Domain Decomposition Operator Splittings for the Solution of Parabolic Equations

Transcription

1 University of Wyoming Wyoming Scolars Repository Matematics Faculty Publications Matematics Domain Decomposition Operator Splittings for te Solution of Parabolic Equations T. P. Matew Peter Polyakov University of Wyoming, G. Russo J. Wang Follow tis and additional works at: ttp://repository.uwyo.edu/mat_facpub Part of te Matematics Commons Publication Information Matew, T. P.; Polyakov, Peter; Russo, G.; and Wang, J "Domain Decomposition Operator Splittings for te Solution of Parabolic Equations." SIAM Journal on Scientific Computing 19.3, Tis Article is brougt to you for free and open access by te Matematics at Wyoming Scolars Repository. It as been accepted for inclusion in Matematics Faculty Publications by an autorized administrator of Wyoming Scolars Repository. For more information, please contact

2 SIAM J. SCI. COMPUT. c 1998 Society for Industrial and Applied Matematics Vol. 19, No. 3, pp , May DOMAIN DECOMPOSITION OPERATOR SPLITTINGS FOR THE SOLUTION OF PARABOLIC EQUATIONS T. P. MATHEW, P. L. POLYAKOV, G. RUSSO, AND J. WANG Abstract. We study domain decomposition counterparts of te classical alternating direction implicit ADI and fractional step FS metods for solving te large linear systems arising from te implicit time stepping of parabolic equations. In te classical ADI and FS metods for parabolic equations, te elliptic operator is split along coordinate axes; tey yield tridiagonal linear systems wenever a uniform grid is used and wen mixed derivative terms are not present in te differential equation. Unlike coordinate-axes-based splittings, we employ domain decomposition splittings based on a partition of unity. Suc splittings are applicable to problems on nonuniform meses and even wen mixed derivative terms are present in te differential equation and tey require te solution of one problem on eac subdomain per time step, witout iteration. However, te truncation error in our proposed metod deteriorates wit smaller overlap amongst te subdomains unless a smaller time step is cosen. Estimates are presented for te asymptotic truncation error, along wit computational results comparing te standard Crank Nicolson metod wit te proposed metod. Key words. domain decomposition, alternating direction implicit metod, fractional step metod, operator splitting, parabolic equation, partition of unity AMS subject classifications. 65N20, 65F10 PII. S Introduction. Te numerical solution of parabolic partial differential equations by implicit time stepping procedures requires te solution of large systems of linear equations. Tese linear systems need only be solved approximately, provided tat te inexact solutions obtained by using approximate solvers preserve te stability and local truncation error of te original sceme. Toug iterative metods wit preconditioners are a popular way for solving suc linear systems, see [5, 6, 3, 8, 15, 33, 4, 7], in tis paper we consider only approximate solvers tat do not involve iteration at eac time step. Suc approximate noniterative solution metods for parabolic equations include te alternating direction implicit ADI metods of Peaceman and Racford [22] and Douglas and Gunn [12], te fractional step metods FS of Bagrinovskii and Godunov [1], Yanenko [34], and Strang [26], and also more recent one-iteration domain decomposition solvers of Kuznetsov [16, 17], Meurant [21], Dryja [13], Blum, Lisky, and Rannacer [2], Dawson, Du, and Dupont [9], Laevsky [19, 18] and Vabiscevic [28] and Vabiscevic and Matus [29], and Cen and Lazarov [20]. Te metod proposed ere uses te same framework as te classical ADI and FS metods for solving parabolic equations of te form u t + Lu = f using an operator splitting L = L L q ; owever, te splittings cosen ere are based on domain decomposition, unlike classical splittings along coordinate directions. Furtermore, tey are applicable to problems wit mixed derivative terms and on nonuniform grids. Te basic idea is simple. Given a smoot partition of unity {χ k } k=1,...,q subordinate to a decomposition of te domain, as described in section 4, an elliptic operator L, Received by te editors June 26, 1997; accepted for publication in revised form July 12, ttp:// Department of Matematics, University of Wyoming, Laramie, WY matew@ledaig.uwyo.edu, polyakov@ledaig.uwyo.edu, junping@scwarz.uwyo.edu. Dipartimento di Matematica, Universita dell Aquila, Via Vetoio, Loc Coppito, L Aquila, Italy russo@smaq20.univaq.it. 912

3 DOMAIN DECOMPOSITION OPERATOR SPLITTINGS 913 suc as te two-dimensional Laplacian, is split as a sum of several simpler operators: Lu = Δu =L L q u, were L i u = χ i u. Given a splitting, an approximate solution of te parabolic equation is obtained by solving several simpler evolution equations of te form u t + L k u = f k ; see section 3. By comparison, classical splittings, see Rictmyer and Morton [23], of L = Δ are of te form L 1 = 2 x and L 2 2 = 2 y. 2 Our goal in tis paper is to derive truncation error estimates of te proposed metods and to discuss teir stability properties. Te rest of te paper is outlined as follows. In section 2, we describe a parabolic equation and its discretization. In section 3, we describe te classical ADI and FS metods in matrix language. In section 4, we describe domain decomposition operator splittings using a partition of unity and derive asymptotic truncation error bounds and stability properties for te proposed metod. A euristic comparison is ten made between te Crank Nicolson metod and te ADI metod wit domain decomposition splitting. Finally, in section 5, numerical results are presented. 2. A parabolic equation and its discretization. We consider te following parabolic equation for ux, y, t onadomainω R 2 : u t = ax, y u cx, yu + fx, y, t, in Ω [0,T], ux, y, 0 = u 0 x, y, in Ω, ux, y, t = 0, on Ω [0,T], were ax, y isa2 2 symmetric positive definite matrix function wit C 2q 1 Ω entries, cx, y 0isinC 2q Ω, fx, y, tisac 2q Ω [0,T] forcing term, and u 0 x, y is a C 2q Ω initial data. Here q is an integer to be specified later in section 4. Wit tese smootness assumptions, te exact solution ux, y, t will be in C 2q Ω [0,T]. We will denote te elliptic operator by Lu = ax, y u + cx, yu, andte parabolic equation by u t + Lu = f Discrete problem. Let A denote te symmetric positive definite matrix obtained by second-order finite difference discretization of te elliptic operator L on a grid on Ω wit mes size andgridpointsx i,y j : A U ij = Lux i,y j +O 2, were U ij ux i,y j,t denotes te discrete solution. To discretize in time, we apply an implicit sceme suc as te backward Euler or Crank Nicolson sceme, wic results in a large sparse linear system to be solved. For example, te backward Euler metod yields 1 U n+1 U n = A U n+1 + F n+1, or I + τa U n+1 = U n + τf n+1, τ were τ is te time step, U n = {Ux i,y j,nτ} denotes te vector of discrete unknowns at time t n = nτ, andf n = {fx i,y j,nτ} is te discrete forcing term at time t n.as is well known, see [23], te above sceme is unconditionally stable in te Euclidean norm. and convergent wit error: u U be = O τ+o 2. Tus, for transient problems τ = O 2 provides an optimal coice of time step τ for te backward Euler metod.

4 914 T.MATHEW,P.POLYAKOV,G.RUSSO,ANDJ.WANG 2 Similarly, te Crank Nicolson metod yields U n+1 U n τ = A U n+1 + U n 2 + F n+1 + F n 2, wic requires te solution of te linear system I + τ 2 A U n+1 = I τ F 2 A U n n+1 + F n 3 + τ. 2 Te Crank Nicolson metod is also unconditionally stable, see [23], wit error: u U cn = O τ 2 + O 2 yielding second-order convergence wen τ = O. We note tat bot of te above discretizations can be written in te form 4 I + ατa U n+1 + BU n = g n, for suitable coices of constants α, matrices B, and forcing terms g n. In particular, te coefficient matrices can be written in te form I + ατa for α =1orα = 1 2,and tey are symmetric positive definite. Furtermore, since it is known tat te largest eigenvalue of A satisfies λ max A =O 2, it follows tat te condition number of I + ατa is bounded by 1 + cτ 2, for some positive constant c independent of τ and. Tus, for 2 τ 1 systems 1 and 3 are ill conditioned, yet better conditioned tan A. Consequently, if iterative metods are used, preconditioners suc as multigrid [4, 15] or domain decomposition metods may be needed to reduce te number of iterations; see, for example, [5, 6, 3, 8, 33, 7]. However, as mentioned earlier, system 4 for U n+1 needs only to be solved approximately witin te local truncation error. In tis paper we only seek solvers tat provide approximate solutions at te cost of one iteration per time step. Suc approximate solvers will modify te sceme 4 and alter its stability and truncation error. In order to distinguis tis wit te modified scemes, we will encefort refer to sceme 4 as te original sceme. In sections 3 and 4, we discuss one-step approximate solvers, and estimate teir truncation errors using te framework of te classical ADI and FS metods. We also describe conditions under wic te stability of te original sceme is preserved. Given te original discretization 4, we define its local truncation error T orig as follows: 5 T orig =I + ατa u n+1 + Bu n g n, were u is te exact solution of te parabolic equation restricted to te grid points. Discretization 4 is said to be stable in a norm if te following olds for some numbers c 1,c 2 : U n c 1 τ U n + c 2 g n. If c 1 and c 2 are independent of and τ, ten te discretization is said to be unconditionally stable, elseconditionally stable. 3. ADI and FS metods. Given an evolution equation of te form u t +Lu = f, te basic idea in te ADI and FS metods is to approximately solve it using an operator splitting L = L L q, were several simpler evolution equations of te form w t + L i w = f i are solved wit suitably cosen f i to provide an approximate solution w u to te original equation.

5 DOMAIN DECOMPOSITION OPERATOR SPLITTINGS 915 We will encefort restrict our discussion to te matrix case, were a discretization in space of u t + Lu = f yields U t + A U = F, and a subsequent discretization in time yields system 4. Corresponding to an operator splitting L = L L q, were q > 1, we assume a matrix splitting of A as a sum of symmetric positive semidefinite matrices A i : 6 A = A A q, were A i 0. It will be assumed tat te matrices A i ave a simpler structure tan A,inte sense to be explained below. In matrix language, given te splitting {A i }, bot te ADI and FS metods approximately solve systems of te form I + ατa w = z by solving several systems of te form I + ατa i w i = z i ; see sections 3.1 and 3.2. Hence, te A i must be suitably cosen so tat te latter systems are easier to solve tan te original system. As an example of splittings, we briefly describe te classical splittings, see [23], of an elliptic operator along te x and y coordinate directions wen no mixed derivative terms are present. Domain decomposition splittings are described in section 4. For classical splittings, te two-dimensional Laplacian L = Δ is split as a sum of L 1 = 2 x 2 and L 2 = 2 y 2.Inmatrixterms A = Δ = D xx D yy, were Δ denotes te discrete Laplacian and D xx and D yy denote te discretization of 2 x 2 and 2 y 2, respectively. Bot A 1 = D xx and A 2 = D yy will be tridiagonal matrices after a suitable ordering of te unknowns if te grid is uniform and ence I + ατa i will also be tridiagonal and solvable in linear time on uniform grids! Unfortunately, suc splittings are not possible if mixed derivative terms are present and tridiagonality can be lost if te grid is nonuniform. By contrast, te domain decomposition splittings described in section 4 are applicable in more general cases, toug te resulting linear systems are not tridiagonal ADI metod. Tis metod was originally proposed by Peaceman and Racford [22] and Douglas [11, 10], and was furter developed by several autors. We follow ere te general formulation described by Douglas and Gunn [12], were additional references may be found. Te reader is also referred to [30, 31, 32, 14, 24, 27]. In matrix language, given a splitting A = A A q and an approximate solution W n U n to te discrete solution at time t n, an approximate solution W n+1 U n+1 of te linear system I + ατa U n+1 = g n BU n is obtained by solving systems of te form I + ατa i w i = z i as follows. ADI ALGORITHM. 1. Given W n, solve for W n+1 1 : I + ατa 1 W n For i =2,...,q, solve for W n+1 i : q ατa j W n + BW n = g n. j=2 I + ατa i W n+1 i W n+1 i 1 ατa iw n =0.

6 916 T.MATHEW,P.POLYAKOV,G.RUSSO,ANDJ.WANG 3. Define te ADI approximate solution at time t n+1 =n +1τ to be W n+1 = W n+1 q. We note tat in order to compute W n+1 in te ADI sceme, we need to solve a total of q linear systems, eac of te form I + ατa i of size n n tesizeofa. Te following stability results for te ADI metod can be found in [12]. THEOREM 3.1. If te matrices A i are symmetric positive semidefinite for i = 1,...,q,ifB is symmetric, and if A i and B commute pairwise, ten te ADI sceme is unconditionally stable. In te noncommuting case, if q =2and if te matrices A i are symmetric positive semidefinite, and if is any norm in wic te original sceme 4 is stable, ten te ADI sceme is stable in te norm u I+ατA 2 u. For q 3, if te matrices A i are positive semidefinite, ten te ADI sceme can be sown to be conditionally stable. For q 3 in te noncommuting case, examples are known of positive semidefinite splittings for wic te ADI metod can loose unconditional stability; see [12]. Neverteless, te above stability results are a bit pessimistic and instability is only rarely encountered in practice. Next, we describe te truncation error terms for te ADI metod, derived in [12]. Te magnitude of tese truncation terms will be estimated in section 4.3 for domain decomposition splittings. THEOREM 3.2. Given te discrete problem I + ατa U n+1 + BU n = g n, wit local truncation error T orig see 5, te ADI solution W n+1 solves te following perturbed problem: q I + ατa W n+1 + α m τ m A σ1 A σm W n+1 W n 7 m=2 +BW n = g n, 1 σ 1< <σ m q wic introduces te following additional terms in te local truncation error: 8 q T adi = T orig + α m τ m A σ1 A σm u n+1 u n, m=2 1 σ 1< <σ m q were u is te exact solution of te parabolic equation restricted to te grid points. Proof. Te proof is given in [12]; owever, for completeness we include it ere. From step 2 of te ADI algoritm, we obtain tat W n+1 i satisfies I + ατa i W n+1 i W n =W n+1 i 1 W n for i =2,...,q. By recursively applying tis, we deduce tat 9 I + ατa 2 I + ατa q Wq n+1 W n =W1 n+1 W n. Now, from step 1 of te ADI algoritm, we deduce tat 10 I + ατa 1 W n+1 1 W n =g n BW n q j=2 ατa jw n I + ατa 1 W n = g n BW n I + ατa W n.

7 DOMAIN DECOMPOSITION OPERATOR SPLITTINGS 917 Multiplying 9 by I + ατa 1 and substituting 10 for te resulting rigt-and side, we obtain 11 Noting tat I + ατa 1 I + ατa q W n+1 q W n =g n BW n I + ατa W n. I + ατa 1 I + ατa q =I + ατa + q α m τ m m=2 1 σ 1< <σ m q A σ1 A σm, 7 follows. Explicit estimates of tese truncation error terms for te case of domain decomposition splittings are given in section FS metods. Tese metods were originally developed by Bagrinovskii and Godunov [1] and Yanenko [34] and furter developed by Strang [26] and oter autors. Given a matrix splitting A = A A q, te FS metods approximate I + ατa by a product of several matrices of te form I + ατa i. For example, te first-order FS metod uses te approximation 12 I + ατa =I + ατa 1 I + ατa q +Oτ 2, wic can be formally verified by multiplying all te terms, bearing in mind tat te matrices A i may not commute. Given 12, an approximate solution W n+1 U n+1 of 13 I + ατa U n+1 + BU n = g n can be obtained by solving 14 I + ατa 1 I + ατa q W n+1 = g n BW n. We summarize below te first-order fractional step algoritm to approximately solve I + ατa U n+1 = g n BU n wit W n+1 fs U n+1. FIRST-ORDER FS ALGORITHM. 1. Let z 1 = g n BW n and for i =1,...,q solve I + ατa i w i = z i, and define z i+1 = w i. 2. Let W n+1 fs = w q. Toug 14 is a second-order approximation locally in τ of 13, it is only firstorder accurate globally, due to accumulation of errors; see [23]. Te following result concerns te stability of te first-order FS metod. THEOREM 3.3. If A i are symmetric and positive semidefinite, and if B 1 in te Euclidean norm., ten te first-order FS metod is stable wit W n+1 W n + c g n for some c>0 and independent of τ and. Proof. SinceW n+1 =I +ατa q 1 I +ατa 1 1 g n BW n and since B 1 in te spectral norm, we only need verify tat I + ατa q 1 I + ατa 1 1 1

8 918 T.MATHEW,P.POLYAKOV,G.RUSSO,ANDJ.WANG in te spectral norm. But tis follows easily: since eac of te terms I + ατa i are symmetric positive definite wit eigenvalues greater tan one, te spectral norms of teir inverses are eac bounded by one. Te following result concerns te local truncation error of te first-order FS metod. THEOREM 3.4. Te first-order FS approximate solution W n+1 of I + ατa U n+1 = g n BU n solves 15 I + ατa W n+1 + q m=2 α m τ m 1 σ1< <σm q A σ 1 A σm W n+1 = g n BW n. Te local truncation error T fs of te first-order FS metod as te following terms in addition to te terms in te local truncation error T orig wen exact solvers are used: 16 q T fs = T orig + α m τ m A σ1 A σm u n+1. m=2 1 σ 1< <σ m q Here u n+1 is te exact solution of te parabolic equation at time n +1τ restricted to te grid points. Proof. Since te product q I + ατa 1 I + ατa q =I + ατa + α m τ m A σ1 A σm, m=2 1 σ 1< <σ m q te proof immediately follows by replacing I + ατa byi + ατa 1 I + ατa q in te original discretization. For a given splitting {A i }, te above local truncation error terms are Oτ 2 toug te constant in te leading-order term τ 2 can vary for different coices of splittings. However, te global error will be Oτ, see [23], due to accumulation of errors, and consequently, te above first-order FS metod is suitable only for globally first-order scemes suc as backward Euler but not for globally second-order discretizations suc as Crank Nicolson. For globally second-order accurate scemes suc as Crank Nicolson, second-order fractional step approximations can be obtained by using Strang splitting [26]. We outline te main idea for a splitting involving two matrices: A = A 1 + A 2. Te aim is to approximate e ατa by a locally tird-order accurate approximation as follows: 17 e τa = e τ 2 A1 e τa2 e τ 2 A1 + Oτ 3. Ten, as in te Crank Nicolson metod, a locally tird-order approximation in τ of te terms e ατai can be used to replace eac of te exponentials e ατai = I + ατ 1 2 A i I ατ 2 A i + Oτ 3. Applying tis to eac of te terms in 17 yields te implementation of Strang splitting. We state witout proof te following.

9 DOMAIN DECOMPOSITION OPERATOR SPLITTINGS 919 THEOREM 3.5. Strang splitting provides unconditionally stable approximations wen A i are symmetric positive semidefinite: A i 0. For any fixed coice of positive semidefinite matrices A 1 and A 2, te truncation error for Strang splitting is tirdorder accurate locally in τ, and globally second-order accurate in τ. Since Strang splitting provides a globally second-order accurate approximation in time, it can be applied to inexactly solve te Crank Nicolson sceme. However, it is more expensive and complicated to generalize wen tere are more tan two matrices in te splitting, and it requires more linear systems to be solved tan te first-order FS metod. In te applications we consider, te ADI metods wic were described earlier are preferable and generate smaller truncation errors, toug tey are not guaranteed to be unconditionally stable witout additional assumptions. 4. Domain-decomposition-based operator splittings. In tis section, we describe domain-decomposition-based splittings A = A A q of te discretization A of L suc tat I + ατa i w i = z i can be solved at te cost of solving a problem on a subdomain. For suc splittings, we provide in section 4.3 an explicit estimate on ow te truncation error depends on te overlap β amongst te subdomains. For related algoritms, we refer te reader to Dryja [13], Laevsky [19, 18], Vabiscevic [28] and Vabiscevic and Matus [29], Kuznetsov [17, 16], Blum, Lisky, and Rannacer [2], Meurant [21], Dawson, Du, and Dupont [9], and Cen and Lazarov [20]. As noted in te introduction, given a smoot partition of unity {χ k x, y} k=1,...,q subordinate to a collection of open subdomains {Ω k } k=1,...,q wic cover Ω, we split an elliptic operator L as a sum of several simpler operators as follows: 18 Lu = ax, y u + cx, yu = q k=1 χ kx, yax, y u+χ k x, ycx, yu = L 1 u + + L q u, were L k u = χ k x, yax, y u+χ k x, ycx, yu. In wat follows, we will denote te coefficients of L k by a k x, y =χ k x, yax, y andc k x, y =χ k x, ycx, y. For numerical purposes, we could replace a smoot partition of unity by one wic is piecewise smoot. In suc a case, it may be necessary to define te operators L k weakly using a variational approac. Te matrices A k are obtained by discretization of L k Partition of unity. Givenanoverlappingcollectionofopen subregions {Ω k } k=1,...,q wose union contains Ω, a partition of unity subordinate to tis covering is a family of q nonnegative, C Ω smoot functions {χ k x, y} k=1,...,q wose sum equals one and suc tat χ k x, y vanises outside Ω k : 0 χ k x, y 1, χ 1 x, y+ + χ q x, y =1, supp χ k x, y Ω k. For numerical purposes, we will consider χ k x, y wic are not necessarily C Ω but are continuous and piecewise smoot. We will refer to suc a partition of unity as piecewise smoot. For more information on partitions of unity, see Sternberg [25]. Construction of piecewise smoot partitions of unity. Given a collection of overlapping subdomains {Ω k } k=1,...,q wic form a covering of Ω, we wis to describe ow a piecewise smoot partition of unity can be constructed. However, first we describe ow to construct an overlapping covering {Ω k } k=1,...,q.

10 920 T.MATHEW,P.POLYAKOV,G.RUSSO,ANDJ.WANG Ω 1 Ω 2 Ω 3 Ω 4 Ω 1 Ω 3 Ω 5 Ω 6 Ω 7 Ω 8 Ω 9 Ω 10 Ω 11 Ω 12 Ω 9 Ω 11 Ω 13 Ω 14 Ω 15 Ω Subdomains Color 1 FIG. 1.16nonoverlapping subdomains Ω i ; enlarged subdomains Ω k of color Let {Ω k } k=1,...,q denote a nonoverlapping collection of subregions of Ω, eac of widt H, witω= q i=1 Ω i and Ω i Ω j = for i j. For example, te subdomains Ω k can be cosen to be te coarse elements in a coarse grid triangulation of Ω wit mes size H; see Figure Next, enlarge eac subdomain Ω k to Ω k to contain all points in Ω witin a distance β of Ω k,wereβ>0; see Figure 1. Ten {Ω k } will form an overlapping covering of Ω: Ω = q k=1 Ω k. We note ere tat te subdomains {Ω k } can often be grouped into a small number of colors so tat any two subdomains of te same color are disjoint, see Figure 1, for a coloring of 16 overlapping subdomains into four colors q = 4, only color 1 is indicated. Once {Ω k } k=1,,q are formed, a piecewise smoot partition of unity {χ k } k=1,...,q can be constructed as follows: 1. Inside Ω k,letω kx, y denote te distance of x, y to te boundary Ω k : ω k x, y = { distx, y, Ω k, x, y Ω k, 0, x, y Ω k. Note tat 0 ω k x, y is a continuous function wit support in Ω k. 2. Ten define χ k x, y by normalizing te ω k x, y in order tat teir sum equals one: χ k x, y ω k x, y q j=1 ω jx, y for k =1,...,q. Te above constructed {χ k x, y} will be continuous and piecewise smoot wit discontinuous derivatives across k Ω k. To obtain C Ω smoot partitions of unity, we refer te reader to [25]. Before we proceed furter, we define te Hölder norms for integer k 0 as follows: u Ck Ω sup max α u, α k Ω

11 DOMAIN DECOMPOSITION OPERATOR SPLITTINGS 921 wereweaveusedtemulti-indexnotation α = α x α1 y α2, α =α 1,α 2. For smoot partitions of unity wit overlap β, te following result olds. PROPOSITION 4.1. Given a suitable collection of overlapping subregions, {Ω k } k=1,...,q wit overlap β, tere exists a smoot partition of unity {χ k x, y} k=1,...,q satisfying 19 χ k Cm Ω c m β m in te Hölder C m Ω norm, were c m > 0 is a constant independent of k and β but dependent on te geometry of te region. Proof. We sketc te main ideas. 1. First, map Ω to an enlarged region Ω using te cange of variables x, ỹ = x/β, y/β. Tis mapping enlarges Ω, and consequently all its subregions {Ω k }, by a factor of β 1. Denote by { Ω k } k=1,...,q te corresponding covering of Ω. Tese subregions will ave an overlap parameter of size O1. 2. Next, construct a C Ω partition of unity { χ k } k=1,...,q on Ω, subordinate to te covering { Ω k } k=1,...,q; see for instance Sternberg [25]. Tis partition of unity on Ω defines a corresponding C Ω partition of unity on Ω as follows: χ k x, y χ k x, ỹ for k =1,...,q. 3. By repeated applications of te cain rule, it follows tat α χ k x α1 y α2 = β α α χ k x α1 ỹ α2. Using te definition of te Hölder norm it follows tat χ k Cm Ω β m χ k Cm Ω = c mβ m, were c m is defined by c m χ k C m Ω. We omit furter details. For numerical implementation, smoot partitions of unity can be replaced by piecewise smoot ones based on te distance functions w k x, y. See section 5 for an example of an alternative partition of unity for rectangular subdomains Domain decomposed matrix splittings. As mentioned before, given a partition of unity {χ k } k=1,...,q as in section 4.1, a domain decomposition splitting of te operator Lu = ax, y u cx, yu is obtained as L k u = a k x, y u+ c k x, yux, y. By construction, q k=1 L k = L, and eac L k are self-adjoint. A k are ten defined as suitable discretizations of te differential operators L k,andso A = q k=1 A k. Wenotetatifcx 0, ten eac L k and A k will be positive semidefinite. Te A k constructedabovecantenbeusedinteadiandfsmetods described in section 3, were te solution of I +ατa k w = z corresponds to solving a problem on subregion Ω k. Additionally, by coloring, problems on disjoint subregions of te same color can be solved in parallel.

12 922 T.MATHEW,P.POLYAKOV,G.RUSSO,ANDJ.WANG 4.3. Truncation errors of te ADI and FS metods for domain decomposition splittings. As described in sections 3.1 and 3.2, wen te ADI and FS metods are used to approximately solve a discretization 20 I + ατa U n+1 = g n BU n, te resulting approximate solutions W n+1 adi and W n+1 fs solve te perturbed problems 7 and 15, respectively. Tese perturbed problems correspond to new discretizations of te original parabolic equation u t + Lu = f wit additional truncation terms of te form τ m A σ1 A σm W n+1 W n andτ m A σ1 A σm W n+1 for te ADI and FS metods, respectively. Our goal in tis section is to estimate te magnitude of tese additional terms for te case of domain decomposition splittings. As is well known, see for instance Rictmyer and Morton [23], once te local truncation errors T adi and T fs are estimated, te global errors E adi t = ut W adi t and E fs t = ut W fs t will follow by Lax s convergence teorem, if te scemes are stable. We note, owever, tat te global error will be larger tan te local truncation errors by a factor of τ 1 due to te accumulation of truncation errors. Since te local truncation errors T orig for te case of backward Euler or Crank Nicolson wit exact solvers are known, our estimates will focus on te new truncation terms A σ1 A σm W and A σ1 A σm W n+1 W n. For simplicity, we will assume tat cx, y =0andtatax, y is a smoot scalar function. Te analysis for te cases wen cx, y 0 and for smoot matrix functions ax, y is similar. Our analysis will require te following smootness assumptions for te functions ax, y, W x, y, t, and {χ k x, y}; ax, y sould be in C 2q 1 Ω, W x, y,. sould be in C 2q 0 Ω, and {χ k x, y} sould be in C 2q 1 Ω. Wit some abuse of notation, W will also refer to W = {W x i,y j,.}, te vector of nodal values of W x, y,. on te grid points of Ω. Our estimates will be based on two simple observations, te details of wic will be provided subsequently. 1. Te finite difference approximations A σ1 A σm W wic occurs in T fs and A σ1 A σm W n+1 W n wic occurs in T adi can be sown to be similar to teir continuous counterparts L σ1 L σm W and L σ1 L σm W n+1 W n, respectively. Tis result is obtained by repeated applications of te integral form of te mean value teorem to te difference quotients. 2. Using step 1, te expressions A σ1 A σm W and A σ1 A σm W n+1 W n can be estimated in terms of te maximum values of te derivatives of W and te derivatives of te coefficients {a k x, y}. Te derivatives of {a k x, y} can be estimated in terms of te derivatives of ax, y and of te partition of unity functions {χ k x, y}. Using te bounds 19 for te partition of unity, we can ten obtain tat A σ1 A σm W ij Caβ 2m+1 W C 2m Ω, were Ca is a positive constant tat depends on te coefficients ax, y and its derivatives. Similar estimates will be valid for A σ1 A σm W n+1 W n but wit an additional factor of τ due to te difference in time W n+1 W n. Before we proceed, we introduce some notation. In addition to multi-index derivatives, α = α x α 1 y α 2, we will use multi-index integrals I x F x i,y j = 1 xi+ x i F ξ,y j dξ.

13 DOMAIN DECOMPOSITION OPERATOR SPLITTINGS 923 Iterated integrals are defined as follows: We also define I x, 1 2 I xx F x i,y j = 1 xi 2 x i as ξ+ ξ F η, y j dηdξ. I x, F x 1 i,y j = 1 2 xi+ 2 x i 2 F ξ,y j dξ. Similar integrals are defined in y. We now outline te details of step 1. By repeatedly applying te integral form of te mean value teorem or te fundamental teorem of calculus, one can express all te difference quotients in A σ1 A σm W in terms of integrals of derivatives of W and integrals of te derivatives of te coefficients a σk x, y. We ave te following result. LEMMA 4.2. Let W C0 2m Ω and let te expansion of L σ1 L σm W be of te form L σ1 L σm W = C α1,...,α m+1 α1 a σ1 αm a σm αm+1 W, α 1,...,α m+1 α 1 1 α α m+1 =2m were eac C α1,...,α m+1 =0or 1. Ten,forW x, y te product A σ1 A σm W satisfies A σ1 A σm W ij = α 1,...,α m+1 α 1 1 α α m+1 =2m C α1,...,α m+1 I I I α1 a σ1 I αm a σm I αm+1 W, were eac of te I terms wit at most 2m suc terms results from a suitable coice of single or multiple integrals I. Proof. Te proof will follow by induction on m. For m = 1, te standard fivepoint finite difference approximation A k W i,j of L k W x i,y j satisfies 21 A k W i,j = a kx i+ 2,yj Wi+1,j W i,j a k x i 2,yj Wi,j W i 1,j a kx i,y j+ 2 Wi,j+1 W i,j a k x i,y j 2 Wi,j W i,j 1, 1 xi+ A k W i,j = 2 a k x i x ξ,y jdξ 2 a k x i 2,y 1 xi j 2 x i 1 yj+ 2 a k y j y x i,ξdξ 2 a k x i,y j 2 1 yj 2 y j 1 ξ+ ξ 1 ξ+ ξ x ξ,y jdξ 2 W x η, y 2 j dηdξ W y x i,ξdξ 2 W y x 2 i,ηdηdξ. xi+ W x i yj+ y j Tis follows trivially by repeatedly applying te fundamental teorem of calculus to te difference quotients in te finite difference discretization. For instance, 1 W i+1,j

14 924 T.MATHEW,P.POLYAKOV,G.RUSSO,ANDJ.WANG xi+1 W W i,j = 1 x i x ξ,y jdξ, and similarly 1 W i,j+1 W i,j = 1 Next, we subtract and add te terms a k x i 2,y j 1 xi+1 2 x i yj+1 W y j y x i,ξdξ. W x ξ,y jdξ and a k x i,y j yj+1 W 2 y j y x i,ξdξ to te resulting terms in te discretization and apply te fundamental teorem of calculus again. Result 21 follows. We can restate equation 21 using te multi-index derivative and integral notation as 22 A k W i,j = I x, 1 2 x a k I x xw a k x i 2,y j I xx 2 xw I y, 1 y a k I y yw a k x i,y j 2 I yy 2 yw. 2 But tis is just an expression for L k W wit multi-index integrals applied to its terms. Consequently, our proof for te case m = 1 is complete. For m 2, we use te induction ypotesis and assume tat te result olds true for m 1. We will ten sow tat te result olds for m. To do tis, define V = A σ2 A σm W. Ten, by te induction ypotesis V = C α2,...,α m+1 I I I α2 a σ2 I αm a σm I αm+1 W. α 2,...,α m+1 α 2 1 α α m+1 =2m 2 Now, A σ1 V can be calculated by using 22 wic involves first- and second-order partial derivatives of V x, y locally. Te resulting expression can be simplified by te following observation. Te partial derivative operators α commute wit all te integral operators I x, Iy,etc. Te commutativity of x and I y follows from te differentiation rule for integrals depending on a parameter. Te commutativity of x and I x can be seen by te following example: x IF x 1 x+ x, y j = x F ξ,y j dξ = 1 x+ ξ F ξ,y j dξ = I x x F x, y j. x Wen iger-order partial derivatives and iterated integrals are involved, tey can be andled by repeated applications of te above. Consequently, all te partial derivatives can be moved inside te integrals. Wen all te partial derivatives are moved inside te integrals, we obtain an expression similar to L σ1 L σm W but containing integrals of te terms. Tis gives te desired result for m. Tis completes te sketc of te proof of step 1. We obtain step 2 immediately from Lemma 4.2 by applying te mean value property of integrals. LEMMA 4.3. Te discretization A σ1 A σm W ij satisfies A σ1 A σm W ij C α1,...,α m+1 a σ1 C α 1 a σm C αm W C α m+1, α 1,...,α m+1 α 1 1 α α m+1 =2m were C α1,...,α m+1 =0or 1. Proof. Te mean value property of integrals yields 1 xi+ F ξ,y j dξ max F ξ,y j. ξ [x i,x i+] x i x

15 DOMAIN DECOMPOSITION OPERATOR SPLITTINGS 925 Applying tis to all te integrals I in Lemma 4.2, we obtain tat α 1,...,α m+1 α 1 1 α α m+1 =2m α 1,...,α m+1 α 1 1 α α m+1 =2m A σ1 A σm W ij C α1,...,α m+1 α1 a σ1 C0 Ω αm a σm C0 Ω αm+1 W σm C0 Ω C α1,...,α m+1 a σ1 C α 1 Ω a σ m C αm Ω W σm C α m+1 Ω. Tis gives us te desired result. Using Leibnitz s rule for partial differentiation of a k x, y =χ k x, ya k x, y and applying bounds 19 for te partition of unity, we deduce tat a k x, y Cp Ω Ca a Cp Ω χ k Cp Ω Caβ p. Here Ca is a constant tat depends only on te coefficients ax, y and some of its derivatives. By substituting tese results in Lemma 4.3, we obtain te following. COROLLARY 4.4. If W C0 2m Ω, te discretization A σ1 A σm W ij satisfies A σ1 A σm W ij Caβ 2m+1 W C 2m Ω. Similarly, te discretization A σ1 A σm W n+1 W n ij satisfies A σ1 A σm W n+1 W n ij Caτβ 2m+1 sup t W C 2m Ω. [0,t f ] Tis completes te details of step 2. We are finally able to estimate te local truncation errors and te global errors of te ADI metod wit domain decomposition splitting. THEOREM 4.5. Let te exact solution ux, y, t of te parabolic equation be in C 2q 0 Ω for eac t [0,t f ] and in C 1 [0,t f ] for eac x, y. Ten te local truncation error T adi of te ADI metod wit domain decomposition splitting satisfies 1 T adi L2 Ω T orig L2 Ω + Caτ 3 β 3 + τ q 2 τ + + β5 β 2q 1 sup t u C 2q Ω, [0,t f ] were T orig is te local truncation error of te original sceme 4 wen exact solvers are used. Wenever te ADI sceme is stable, te error E adi t =ut W adi t satisfies 23 1 E adi t L2 Ω E orig t L2 Ω+Caτ 2 β 3 + τ q 2 τ + + β5 β 2q 1 sup t u C 2q Ω [0,t f ] for t [0,t f ].

16 926 T.MATHEW,P.POLYAKOV,G.RUSSO,ANDJ.WANG Proof. By 8, te local truncation error satisfies T adi = T orig + q m=2 α m τ m 1 σ 1< <σ m q A σ1 A σm u n+1 u n. Considering te Euclidean norm of te above expression, and estimating te additional terms using Corollary 4.4, we obtain T adi T orig + nca q m=2 τ m+1 β 2m+1 sup [0,tf ] t u C 2m Ω T orig + ncaτ 3 1 β + τ 3 β + + τ q 2 5 β sup 2q 1 [0,tf ] t u C 2q Ω, were n is te number of grid points in Ω. Multiplying te entire expression by and using tat is equivalent to L2 Ω, we can replace T adi and T orig by T adi L2 Ω and T orig L2 Ω, respectively. Te desired estimate for te local truncation error ten follows by noting tat n is proportional to te square root ofteareaofω,wiccanbeabsorbedinca. Te error follows from te local truncation error by an application of te Lax convergence teorem. Remark 1. Eac of te terms α1 a σ1 αm a σm αm+1 W will be zero in most of Ω, more specifically, outside m k=1 Ω σ k, and our proof as not taken tis into account. Tis is due to te compact support of te coefficients a σk x, y inω σ k. However, we expect tat an estimate taking tis into account will be muc more complicated. Consequently, our teoretical estimate for te truncation errors are pessimistic wen compared wit actual numerical results from section 5. Remark 2. For a fixed coice of subdomains {Ω k } and partition of unity {χ k}, te overlap β is fixed. Consequently, te above truncation error bounds for te ADI metod wit domain decomposition splitting is asymptotically second order in time, i.e., as τ 0 ut W adi t L2 Ω Ca 2 + τ q 2 τ + + β3 β 2q 1 sup t u C 2q Ω. [0,t f ] Remark 3. As te number of subdomains increase, and teir diameters H and overlap β decrease i.e., β,h 0, te accuracy of te proposed ADI metod deteriorates. However, since all te terms involving β aremultipliedbypowersofτ, wecan coose a smaller time step τ adi for te ADI metod so tat te resulting global error is O 2, te same as for te Crank Nicolson metod wit exact solvers. A simple calculation yields tat substituting produces a global error E adi t satisfying τ adi β 2q 1 q, were β<1, E adi t L2 Ω β 4q 2 q 2 sup t u C 2q Ω. [0,t f ] + 2 β 4q 2 q q β 2q2 q q 2q+1 sup t u C 2q Ω, [0,t f ]

17 DOMAIN DECOMPOSITION OPERATOR SPLITTINGS 927 However, since τ adi <, te ADI metod will require more time steps tan te Crank Nicolson metod. For a euristic comparison of te complexity of te ADI metod wit te standard Crank Nicolson metod, see section 4.4. Te truncation errors of te first-order FS metod can be analyzed similarly. Tey are larger tan te corresponding errors for te ADI metod by a factor of τ 1. THEOREM 4.6. Te local truncation error T fs of te FS metod wit domain decomposition splitting satisfies 1 T fs L2 Ω T orig L2 Ω + Caτ 2 β 3 + τ q 2 τ + + β5 β 2q 1 sup t u C 2q Ω, [0,t f ] were u is te exact solution of te partial differential equation. satisfies Te global error E fs t L 2 Ω ut W fs t L 2 Ω 1 E orig t L 2 Ω + Caτ β + τ 3 β + + τ q 2 5 β sup 2q 1 [0,tf ] t u C 2q Ω. Proof. Te proof follows by an application of Lemma 4.3 to te truncation error of te first-order FS metod A comparison of te Crank Nicolson metod wit te proposed ADI metod. In tis section, we euristically compare te work complexity total floating point operations of te Crank Nicolson metod using exact solvers wit tat of te ADI metod using domain decomposition splitting. Our comparison will be based on calculating te cost of eac metod for computing te solution up to time t f and to te same specified accuracy. In order to obtain an explicit comparison, we make te following assumptions: 1. An algebraic direct solver is used to solve te linear systems occurring in bot te Crank Nicolson metod and in te ADI metod wit domain decomposition splitting. Te cost of te direct solver to solve a linear system in n unknowns is assumed to be C α n α,were1 α Te number of subdomains in te ADI metod is cosen to be N d,forsome positive integer N, wered is te space dimension d =2ord =3. Te diameter H and overlap β of eac subdomain Ω k is cosen to be proportional to H N 1 and β N Te number of unknowns in eac subdomain Ω k is assumed to be n/n d M, were M>1 is some magnification factor tat enlarges te number of unknowns in eac subdomain. 4. Tetimestepτ cn of te Crank Nicolson metod is cosen to be τ cn =. In order to ensure tat bot te Crank Nicolson metod and te ADI metod ave similar errors, see Remark 3 from section 4.3, te time step τ adi of te ADI metod is cosen to be τ adi β 2q 1 q. For te above coices, bot metods ave global errors satisfying E cn E adi 2.

18 928 T.MATHEW,P.POLYAKOV,G.RUSSO,ANDJ.WANG Finally, since we cose β N 1 in assumption 2, te ADI time step becomes τ adi N 2q 1 q. For bot metods, te cost of computing te solution up to time t f satisfies Cost = Number of time steps Cost per time step. For te Crank Nicolson metod we obtain Cost of CN = tf C α n α, were n is te number of space unknowns. For te ADI metod wit domain decomposition splitting, we obtain tf Cost of ADI = τ adi N d C Mn α α, N d t f N d C Mn α N 2q 1 α, N d q tf C αn α M α N 2q 1 q αd+d, Cost of CN M α N 2q 1 q αd+d, were M>1iste constant magnification factor and q is te number of colors in te domain decomposition splitting; see section 4.1. Asymptotically, wen 1 N d <n, we obtain tat M α N 2q 1 q αd+d < 1, if 1 αd+ 2q 1 < 0, q or equivalently if α>1+ 2q 1 qd. Tus, in two dimensions d = 2, te ADI metod will cost asymptotically less tan Crank Nicolson if te direct algebraic solver costs C α n α wit α>2 1 2q. In tree dimensions, te ADI metod will be preferable if α> q. Remark 4. Numerical tests in section 5 indicate tat a more reasonable time step for te ADI metod is τ adi β, wic leads to similar global errors for te ADI and Crank Nicolson metods. For suc coices of time steps, te ADI metod will be competitive wit te Crank Nicolson metod if te complexity of te direct solvers is C α n α were { 3 2 d =2, were Ω R d. α> 4 3 d =3, 5. Numerical results. We report ere on te results of numerical tests conducted using te generalized ADI sceme and te first-order FS metod described in section 3, using te domain decomposition splittings of section 4. Our goal in tese

19 DOMAIN DECOMPOSITION OPERATOR SPLITTINGS 929 TABLE 1 Two nonoverlapping subdomains: Errors at time t f =1;τ cn = τ adi = CN ADI FS tests was to compare te accuracies at a fixed time t f,sayt f =1,ofteCrank Nicolson solution wen exact solvers are used wit tat of te ADI and first-order FS solutions. Accordingly, we tabulate te global errors for te exact solver-based Crank Nicolson metod and te ADI and first-order FS metods. To enable computing te global error, we use te eat equation wit a known exact solution ux, y, t =e t sinπxsinπy on te unit square Ω = [0, 1] 2 : 24 u t = Δu +1+2π 2 e t sinπxsinπy on Ω [0,T], ux, y, t =0 = sinπxsinπy on Ω, ux, y, t = 0 on Ω [0,T]. Te standard five-point finite difference sceme was used to discretize in space on a uniform grid of mes size. Te Crank Nicolson metod was used to discretize in time wit time steps τ cn as indicated in te tables. In te tables, we tabulate te scaled Euclidean norm scaled according to area of te global error at time t f =1for te Crank-Nicolson solution using exact solvers denoted by CN, and Crank Nicolson wit inexact ADI-based solvers denoted by ADI, and Crank Nicolson wit inexact first-order fractional step solvers denoted by FS. Partitions of unity: Te subdomains Ω k cosen in our experiments were rectangular. For te case of overlapping subregions, we constructed piecewise smoot partitions of unity as follows. For a model rectangle Ω k =[0,a] [0,b] we define ω k x, y =sinπx/asinπy/b insideω k and zero outside it. Ten, {χ kx, y} is derived from {ω k x, y} as described in section 4.1. Table 1 contains te results for te limiting case of zero overlap, i.e., β =0. Tis case does not strictly fit in te teoretical framework developed in te paper. We consider two nonoverlapping subdomains Ω 1 = Ω 1 = [0, 1 2 ] [0, 1] and Ω 2 =Ω 2 =[ 1 2, 1] [0, 1], and we define χ ix, y =χ Ωi x, y, for i =1, 2, i.e., te caracteristic or indicator functions of te subdomains. In tis case, te operators L k can only be defined weakly, in a variational sense. However, te finite difference approximations were well defined witout additional modification since te grid we cose was aligned wit te interface. Toug te overlap β = 0, te numerical results indicate tat te domain-decomposition-based ADI solution as, approximately, only trice as large errors compared to te solution of Crank Nicolson wit exact solvers. Te FS solution, owever, ad about 100 times larger error tan te CN solution. Table2containsteresultsfortwo overlapping subdomains: Ω 1 =[0, 3 4 ] [0, 1] and Ω 2 =[ 1 4, 1] [0, 1]. In tis case te overlap β =1/4, and we note tat te domain-decomposition-based ADI solution as error very close to te CN solution using exact solvers. Te FS solution ad approximately 50 times larger error tan tis CN solution. Table3containsteresultsfortecaseof16 overlapping subdomains, as sown in Figure 1. A 4 4 rectangular partition was first formed and eac subrectangle was enlarged to include overlap of β =1/12 1/3 of te subdomain widt. Te subdomains were grouped into q = 4 colors, and one of te four colored subregions is sown in Figure 1. In solving I + ατa k w = z, parallelization is possible wit

20 930 T.MATHEW,P.POLYAKOV,G.RUSSO,ANDJ.WANG TABLE 2 Two overlapping subdomains: Errors at time t f =1;τ cn = τ adi = CN ADI FS TABLE 3 16 overlapping subdomains: H 1 =4; τ cn = τ adi = CN ADI FS TABLE 4 Many overlapping subdomains: fixed 1 =64and τ cn =. Varying τ adi, H, andβ τ adi /4 /8 /16 H 1 H ADI CN different solvers on disjoint subregions of te kt color. Te results indicate tat te domain-decomposition-based ADI solution as approximately seven times larger error tan te CN solution wit exact solvers. Tis was expected due to te larger β 1 term in te truncation error formula of 23. Te FS solution ad approximately 26 times larger error tan tis CN solution. Table 4 tabulates te global error for a fixed grid of size 1 = 64 on wic 16, 64, and 256 subdomains were cosen. According to te asymptotic error bounds of equation 23, te global error sould remain approximately O 2 ifτ adi = β 2q 1 q. We cose τ adi = β, wic is more optimistic tan te teoretical bounds 23. Tus, as te number of subdomains increased, te cost of calculating te solution also increased proportionally to te inverse of te subdomain widt. Te results indicate tat te error in te ADI solution remained approximately constant and close to te error for te CN solution wit exact solvers Discussion. Tenumericalresultsindicatetatintecaseoftwosubdomains wit reasonable overlap, te global error of te ADI metod wit domain decomposition splitting is similar to errors for te Crank Nicolson metod wit exact solvers. However, for te case of many subdomains, te terms β 1 in te error 23 causes a deterioration in te global error. If a smaller time step τ adi = β is used in te ADI metod, ten te resulting global error of te ADI solution remains close to te error for CN wit exact solvers and τ cn =. Our euristic study of te complexity of tese metods in section 4.4 leads us to note tat if algebraic direct solvers of complexity C α n α are used to solve all linear systems, ten te ADI metod wit domain decomposition splitting can be competitive wit te Crank Nicolson metod provided α>1+ 2q 1 qd,weredis te space dimension.

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