract. In tis paper, we study robust iterative solvers for finite element systems resulting in approximation of steady-state Ricards equation in porous media wit igly eterogeneous conductivity fields. It is known tat in suc cases te contrast, ratio between te igest and lowest values of te conductivity, can adversely affect te performance of te preconditioners and, consequently, a design of robust preconditioners is important for many practical applications. Te proposed iterative solvers consist of two kinds of iterations, outer and inner iterations. Outer iterations are designed to andle nonlinearities by linearizing te equation around te previous solution state. As a result of te linearization, a large-scale linear system needs to be solved. Tis linear system is solved iteratively (called inner iterations), and since it can ave large variations in te coefficients, a robust preconditioner is needed. First, we sow tat under some assumptions te number of outer iterations is independent of te contrast. Second, based on te recently developed iterative metods (see [15, 17]), we construct a class of preconditioners tat yields convergence rate tat is independent of te contrast. Tus, te proposed iterative solvers are optimal wit respect to te large variation in te pysical parameters. Since te same preconditioner can be reused in every outer iteration, tis provides an additional computational savings in te overall solution process. Numerical tests are presented to confirm te teoretical results. 1. Introduction In tis paper, we study robust preconditioners for solving finite element approximations of nonlinear flow equations in eterogeneous media. Our motivation stems from Ricards equation ([28]) wic describes te infiltration of water into a porous media wose pore space is filled wit air and water. In many cases, te eterogeneous porous media is caracterized by large variations of te conductivity. For example, in natural porous formations it is common to ave several orders of magnitude of variations in te conductivity values. A ig contrast, expressed as te ratio between ig and low conductivity values, brings an additional scale into te problem. A design of robust preconditioners tat converge independent of small scales and ig-contrast of te media for nonlinear problems is a callenging task. In tis paper, we address tis problem for te model of two-pase flow in porous media, te steady-state Ricards equation. Te Ricards equation as te form (1) D t θ(u) div(k(x, u) (u + x 3 )) = f, x, were θ(u) denotes te volumetric fluid content, and k(x, u) k 0 > 0 is te relative ydraulic conductivity and k 0 is a constant. We assume tat suitable initial and boundary data are provided. Te dependence of te volumetric water content and te relative ydraulic conductivity from te pressure ead is establised experimentally by assuming some functional form. Tere is a large number of functional forms used by ydrologists and soil scientists. In our numerical experiments, we use tree popular among te soil scientists models, namely, Haverkamp, van Genucten models, and Exponential (see, e.g. [7, 21, 33, 27]). Date: beginning August 18, 2010, today is: May 21, Key words and prases. FE metod, nonlinear permeability, igly eterogeneous media, ig contrast media. 1

2 2 Y. EFENDIEV, J. GALVIS, S. KI KANG, AND R.D. LAZAROV In tis paper, we are interested in robust preconditioners for te finite element system resulting from te discretization of nonlinear equations wen k(x, u) is eterogeneous wit respect to space. We consider te steady-state Ricards equation (2) div(k(x, u) (u + x 3 )) = f, x, were k(x, u) as ig variations in x. In many practical cases, te eterogeneous portion of te relative permeability is given by a spatial field tat does not depend on u, i.e., k(x, u) = k(x)λ(u). By denoting, u+x 3 as a new variable and assuming λ is smoot, we can write te above equation as (3) div(k(x)λ(x, u) u) = f, x, were k(x) is a eterogeneous function, wile λ(x, u) is a smoot function tat varies moderately in bot x and u. Robust preconditioners for a finite element approximation of Equation (3) wit suc coefficients will be studied in te paper. We note tat coarse-grid approximations of Ricards equation are discussed in literature (e.g., [8, 1, 13]). Various iterative metods for solving nonlinear equations ave been proposed and studied in te past, e.g. [5, 6, 10, 24, 31, 34]. For example, in [5, 31], a nonlinear iterative procedure as been proposed and its optimality as been establised, in [24], multilevel iterative metods ave been studied for Ricards equation, in [10], two-level domain decomposition metods ave been proposed and analyzed. To te best of our knowledge, te tecniques developed in te previous works ave not considered te case of igly eterogeneous conductivity fields, wic is te main objective of tis paper. Te proposed iterative procedure involves outer iterations and inner iterations, a tecnique tat is commonly used in te literature. Outer iterations are designed to andle nonlinearities by linearizing te equation around te previous state. Te simplest is Picard iteration tat is described by div(k(x)λ(x, u n ) u n+1 ) = f, were n denotes te outer iteration number. For every outer iteration n, a linear problem needs to be solved. For te solution of te linear problem, we employ two-level domain decomposition preconditioners witin conjugate gradient (CG) iterative tecnique. Bot inner and outer iteration can, in general, depend on te contrast and small scales. One of our main goals is to construct an iterative process tat converges independently of bot, te small scales and te contrast. In particular, we sow tat te robust iterative tecniques designed for a linear system can be re-used for every outer iteration if λ is a smoot function. Terefore, it is important to use efficient preconditioners for solving linear systems arising in approximation of problems wit igly eterogeneous coefficients. Suc preconditioners, designed in te earlier works [15, 17, 18], are discussed below and described in Section 3. For every outer iteration, te resulting linear system on te fine scale is solved using a twolevel domain decomposition preconditioner (e.g., [32, 25]), wic involves local (subdomain) and global (coarse) problems. Te number of iterations required by domain decomposition preconditioners is typically affected by te contrast in te media properties (e.g., [25, 32]) tat are witin eac coarse grid block. Because of te complex geometry of fine-scale features, it is often impossible to separate low and ig conductivity regions into different coarse grid blocks. Consequently, witout proper preconditioner, te number of iterations can be very large, wic substantially reduces te efficiency of te iterative metod, particularly for nonlinear flows. In tis paper, for every outer iteration we use te preconditioners designed in [15, 17]. Te main idea of tese preconditioners consists of augmenting te coarse space in te domain decomposition metods. In particular, a coarse space based on local spectral problems using multiscale functions is constructed. We prove tat wen te coarse space in te domain decomposition metods includes tese eigenfunctions, te condition number of te preconditioned matrix is bounded independently of te contrast. Te coice of multiscale spaces is important

3 ROBUST DOMAIN DECOMPOSITION METHOD FOR NONLINEAR FLOWS 3 to acieve small dimensional coarse spaces. By incorporating small-scale localizable features of te solution into initial multiscale basis functions, we ave sown tat one can acieve small dimensional coarse spaces witout sacrificing te convergence properties of te preconditioners. Initial multiscale spaces can employ constructions proposed for multiscale finite element metods in [12, 14, 22, 23]. We sow tat bot, te number of outer iterations and te number of inner iterations, are bounded independently of pysical parameters, suc as te contrast and small spatial scales. We first prove tat under some assumptions te number of outer iterations depends on te contraction constant tat is independent of te contrast in te conductivity field. Our reasoning takes into account te ig variations of te contrast in te conductivity field and follows te standard for suc nonlinear problems tecnique, e.g., [4]. As for inner iterations, we use twolevel preconditioners developed in [15, 17] tat provide, independent of te contrast, condition number for every outer iteration. We use te same preconditioner for every outer iteration repeatedly witout sacrificing te convergence of te overall metod. We note tat one can use Kircoff s potential (see, e.g. [7, p ]) to transform te original equation into a linear equation for te potential. However, tis tecnique becomes cumbersome wen λ(x, u) depends on x and does not ave an explicit form (e.g., given via a grap interpolation). Moreover, te difficulty of inversion of Kircoff s potential still needs to be performed and te extensions to time-dependent problems can become complicated. We test our metodology on a number of numerical examples for various nonlinear models. We consider two different eterogeneous permeability fields and vary te contrast over four orders in magnitude. Our numerical results sow tat te number of outer iterations does not depend on te contrast. Moreover, te number of inner iterations on every outer iteration does not depend on te contrast if an appropriate preconditioner is cosen. We also test two-level domain decomposition preconditioner wen te coarse space includes only te initial multiscale basis functions. In tis case, te number of iterations at every outer iteration grows as te contrast increases. Te paper is organized as follows. In Section 2, we introduce te problem. Section 3 is devoted to te description of robust preconditioners. Some of te proofs are presented in te Appendix. In Section 4 we present numerical results and, finally, in Section 5 we draw some conclusions. 2. Problem setting 2.1. Weak Formulation. We multiply equation (3) by a test function v H0 1 () and integrate over te domain. After applying te divergence teorem, we get tat te solution u satisfies te following integral identity k(x)λ(x, u) u vdx = fvdx, for all v H0 1 (). Now we define te space V = H0 1 (), set of all functions wit square integrable generalized derivatives of first order vanising on te boundary, te form a(, ; ) (4) a(u, v; w) = k(x)λ(x, w) u vdx, and te functional F ( ) (5) F (v) = fvdx.

4 4 Y. EFENDIEV, J. GALVIS, S. KI KANG, AND R.D. LAZAROV Ten te variational form of (3) is to find u V suc tat (6) a(u, v; u) = F (v), for all v V Finite Element Discretization. Let T be a triangulation of te domain into a finite number of triangular (tetraedral) elements. We assume tat T is quasiuniform and regular; see [9]. Let V be te finite dimensional subspace of V of piece-wise polynomials wit respect to T. Let u V be a solution of te following discrete problem. (7) a(u, v; u ) = F (v), for all v V. We know tat under suitable conditions, one can ensure te existence of a solution to te above equation. Define te nonlinear map T : V V by (8) a(t u, v; u ) = F (v), for all v V. Tis is well defined, since u V A nonlinear fixed point iteration. In tis section we describe a robust numerical metod to approximate te numerical solutions of te Ricards equation (7). We use a fixed point iteration based on te contractivity of te mapping T defined in (8). Te numerical solution u can be approximated to an arbitrary accuracy using Picard iteration. Starting wit an initial guess u 0 V, we define te nonlinear fixed point iteration by u n+1 = T u n. Tat is, given u n, te next approximation un+1 is te solution of te linear elliptic equation (9) a(u n+1, w; u n ) = F (w), for all w V. In order to define te solution metod, we reformulate te problem (9) in terms of te linear operator A n : V V defined for any given u n V as (10) a(v, w; u n ) = (An v, w), for all v, w V, were (, ) is te standard L 2 -inner product in V. In a similar manner, we present te linear functional F (w) in te form (11) F (w) = (b, w), for all w V. Obviously, b is te L 2 -projection of te rigt and side f of (3) on V. Ten te equation (9) can be rewritten in te following operator form (12) A n u n+1 = b. Note tat equation (9) (and its operator counterpart (12)) is an approximation of te linear equation div(k(x)λ(x, u n ) un+1 ) = f wit u n being te previous iterate. It is known tat te presence of te ig-contrast coefficient k(x) makes it computationally difficult to construct appropriated robust linear solvers for computing u n+1. Moreover, taking into account te contractivity of te operator T, in order to get a robust metod to compute te solution of te Ricards equation (8), we only need a robust metod for solving te linear problem (9). Because of te small scales and ig contrast in te conductivity field, te solution of tis system (of size proportional te fine grid points) is proibitively expensive. Terefore, an adequate robust iterative metod is needed. Te construction of robust solvers for ig-contrast linear elliptic equation as been considered by many autors. We will use as a preconditioner a two-level domain decomposition metod proposed in [15, 17, 18], wic involves solutions of appropriate local spectral problems.

5 ROBUST DOMAIN DECOMPOSITION METHOD FOR NONLINEAR FLOWS 5 If B 1 is te preconditioner, our goal is to ave te condition number of B 1 A n bounded independent of te contrast and n (i.e, independent of u n ). Now we describe a construction of suc preconditioner for (12), wic will give a robust, wit respect to te contrast, metod for Ricards equation. 3. Finite element discretization and two level domain decomposition preconditioner 3.1. Finite element approximation and local spaces. First, we provide an overview of te use of domain decomposition tecniques for constructing preconditioners for multiscale finite element approximations of ig-contrast elliptic equations (cf., [15, 17, 18, 19, 20]). For an extension to multilevel metods, we refer to [16]. Next, we briefly describe a two-level domain decomposition setting tat we use and introduce te local spaces and te coarse space. Let T H and T be coarse and fine partitions of into finite elements K (or nonoverlapping subdomains) tat consists of triangles, quadrilaterals, etc.. We assume tat te coarse elements of T H consist of a number of fine elements from T. Practically, we first introduce te coarse-grid T H and ten obtain te fine grid T by partitioning eac coarse element into a number of smaller ones. Let χ i be te nodal basis of te standard finite element space wit respect to te coarse triangulation T H. We denote by N v te number of coarse nodes, by {y i } Nv i=1 te vertices of te coarse mes T H, and define a neigborood of eac node y i by (13) ω i = {K j T H ; y i K j }. Let V0 (ω i) V be te set of finite element functions wit support in ω i and Ri T : V0 (ω i) V denote te extension by zero operator. We define, for later use, te one level additive preconditioner (e.g. [25, 32]) (14) B 1 1 = N v i=1 R T i (A 0 i ) 1 R i, were te operators A 0 i : V 0 (ω i) V 0 (ω i) are defined by (15) (A 0 i v, w) = a(v, w; u 0 ), for all v, w V0 (ω i ), i = 1,..., N v. Te application of te preconditioner B1 1 involves (A 0 i ) 1 wic means solving local problems subdomain-wise in eac iteration. Te operator A 0 i, defined by te bilinear form a(, ; u0 ) restricted to V0 ( i ), is local and invertible Coarse space construction. For given M c number of linearly independent functions {Φ i } Mc i=1 associated wit te coarse mes T H (tese will be introduced later), we define a coarse space V 0 by (16) V 0 = span{φ i } Mc i=1. Below we sall give tree coices of sets {Φ i } Mc i=1, tat ave been already used in te construction of a robust preconditioner for A n. Tese are: (1) multiscale coarse space (see, e.g. [12] and te references terein), (2) energy minimizing coarse space (see, e.g. [35]), and (3) a coarse space wit local spectral information, (see, e.g. [15, 17, 18]). On an abstract level, te main assumption is tat Φ i V, but te support of eac Φ i is related to te coarse mes T H so tat M c << dim V. Below we refer to te Φ i s as coarse-scale basis functions. Te coarse space V 0 defines an operator A c : V 0 V 0, (A c v, w; u 0 ) = a(v, w; u0 ), v, w V 0.

6 6 Y. EFENDIEV, J. GALVIS, S. KI KANG, AND R.D. LAZAROV Note tat if R T c : V 0 V is te natural interpolation operator, ten we ave (17) A c = R c A 0 R T c wit A 0 defined by (10) for n = 0. Note tat te operator A c uses te initial guess u 0 V and is constructed only once at te beginning of te fixed point nonlinear iteration. Likewise, te coarse basis functions {Φ j } Mc j=1 are related to te form a(, ; u 0 ) and are constructed only one time. Tese can be regarded as a preprocessing step. Once te coarse space V 0 is constructed and te coarse-scale operator A c is defined, we can use te two level additive preconditioner of te form N v (18) B 1 = Rc T A 1 c R c + Ri T (A 0 i ) 1 R i = Rc T A 1 c R c + B1 1. i=1 Te preconditioner B 1 involves solving one coarse-scale system and N v local problems in eac overlapping subdomain ω i, i = 1,..., N v. Te goal is to reduce te number of iterations in te iterative procedure, e.g., a preconditioned conjugate gradient. An appropriate construction of te coarse space V 0 plays a key role in obtaining robust iterative domain decomposition metod. In te next Section 3.3 we present examples of suc coarse space constructions. We summarize te fixed point iteration in Algoritm 1. Algoritm 1 Fixed point iteration 1: Initialize: Coose u 0 V and compute te residual r 0 = b A 0 u 0. 2: Construct te coarse basis {Φ j }, te coarse space V 0 in (16), and te coarse operator A c in (17). 3: for n = 1, 2,... until convergence do 4: Set te linear system A n u n+1 = b (see (12)). 5: Using PCG wit preconditioner B 1 in (18) solve te linear system in 4: to get u n+1 6: Compute te residual r n+1 = b A n+1 u n+1. 7: end for. Remark 1. In te general domain decomposition metod setting te overlapping subdomains {ω i } could be cosen independently of te coarse triangulation T H. However, for te purpose of tis paper, we will only consider te partition introduced above Some multiscale coarse spaces. In tis subsection we review several possibilities for construction of coarse basis functions tat ave been used to design two-level preconditioners tat are robust wit respect to te contrast Linear boundary conditions multiscale coarse spaces. Let χ H i be te nodal basis of te standard finite element space wit respect to te coarse triangulation T H. We define multiscale finite element basis function χ ms i tat coincides wit χ H i on te boundaries of te coarse partition. Namely, for eac K ω i (19) k χ ms i vdx = 0, v V V0 (ω i ) and χ ms i = χ H i on K. K Tis means tat χ H i is an approximation in te fine-grid space of te boundary value problem (20) div(k χ ms i ) = 0 in K ω i, χ ms i = χ H i in K, for all K ω i, were K is a coarse grid element witin ω i. Ten we define (21) V ms 0 = span{χ ms i }.

7 ROBUST DOMAIN DECOMPOSITION METHOD FOR NONLINEAR FLOWS 7 Note tat multiscale basis functions coincide wit standard finite element basis functions on te boundaries of coarse grid blocks, wile are oscillatory in te interior of eac coarse grid block. Even toug te coice of χ H i can be quite arbitrary, our main assumption is tat te basis functions satisfy te leading order omogeneous equations wen te rigt and side f is a smoot function (e.g., L 2 integrable). We remark tat te MsFEM formulation allows one to take advantage of scale separation. In particular, K can be cosen to be a volume smaller tan te coarse grid. Various oter boundary conditions ave been introduced and analyzed in te literature, see [12] and references terein. For example, in [23], reduced boundary conditions are found to be efficient in many porous media applications Energy minimizing coarse spaces. Coarse basis functions can be obtained by minimizing te energy of te basis functions subject to a global constraint (see, [35]). More precisely, one can use te partition of unity functions {χ em i } Nv i=1, wit N v being te number of coarse nodes, tat provide te least energy. Tis can be accomplised by solving N v (22) min k χ em i 2, ω i i=1 subject to te constraint i χem i = 1 wit supp(χ em i ) ω i, i = 1,..., N v. Note tat i χem i = 1 is a global constraint toug it is not tied to any particular global fields unlike te metods discussed previously. One can solve (22) following a procedure discussed in [35] and ten define te coarse space (23) V em 0 = span{χ em i }. We note tat te computation of tese basis functions requires te solution of a global linear system, a procedure more expensive compared to te local computation of multiscale finite element basis functions wit linear boundary conditions χ ms i A coarse space wit local spectral information. We motivate te coice of te coarse spaces based on te analysis presented in [15, 17, 18]. First, we briefly review te results of [15, 17, 18]. For fixed ω i consider te eigenvalue problem (24) div(k ψ ω i l ) = µω i l kψ ω i l, were µ ω i l and ψ ω i l are eigenvalues and eigenvectors in ω i and k is defined by Nv 1 (25) k = H 2 k χ in j 2. j=1 We recall tat χ in j (simply denoted by χ j in furter discussions) are te initial multiscale basis functions (eiter multiscale basis functions wit linear boundary conditions or energy minimizing basis functions) and N v is te number of te coarse nodes. Te eigenvalue problem considered above is solved wit zero Neumann boundary condition and understood in a discrete setting. Assume eigenvalues are given by µ ω i 1 µω i 2... Basis functions are computed by selecting a number of eigenvalues (starting wit small ones) and multiplying corresponding eigenvectors by χ i. Tus, multiscale space is defined for eac i as te span of χ i ψ ω i l, l = 1,..., L i, were L i is te number of selected eigenvectors (see Figure 1 for an illustration).

8 8 Y. EFENDIEV, J. GALVIS, S. KI KANG, AND R.D. LAZAROV Figure 1. Illustration of basis construction We note tat {ω i } yi T H is a covering of. Let {χ i} Nv i=1 be a partition of unity subordinated to te covering {ω i } suc tat χ i V0 (ω i) and χ i 1 H, i = 1,..., N v. Define te set of coarse basis functions (26) Φ i,l = I (χ i ψ ω i l ), for 1 l L i and 1 i N v, were I is te fine-scale nodal value interpolation and L i is an integer number specified for eac i = 1,..., N v. Note tat in tis case, tere migt be several basis functions per coarse node. Te number of basis functions per node is defined via te eigenvalue problem (24). Denote by V 0 te local spectral multiscale space (27) V lsm 0 = span{φ i,l : 1 l L i and 1 i N v } Condition number estimates. In tis section, we present a teoretical result wic sows tat te number of outer iterations is independent of te contrast. First, for a given K > 0 we introduce te ball (28) V K,p := {v V : v W 1 p K}. Te following tree assumptions are used in te proofs of Teorems 1, 2, and 3. Assumption 1. (A) C 0 k(x) M, were C 0, and M is a constant. (B) Te function λ(x, u) satisfies te following conditions. (a) λ(x, u) is Lipscitz continuous wit respect to u, i.e., tere exists a constant C 1 suc tat λ(x, u) λ(x, v) C 1 u v, for all u, v V, x, (b) λ(x, u) is bounded above, i.e. tere is a constant C suc tat λ(x, u) C for all x and u L () (c) λ(x, u) is bounded below, i.e. tere is a constant C 2 suc tat 0 < C 2 λ(x, u) for all x and u V. (C) See (43). Under tese assumptions, we sow te following teorems concerning te existence of te solution and te boundedness of te contraction constant. Teorem 1. Under te Assumption 1 (A) and (B), tere are constants α <, 0 > 0 and ɛ > 0 suc tat for all 0 < 0 and u V a(u, v ; ) (29) u W 1 p () α sup, wit a(u, v; ) = k u v dx, 0 v V v W 1 q ()

9 ROBUST DOMAIN DECOMPOSITION METHOD FOR NONLINEAR FLOWS 9 wenever 2 p ɛ, q is te dual index to p, 1 p + 1 q = 1 and W 1 q () is a semi-norm in W 1 q (). Teorem 2. Let te Assumption 1 (A), (B), and (C) old. Ten (a) tere exists K > 0, p > 2, 0 > 0, and δ > 0 suc tat for all F wit F W 1 δ, T p maps V K,p into itself for all 0 < 0 and by Browder fixed point Teorem, tere exists a solution ũ of equation (7) and it satisfies (30) T ũ = ũ. (b) Te map T : V K,p te contrast. V K,p is a contraction and te contraction constant is independent of Teorem 3. Under te assumptions of Teorem 1, we ave cond(b 1 A n ) C, were C is independent of te contrast. Te proofs of tese teorems are presented in Appendix A. 4. Numerical results In tis section we present some representative numerical examples. We solve te Ricards equation (7) in = [0, 1] [0, 1] wit f(x) = 1 and omogeneous Diriclet boundary conditions. We consider several models for te ydraulic conductivity: te Haverkamp, van Genucten, and Exponential model, (see, e.g. [7, 21, 33, 27]), as introduced below. Te coarse mes T H is obtained by dividing into a mes. Te fine triangulation is obtained by dividing eac coarse-mes element into squares and furter dividing eac square into two triangles. Tus, te fine-mes step size is = 1/100. In all te numerical experiments we use te initial approximation for te iterative process u 0 tat solves (31) a(u 0, v; 0) = F (v), for all v V. We apply te Algoritm (1). As stated in Algoritm (1) we use te preconditioner B 1 in (18) wit tree different coarse spaces: (1) V0 ms described in Section In tis case B 1 is denoted by B 1 (2) V0 em described in Section In tis case B 1 is denoted by B 1 ms; em; lsm. (3) V0 lsm described in Section In tis case B 1 is denoted by B 1 We study te performance of Algoritm 1 wit initial guess u 0 and preconditioners B 1 em, and B 1 lsm. We consider different permeabilities wit complex ig-contrast configurations, see Figure 2. A number of parameter values in te nonlinearity of te ydraulic conductivity are tested in our simulations. In particular, for eac experiment we cose a different set of parameters for te model and a set of contrast values for te ydraulic conductivity. We note tat, for eac outer iteration in Algoritm 1 we ave a PCG iteration. Te inner PCG iteration is convergent wen te initial residual is reduced by a factor of tol in = 1e 10 wile te outer tolerance is set to tol out = 1e 8. We consider te following indicators for te performance of te preconditioners: ms, B 1 Coarse space dimension; Te number of outer iterations of te nonlinear fixed point iteration (R-iter); Te maximum and minimum number of inner PCG iterations over all outer iterations (CG-iter) and te estimated maximum condition numbers (Cond). We also verify numerically our main assumption in te proof of Teorem 1. Tat is, for every outer iteration update we compute k u p p = D ( k u ) p dx, p = 1, 2, 3,..., 10. We observe tat tis quantity remains bounded in all experiments.

10 10 Y. EFENDIEV, J. GALVIS, S. KI KANG, AND R.D. LAZAROV Figure 2. (Left): Conductivity field 1. Blue designates te regions were te coefficient is 1 and oter colors designates te regions were te coefficient is a random number between η and 10 η. (Rigt): Conductivity field 2. Blue designates te regions were te coefficient is 1 and red designates te regions were te coefficient is η Haverkamp model. First, we will study te Haverkamp model. In tis model, (see, e.g. [21]), te ydraulic conductivity is given by A (32) k(x, u) = k s (x) A + ( u /B) γ. We present te first set of numerical results in Tables 1 and 2. We use te preconditioner Bms 1 based on te coarse space V0 ms. We observe from tese tables tat te numbers of outer iterations do not cange wen te contrast value η increases. However, te condition number of te preconditioned system grows as η increases. We also observe tat te quantity k u p p, p = 1, 2, 3,..., 10, tat is related to te number of outer iterations, is bounded. We observe tat te number of outer iterations is larger wen B and γ (see (32)) decrease. Tis is because te smaller values of B and γ increase te magnitude of te conductivity tat comes from its nonlinear component. Comparing Tables 1 and 2 tat use different conductivity fields, we see tat te condition numbers in Table 2 are smaller tan te condition numbers in Table 1. Tis is because conductivity field 2 (see Figure 2) as simpler eterogeneity structure compared to conductivity field 1. Next, we repeat te above numerical experiments using te preconditioner Bem 1 based on te coarse space V0 em. Numerical results are presented in Tables 3 and 4. We observe tat, as before, te number of outer iterations is fixed wit increasing η. On te oter and, te condition number of te PCG iteration grows as te contrast increases. Tis condition number is muc larger compared to te case wen spectral basis functions are used as presented in te next tables. Furter, we sow te numerical experiment using te preconditioner B 1 lsm based on te spectral coarse space V0 lsm. Numerical results are presented in Tables 5 and 6. As before, we observe tat te number of outer iterations is independent of te contrast. We observe tat te condition number is also independent of te contrast. Note tat te condition number is substantially smaller tan te one of te preconditioned system using Bms 1 or Bem. 1 In general, te number of inner PCG iterations is muc smaller compared to tose wen oter coarse spaces are used van Genucten Model. Next, we consider te Van Genucten model (see [33]) tat is one of widely used empirical constitutive relations. In tis model, te ydraulic conductivity is

11 ROBUST DOMAIN DECOMPOSITION METHOD FOR NONLINEAR FLOWS 11 A = 1, B = 1, γ = 1 A = 1, B = 0.01, γ = , e , e , e , e , e , e , e , e Table 1. Numerical results for preconditioner Bms. 1 Here we use te Haverkamp A model k(x, u) = k(x) A+( u /B) wit k depicted in te left picture of Figure 2. γ Te coarse space dimension is 81. A = 1, B = 1, γ = 1 A = 1, B = 0.01, γ = , e , e , e , e , e , e , e , e Table 2. Numerical results for preconditioner Bms. 1 Here we use te Haverkamp A model k(x, u) = k(x) A+( u /B) wit k depicted in te rigt picture of Figure 2. γ Te coarse space dimension is 81. A = 1, B = 1, γ = 1 A = 1, B = 0.01, γ = , e , e , e , e , e , e , e , e Table 3. Numerical results for preconditioner Bem. 1 Here we use te Haverkamp A model k(x, u) = k(x) A+( u /B) wit k depicted in te left picture of Figure 2. γ Te coarse space dimension is 81. A = 1, B = 1, γ = 1 A = 1, B = 0.01, γ = , e , e , e , e , e , e , e , e Table 4. Numerical results for preconditioner Bem. 1 Here we use te Haverkamp A model k(x, u) = k(x) A+( u /B) wit k depicted in te rigt picture of Figure 2. γ Te coarse space dimension is 81. given by (33) k(x, u) = k s (x) {1 (α u /B)n 1 [1 + (α u ) n ] m } 2. [1 + (α u ) n ] m 2

12 12 Y. EFENDIEV, J. GALVIS, S. KI KANG, AND R.D. LAZAROV A = 1, B = 1, γ = 1 A = 1, B = 0.01, γ = , , , , , , , , Table 5. Numerical results for preconditioner B 1 lsm. Here we use te Haverkamp A model k(x, u) = k(x) A+( u /B) wit k depicted in te left picture of Figure 2. γ Te coarse space dimension is 166. A = 1, B = 1, γ = 1 A = 1, B = 0.01, γ = , , , , , , , , Table 6. Numerical results for preconditioner B 1 lsm. Here we use te Haverkamp A model k(x, u) = k(x) A+( u /B) wit k depicted in te rigt picture of Figure 2. γ Te coarse space dimension is 184. As before, we will present numerical results for all tree coarse spaces. First, in Tables 7 and 8 we present te numerical results for te preconditioner Bms. 1 We observe tat te number of outer iterations is smaller compared to te oter two models. Te number of outer iterations stays te same wile increasing η. On te oter and, te condition number of te linearized system increases as η increases. We observe tat te value k u p p, p = 1, 2, 3,..., 10 is bounded independent of te contrast. Now we compare Table 7 and Table 8 for two different conductivity fields depicted in Figure 2. We observe tat te condition numbers presented in Table 8 is smaller tan tose presented in Table 7 wic is consistent wit our previous observations. Numerical results for te preconditioner Bem 1 are presented in Tables 9 and 10, wile numerical results for te preconditioner B 1 lsm are presented in Tables 11 and 12. As before, we observe tat te number of outer iteration does not cange wit η increasing. However, te condition number of te inner iteration is increasing for Bem, 1 wile te condition number of te inner iteration does not cange (and is muc smaller) for B Exponential Model. Finally, we present numerical results for te exponential model. Here te ydraulic conductivity depend exponentially on te pressure ead u, tat is, lsm. (34) k(x, u) = k s (x)e αu/b. Tis nonlinear equation can also be derived by omogenizing Stokes equation in porous media wen te fluid viscosity exponentially depends on te pressure [27]. We present te first set of numerical results in Tables 13 and 14. First, we use te preconditioner Bms 1 based on te coarse space V0 ms. We observe tat te number of te outer iterations does not cange wen te contrast η increases. However, te condition number of te preconditioned system increases proportional to η. We also observe tat te quantity

13 ROBUST DOMAIN DECOMPOSITION METHOD FOR NONLINEAR FLOWS 13 α = 0.005, B = 1, n = 2, m = 0.5 α = 0.01, B = 1, n = 4, m = , e , e , e , e , e , e , e , e Table 7. Numerical results for preconditioner Bms. 1 Here we use te van Genucten model k(x, u) = k(x) {1 (α( u /B))n 1 [1+(α( u /B)) n ] m } 2 wit k depicted [1+(α( u /B)) n ] m/2 in te left picture of Figure 2. Te coarse space dimension is 81. α = 0.005, B = 1, n = 2, m = 0.5 α = 0.01, B = 1, n = 4, m = , e , e , e , e , e , e , e , e Table 8. Numerical results for preconditioner Bms. 1 Here we use te van Genucten model k(x, u) = k(x) {1 (α( u /B))n 1 [1+(α( u /B)) n ] m } 2 wit k depicted [1+(α( u /B)) n ] m/2 in te rigt picture of Figure 2. Te coarse space dimension is 81. α = 0.005, B = 1, n = 2, m = 0.5 α = 0.01, B = 1, n = 4, m = , e e , e e , e e , e e Table 9. Numerical results for preconditioner Bem. 1 Here we use te van Genucten model k(x, u) = k(x) {1 (α( u /B))n 1 [1+(α( u /B)) n ] m } 2 wit k depicted [1+(α( u /B)) n ] m/2 in te left picture of Figure 2. Te coarse space dimension is 81. α = 0.005, B = 1, n = 2, m = 0.5 α = 0.01, B = 1, n = 4, m = , e e , e e , e e , e e Table 10. Numerical results for preconditioner Bem. 1 Here we use te van Genucten model k(x, u) = k(x) {1 (α( u /B))n 1 [1+(α( u /B)) n ] m } 2 wit k depicted [1+(α( u /B)) n ] m/2 in te rigt picture of Figure 2. Te coarse space dimension is 81.

14 14 Y. EFENDIEV, J. GALVIS, S. KI KANG, AND R.D. LAZAROV α = 0.005, B = 1, n = 2, m = 0.5 α = 0.01, B = 1, n = 4, m = , , , , Table 11. Numerical results for preconditioner B 1 lsm. Here we use te van Genucten model k(x, u) = k(x) {1 (α( u /B))n 1 [1+(α( u /B)) n ] m } 2 wit k depicted [1+(α( u /B)) n ] m/2 in te left picture of Figure 2. Te coarse space dimension is 166. α = 0.005, B = 1, n = 2, m = 0.5 α = 0.01, B = 1, n = 4, m = , , , , Table 12. Numerical results for preconditioner B 1 lsm. Here we use te van Genucten model k(x, u) = k(x) {1 (α( u /B))n 1 [1+(α( u /B)) n ] m } 2 wit k depicted [1+(α( u /B)) n ] m/2 in te rigt picture of Figure 2. Te coarse space dimension is 184. k u p p, p = 1, 2, 3,..., 10 is bounded independent of contrast η. We see tat te number of outer iterations stays te same for bot set of parameters for nonlinearities wic means larger α values do not affect te outer iterations. We observe from Tables 13 and 14 (tese use different conductivity fields) tat te condition numbers in Table 14 are smaller tan te corresponding condition numbers in Table 13. Tis is because conductivity field 2 as simpler subgrid structure compared to conductivity field 1. Next, we repeat te numerical experiment using te preconditioner Bem 1 based on te coarse space V0 em and B 1 lsm presented in Tables 15 and 16 wile te results for V lsm wit coarse space V lsm 0. Numerical results for te coarse space Bem 1 are 0 are presented in Tables 17 and 18. As before, we observe tat te number of outer iterations is independent of te contrast. However, te for space V0 em te condition number increases as we increase te contrast. On te oter and, te condition number is independent of contrast wen V0 lsm is used as a coarse space. Moreover, we observe tat te condition number produced by V0 lsm, is only 6 wile te condition number for V0 em is about 400 for η = In conclusion, B 1 lsm provides a truly independent-of-contrast solver. 5. Conclusions In tis paper, we study robust iterative solvers for finite element discretizations of steadystate Ricards equation. We assume tat te nonlinear conductivity field can be written as a product of a nonlinear function and a eterogeneous spatial function tat as ig contrast. Due to spatial eterogeneities, te number of iterations in an iterative metod, in general, will depend on te contrast. To alleviate tis problem, we design and investigate iterative solvers tat converge independent of te pysical parameters (small spatial scales and large contrast). Te proposed iterative solvers consist of outer and inner iterations, as it is commonly done in

15 ROBUST DOMAIN DECOMPOSITION METHOD FOR NONLINEAR FLOWS 15 α = 1, B = 1 α = 2, B = , e , e , e , e , e , e , e , e Table 13. Numerical results for preconditioner B 1 ms. Here we use te Exponential model k(x, u) = k(x)e α(u/b) wit k depicted in te left picture of Figure 2. Te coarse space dimension is 81. α = 1, B = 1 α = 2, B = , e , e , e , e , e , e , e , e Table 14. Numerical results for preconditioner B 1 ms. Here we use te Exponential model k(x, u) = k(x)e α(u/b) wit k depicted in te rigt picture of Figure 2. Te coarse space dimension is 81. α = 1, B = 1 α = 2, B = , e , e , e , e , e , e , e , e Table 15. Numerical results for preconditioner B 1 em. Here we use te Exponential model k(x, u) = k(x)e α(u/b) wit k depicted in te left picture of Figure 2. Te coarse space dimension is 81. α = 1, B = 1 α = 2, B = , e , e , e , e , e , e , e , e Table 16. Numerical results for preconditioner B 1 em. Here we use te Exponential model k(x, u) = k(x)e α(u/b) wit k depicted in te rigt picture of Figure 2. Te coarse space dimension is 81. te literature. Outer iterations, designed to andle nonlinearities, linearize te equation around te previous solution state. We sow tat tis linearization yields contrast independent iterative

16 16 Y. EFENDIEV, J. GALVIS, S. KI KANG, AND R.D. LAZAROV α = 1, B = 1 α = 2, B = , , , , , , , , Table 17. Numerical results for preconditioner B 1 lsm. Here we use te Exponential model k(x, u) = k(x)e α(u/b) wit k depicted in te left picture of Figure 2. Te coarse space dimension is 166. α = 1, B = 1 α = 2, B = , , , , , , , , Table 18. Numerical results for preconditioner B 1 lsm. Here we use te Exponential model k(x, u) = k(x)e α(u/b) wit k depicted in te rigt picture of Figure 2. Te coarse space dimension is 184. procedure. For inner iterations, we use recently developed iterative metods (see [15, 17]) tat converge independent of te contrast. One of main ingredients of tis approac, te construction of coarse spaces, is discussed in details in te paper. Since te same preconditioner was used for every outer iteration, tis makes te overall solution process quite efficient. Numerical results are presented to confirm te teoretical findings. In te future, we would like to study te time-dependent case and te case wit non-separable nonlinearities and eterogeneities. In te latter, we plan to develop nonlinear local problems tat can identify ig-conductivity regions and include tese features into te coarse space. Appendix A. Proof of Teorems 1, 2, and 3 A.1. Proof of Teorem 1. It was sown in [4] tat for δ > 0 tere exists ɛ > 0 suc tat u, v (35) u W 1 p () (1 + δ) sup, for all 2 p ɛ, 0 v V v W 1 q () were 1 p + 1 q = 1 and δ and ɛ are independent of. Now, we consider a ig-contrast case via a perturbation argument. Define a bilinear form B : Wp 1 () Wq 1 () R by B(u, v) := u, v 1 a(u, v; ). M It follows from Assumption 1 (A) and Hölder s inequality tat ( (36) B(u, v) 1 C ) ( 0 u(x) v(x) dx 1 C ) 0 u M M W 1 p () v W 1 q ().

17 ROBUST DOMAIN DECOMPOSITION METHOD FOR NONLINEAR FLOWS 17 Note tat C 0 /M < 1. Ten, te identity u, v = B(u, v) + 1 a(u, v; ), togeter wit M estimates (35) and (36) yields M ( δ ( 1 C )) 0 u M W 1 p () sup a(u, v ; ). 0 v V v W 1 q () Let δ = C 0 2M C 0, and coose ɛ to be as given in (35) for tis particular coice of δ. Ten, ( M 1 1+δ ( 1 C ) ) 0 M = C 0 /2. Recall tat a(u, v; ) can be very large because of ig contrast. Tis completes te proof. Note tat ɛ and α depend only on te constants C 0, C and M, toug te coercivity bound is independent of te contrast M. A.2. Proof of Teorem 2. (a) For any u V K,p, k(x)λ(x, u ) satisfies te conditions of Teorem 1 wit a constant M 0 suc tat (37) M 0 = sup{k(x)λ(x, s) : s L c p log K}, were is te mes-size of te partition T and c p is te constant in Sobolev s inequality [4], (38) v L () c p log v W 1 p (), for all v W 1 p (). Te constant M 0 exists because of Assumption 1 (A) and (B). Ten, u V K,p implies tat u L () c p log u W 1 p () c p log K and ence sup{k(x)λ(x, u )} M 0. For sufficiently small K (e.g., K = C/c p ) tere is a p > 2 suc tat te inequality (29) in Teorem 1 olds. Ten, a(t u, v ; u ) T u W 1 p () α sup 0 v V v W 1 q () = α sup 0 v V F (v ) v W 1 q () C F W 1 p (). (from Teorem 1) Coose F W 1 p () K/C to get T u W 1 p () K, i.e., T maps V K,p into itself. By Browder fixed point [11], tere exists a solution ũ of equation (7) and it satisfies (39) T ũ = ũ. (b) Now, we sall sow te mapping T is contraction and also tat te contractivity constant is independent of te contrast. Suppose u, v V K,p satisfy a(t u, w; u ) = F (w) and a(t v, w; v ) = F (w). Tus, (40) a(t u, w; u ) a(t v, w; v ) = 0. Since a(,, ) is a bilinear form, from equation (40) we get (41) a(t u T v, w; u ) = a(t v, w; v ) a(t v, w; u ).

18 18 Y. EFENDIEV, J. GALVIS, S. KI KANG, AND R.D. LAZAROV Now, using te definition of a(,, ), te rigt and side of te equation (41) can be written as k(x)(λ(x, v ) λ(x, u )) T v wdx (42) ( k(x)( T v ) 2 λ(x, v ) λ(x, u ) 2 dx ( k(x) q T v 2q dx ( ( ( ( ( k(x)( w) 2 dx k(x) q T v 2q dx k(x)( w) 2 dx ( 2 k(x)( w) 2 dx ( 2q λ(x, v ) λ(x, u ) 2q dx 2 1 (By Hőlder s inequality, q + 1 q = 1) ( 2q C 1 v u 2q 2q dx 2 k(x) q T v 2q dx k(x)( w) 2 dx (By Lipscitz continuity of λ) 2q ( C 1 C 2q 2, (by Sobolev inequality), ( (v u )) 2 2 dx were we ave used te Sobolev inequality u L 2q () C 2q Du L 2 () wit 2q [1, ] for function u wit bounded mean oscillation. Next, we want to bound ( k(x) q T v 2q dx 2q wit some constant wic is independent of te contrast, i.e., te constant doesn t depend on k( ). Now, we make te following assumption, wic is sligtly different tan Assumption 1(C). Assumption 2. Given te equation a(t v, T v, v ) = F (T v ) (see (8)), we assume tat (43) (k(x) T v 2 ) q/2 dx C q F, were C q F 0 as F Wq 1 () 0 for some q > 2. We note tat wen F = 0 ten CF 2 = 0, tus, T v is zero almost everywere. Moreover, if F W 1 2 () is small, ten C2 F is small and C2 F converges to zero as F W 1 2 () goes to zero. Te inequality (43) assumes tat we ave continuity of C q F wit respect to F W 1 2 () for any q > 2 tat is sufficiently close to 2. We note tat T v W 1 q () is bounded by F W 1 q () as sown above. Tis is typically used to sow te contractivity of te map T. Now, we can conclude tat k(x)(λ(x, v ) λ(x, u )) T v wdx (44) ( C ( (v u )) 2 dx were te constant C depends on Lipscitz constant C 1. 2q ( 2 k(x)( w) 2 dx 2 2,

19 ROBUST DOMAIN DECOMPOSITION METHOD FOR NONLINEAR FLOWS 19 Now put w = T u T v, ten left and side of (41) is bounded below, a(t u T v, T u T v, u ) = (k(x)λ(x, u )( (T u T v )) 2 dx (45) C 2 k(x)( (T u T v )) 2 dx. Combine equations (44) and (45), ten we get ( k(x)( (T u T v )) 2 dx C2 1 C ( (v u )) 2 dx Ten, using te Assumption 1 (a), we get ( C ( (T u T v ) 2 dx So we can deduce tat (46) 2 C 1 2 C ( T u T v W 1 2 C C 1 2 C u v W 1 2, ( 2 k(x)( (T u T v )) 2 dx ( (v u )) 2 2 dx. 2. i.e., te mapping T is a contraction if C is cosen sufficiently small (see Assumption 2). A.3. Proof of Teorem 3. From Lemma 1 and Lemma 10 of [17] we ave tat tere is a stable decomposition, tat is, tere exists v 0 V0 lsm, v i V0 (ω i), i = 1,..., N v, suc tat N v ( ) k v 0 + k v i 2 1 C D ω i H 2 k v 2, µ L+1 D i=1 for some positive constant independent of te contrast and µ L+1 = min i µ Li +1. Here we select te first L i smallest eigenvalues of (24). Ten, for a fixed w we ave stable decomposition, N v λ(x, w)k(x) v 0 (x) + λ(x, w)k(x) v i 2 D i=1 ω i ( ) max x D λ(x, w C min x D λ(x, ω) H 2 λ(x, w)k(x) v 2. µ L+1 According to te abstract teory of domain decomposition, see [32, 25], we conclude tat te condition number of te preconditioned matrix is of order ( ) cond(b 1 max x D λ(x, w A) C min x D λ(x, ω) H 2. µ L+1 Furter noting tat te number of nonlinear outer iterations is bounded (see Teorem 2), we conclude tat te proposed iterative procedure converges independent of te contrast. Acknowledgments Te researc of Y. Efendiev, J. Galvis, and R. Lazarov as been supported in parts by award KUS-C , made by King Abdulla University of Science and Tecnology (KAUST). S. Ki Kang and R. Lazarov are also supported in part by te award made by NSF DMS S.K. Kang is grateful to Fraunofer Institute for Industrial Matematics (ITWM) for osting er visit in te Spring D

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