An Efficient Multigrid Solver for a Reformulated Version of the Poroelasticity System

Transcription

1 An Efficient Multigrid Solver for a Reformulated Version of te Poroelasticity System F.J. Gaspar a F.J. Lisbona a, C.W. Oosterlee b P.N. Vabiscevic c a Departamento de Matemática Aplicada, University of Zaragoza, Pedro Cerbuna, 12, Zaragoza, Spain b Delft University of Tecnology, Delft Institute of Applied Matematics (DIAM, Mekelweg 4, 2628 CD Delft, te Neterlands c Institute for Matematical Modelling, Russian Academy of Sciences, 4-A, Miusskaya Sq., Moscow, Russia Abstract In tis paper, we present a robust and efficient multigrid solver for a reformulated version of te system of poroelasticity equations. Te reformulation enables us to treat te system in a decoupled fasion. We sow tat te reformulation boils down to a stabilization term in te iterative sceme, and tat te solution of te original problem is identical to te solution of te reformulated problem. A igly efficient multigrid metod can be developed, confirmed by numerical experiments. Key words: Key words: poroelasticity, reformulation, stability, collocated grid, multigrid 1991 MSC: 65N55, 74F10, 74S10, 65M12 1 Introduction In tis paper we would like to add an application and its efficient numerical treatment to te recently increased interest in saddle point type problems [2]. It is te time-dependent incompressible system of poroelasticity equations [3]. After a semi-discretization in time, te two-dimensional system, in its original Preprint submitted to Elsevier Science 20 July 2005

2 form, can be written as a block 2 2 system, A D G u B p = 0 f. (1 Here, u = (u, v is te deformation (in te x- and y-directions, p is te pore pressure, A is te 2D linear elasticity operator, D is te divergence operator, G is te gradient, and B is a Laplace type operator multiplied by te time step. For very small time steps, te size of te lower diagonal block entry in (1 can become arbitrarily small. So, te system may be viewed as a sort of singularly perturbed system related to A D G u O p = 0 f. (2 Poroelasticity teory addresses te time dependent coupling between te deformation of porous material and te fluid flow inside. Te porous matrix is supposed to be saturated by te fluid pase. Te state of tis continuous medium is caracterized by te knowledge of elastic displacements and fluid pressure at eac point. A penomenological model was first proposed and analyzed by Biot [3], studying te consolidation of soils. Poroelastic models are used to study problems in geomecanics, ydrogeology, petrol engineering and biomecanics [8,5]. In tis paper, we present an efficient multigrid metod for a reformulated version of te system of poroelasticity equations. We sow by means of block matrix manipulations tat system (1 can be brougt in a form wic is favorable for (almost decoupled iterative solution. At te same time, we analyze te reformulated system and prove tat te solution of tis system is identical to te solution of te original system. We sow in 1D tat by te transformation, after discretization, a numerical stabilization term of lower order as been brougt in te discrete version of te lower diagonal block of (1, wic does not influence te order of numerical convergence negatively. By a one-dimensional poroelasticity analysis and by corresponding numerical experiments we sow te stabilizing effect of te term. Numerical 2D experiments confirm te stability, accuracy and te efficient multigrid treatment of te resulting transformed system. Te paper is organized as follows, te poroelastic system, its transformation plus a corresponding solution algoritm are presented in Section 2. Furtermore, in Section 2.2 a previously used stable discretization on a staggered grid is explained, wic is used for comparison wit te current approac. In Section 3 te one-dimensional case is considered. Te discrete problem is 2

3 stabilized and te convergence of te corresponding transformed sceme is proven. In a diagram te relation between te original and te transformed system wit respect to stability, bot in te continuous and in te discrete case is also presented. Section 4 ten details te solution algoritm wit only multigrid metods for scalar equations. Finally Section 5 presents numerical poroelastic experiments indicating te efficiency of te solution algoritm. 2 Matematical Model and Discretization 2.1 Continuous System Te quasi static Biot model for soil consolidation can be formulated as a system of partial differential equations for displacements and te pressure of te fluid. One assumes te material s solid structure to be linearly elastic, initially omogeneous and isotropic, te strains imposed witin te material are small. We denote by u = (u, v, p T te solution vector, consisting of te displacement vector u = (u, v T and pore pressure of te fluid p. Te governing equations read µ u (λ + µgrad div u + α grad p = 0, x Ω, (3 α t (div u κ p = f(x, t, 0 < t T, (4 η were λ and µ are te Lamé coefficients; κ is te permeability of te porous medium, η te viscosity of te fluid, α is te Biot-Willis constant (wic we will take equal one and represents te vectorial Laplace operator. Te quantity div u (x, t is te dilatation, i.e. te volume increase rate of te system, a measure of te cange in porosity of te soil. Te source term f(x, t represents a forced fluid extraction or injection process, respectively, see [3]. For simplicity, we assume ere tat Ω is rigid (zero displacements and permeable (free drainage, so tat we ave omogeneous Diriclet boundary conditions, u(x, t = 0, p(x, t = 0, x Ω. (5 Before fluid starts to flow and due to te incompressibility of te solid and fluid pases, te initial state satisfies div u(x, 0 = 0, x Ω. (6 In te numerical experiments in Section 5 more complicated boundary conditions are cosen. Te incompressible, two-dimensional variant of Biot s consolidation model reads 3

4 (λ + 2µu xx µu yy (λ + µv xy + p x = 0, (λ + µu xy µv xx (λ + 2µv yy + p y = 0, (7 (u x + v y t a (p xx + p yy = f (plus initial and boundary conditions wit a = κ/η > 0. Problem (7 is a limit of te compressible case. Te compressible system is, owever, easier to solve due to an extra contribution to te main diagonal of te matrix related to tis system. We concentrate on a solver for te two-dimensional incompressible case, and consider a model operator L, wic reads (λ + 2µ xx µ yy (λ + µ xy x L = (λ + µ xy µ xx (λ + 2µ yy y. (8 x y ã ( xx + yy L can be interpreted as a stationary variant of (7, i.e., te operator after an implicit (semi- discretization in time. For example, in case of Crank-Nicolson time discretization we ave ã = 0.5a. From (8 one may calculate te corresponding determinant: det (L = µ ( ã(λ + 2µ 2 wit Laplace operator and biarmonic operator 2. Te principal part of det (L is m wit m depending on te coice of λ, µ, and ã. From pysical reasoning, we always ave µ, ã, λ + 2µ > 0, yielding m = 3. Te number of boundary conditions tat must accompany L is m [4,12]. 2.2 Previous Approac: Staggered Grid Pysically, wen a load is applied in a poroelasticity problem, te pressure suddenly increases and a boundary layer appears in te early stages of te time-dependent process. In te case of an unstable discretization, unpysical oscillations can occur in te first time steps of te solution. After tis pase, te solution sows a muc smooter beavior. Te time-dependent operator (8 suffers from stability difficulties. Te coefficient in te L 3,3 -block in (8 is typically, depending on te time step, extremely small. In order to avoid oscillating solutions, te discretization as to be designed wit care. Previously, we adopted a staggered grid discretization in [6] for system (7, using nearest neigbor central finite differences. Pressure points in te staggered grid were located at te pysical boundary, and te displacement points are defined at te cell faces, as often pressure is prescribed at te boundary. Te divergence operator is naturally approximated by a central discretization 4

5 : u : v : p Fig. 1. Staggered location of unknowns for poroelasticity. of te displacements in a cell, see Figure 1. Te discretization of eac equation, centered around te equation s primary unknown, reads in tis case (λ + µ( xx µ (λ + µ( xy ( x u L u = (λ + µ( xy µ (λ + µ( yy ( y v = f, ( x ( y ã p (9 wit f = (0, 0, f T, and te subscripts denote central discrete operators on te staggered grid. An efficient multigrid solver for te system of poroelasticity equations discretized on te staggered grid as been developed in [7,15]. Te multigrid sceme developed solves te system in a coupled fasion, wit a distributive smooter: In order to relax L u = f, a new variable w by u = C w was introduced and system L C w = f as been considered in smooting. Te resulting transformed system is suited for decoupled smooting, i.e., eac equation can be treated separately. Te so-called distributor reads [15] I 0 ( x C = 0 I ( y (λ + µ ( x (λ + µ ( y (λ + 2µ (10 5

6 wit identity I. Ten, te transformed system reads µ 0 0 L C = 0 µ 0 LC 3,1 LC 3,2 ã(λ + 2µ 2 wit (11 LC 3,1 = ( x ã(λ + µ ( ( xxx + ( xyy, (12 LC 3,2 = ( y ã(λ + µ ( ( xxy + ( yyy. (13 Equation-wise smooting for (11 is now an option, starting wit te first equation in te system, etc. All multigrid components related to te coarse grid correction are naturally dictated by te staggered grid arrangement. Te multigrid sceme works very well, wit convergence factors less tan 0.2 [7,15]. However, te coice for a staggered grid is a rigorous one. Staggered grid discretizations may not be easily generalized to curved domains, or to unstructured grids. Te staggered grid coice as only been made, because in te initial pase of a time-dependent problem a boundary layer may occur tat is prone to unpysical oscillations on a collocated grid, witout additional stabilization. Terefore, we perform ere a system transformation and (implicitly, by doing so a stabilization so tat we can discretize on a collocated grid, witout any unpysical oscillations. 2.3 Transformed System Let us rewrite problem (3-(6 as A u t + grad p t = 0, in Ω, (14 div u t κ p = f, η in Ω, (15 u = 0, p = 0, on Ω, (16 div u(x, 0 = 0, x Ω, (17 were A = µ (λ + µgrad div. We transform problem (14-17 to an equivalent problem. Firstly, applying te divergence operator to (14 and te operator (λ + 2µ to (15, adding te resulting equations and taking into account te equality (λ + 2µ div = div A, 6

7 we obtain p t + (λ + 2µκ η 2 p = (λ + 2µ f. (18 Secondly, by applying operator (λ+µ grad to (15 and by adding te resulting equation to (14 we get u p µ + grad t t (λ + µ κ grad p = (λ + µ grad f. η Wit te new variables q = p and v = u, we deal wit te transformed t system: µ v + grad p t + (λ + µ κ grad q = (λ + µ grad f, η (19 q + p = 0, (20 q t (λ + 2µκ q = (λ + 2µ f, η (21 v = 0, p = 0, div v + κ q = f, on Ω. (22 η plus initial conditions. We ave proven te following result: Proposition 1 If (u, p is a solution of problem (14-17 ten (v, p, q is te solution of problem ( In te following proposition we prove tat bot problems really are equivalent. Proposition 2 If (v, p, q is solution of problem (19-22 ten (u, p is solution of problem ( Proof: By applying te divergence operator to (19 and te use of equality div = div, we find µ divv q t + (λ + µ κ q = (λ + µ f (23 η Adding (23 and (21, we obtain µ (divv + κη q f = 0, in Ω. 7

8 By using boundary conditions (22 we deduce equation (15. Equation (14 is obtained by applying te operator (λ + µgrad to (15 and using (19. Note tat problem (19-22 is coupled over te boundary of te domain. Furter te generalization of te transformation to tree dimensions is straigtforward. One additional unknown, te time-dependent displacement in te z- direction, is ten also resolved. Let us consider a semi-discretization in time wit step time = T/M wit M a positive integer. For 1 m M 1 and assuming tat v m, p m and q m are known, te following iterative sceme is proposed. Algoritm I: (1 Solve: ( q q m (λ + 2µ κ η q = (λ + 2µ f, in Ω, div v m + κ η q = f on Ω. (2 Solve p = q in Ω, p = 0, on Ω. (3 Solve µ v + grad p p m + (λ + µ κ t η grad q = (λ + µ grad f, in Ω, v = 0, on Ω. Notice tat te boundary condition in step (1 is lagging beind one time step. Witout additional iteration te sceme presented ere will terefore be of O(. Two important issues regarding Algoritm I are discussed in Sections 3.1 and 4. First of all, te operators to be inverted in te algoritm above are only scalar Laplace type operators, for wic standard multigrid for scalar equations works extremely well (Section 4. Terefore, a igly efficient iterative 8

9 solution process can be defined for te transformed system. Secondly, wen working wit te reformulated system stable numerical solutions are obtained on a standard collocated grid (Section 3.1. So far, we can prove tis stability issue only for te 1D poroelasticity model situation. 3 One-Dimensional Poroelasticity 3.1 1D Transformation In 1D, te governing equations simplify and read (P : (λ + 2µ 2 u x + p = 0, 2 x (24 2 u x t κ 2 p = f, η x2 x (0, 1, 0 < t T, (25 We denote te original problem (24,(25 by (P for reference in Figure 2. For simplicity, we again assume tat Ω is rigid (zero displacements and permeable (free drainage, so tat we ave omogeneous Diriclet boundary conditions, u(0, t = u(1, t = 0, p(0, t = p(1, t = 0, and te initial state satisfies u (x, 0 = 0. (26 x Te equivalent transformed 1D problem, denoted by (P tr, is (P tr : (λ + 2µ 2 u x + p = 0, 2 x (27 q + 2 p = 0, x2 (28 q t (λ + 2µκ 2 q η x = (λ + f 2 2µ 2 x, 2 (29 u = 0, p = 0, 2 u x t + κ q = f, η on Ω. (30 Te 1D transformation is easier tan te 2D/3D cases, as it is not necessary to cancel out te term grad divu from te first equation. 9

10 3.2 Discrete Case We consider a uniform grid on te interval [0, 1], wit step size : ω = {x i x i = i, i = 0, 1..., N}, and denote by ω and ω te interior and te boundary nodes respectively. We consider te Hilbert space H ω of te discrete functions u = (u 0, u 1,..., u N on te grid ω, wit scalar product and norm given by (u, v ω = ( u0 v 0 + u N v N 2 + N 1 i=1 u i v i, u ω = (u, u. In a similar way, let us consider te Hilbert space H ω = {u H ω u 0 = u N = 0}, wit scalar product and associated norm. N 1 (u, v = u i v i, i=1 We define te self adjoint and positive definite operator δ on H ω as 0 i = 0, (δw i = w i+1 2w i + w i 1 i = 1,..., N 1, 2 0 i = N, and we consider te operators A = (λ + 2µδ and B = δ. We also introduce te difference operators, G = D : H ω H ω p 1 i = 0, p i+1 p i 1 (Gp i =, i = 1,..., N 1, 2 p N 1, i = N, wic verify (Gp, u = (p, Du, (u, p H ω H ω. Let U m H ω and P m H ω be approximations to u(x, t m and p(x, t m, were t m = m, m = 0, 1,..., M, M = T. Ten, using an Euler implicit sceme to discretize in time we get te problem 10

11 (P : AU + GP DU DU m = 0, (31 + κ BP = f, m = 0,..., M 1. (32 η wit initial condition DU 0 = 0. (P represents te discretization of te original problem on a collocated grid, wic is not stable. If we follow te transformations made in te continuous case, i.e. apply operator A to te second equation and operator D to te first equation and take into account te initial condition DU 0 = 0, we obtain te transformed discrete problem, denoted by (P tr, and, (P tr : AU 1 + GP 1 = 0, (33 Q 1 = BP 1, (34 CQ 1 + (λ + 2µ κ η BQ1 = (λ + 2µBf 1, (35 D ( U 1 + κ η Q1 = f 1, on ω, (36 C ( Q Q m D AU + GP = 0, (37 Q = BP, (38 + (λ + 2µ κ η BQ = (λ + 2µBf, (39 ( U U m m = 1, 2,..., M 1, + κ η Q = f on ω, (40 were a new difference operator appears, given by: 1 4 (q 2 + 2q 1 i = 1, 1 (Cq i = 4 (q i+1 + 2q i + q i 1, i = 2,..., N 2, 1 4 (q N 2 + 2q N 1, i = N 1, One can easily verify tat DG = CB = BC. In tis way we ave obtained a problem tat is equivalent to problem ( Remark Wit tis sceme we may obtain oscillations because te matrices in (35 and (39 are, in general, no M-matrices. 11

12 If 2 4(λ + 2µκ < we ave an M-matrix. However, tis restriction on te η spatial step is severe if we use small time steps in te initial stage of te consolidation process. In order to stabilize equations (35 and (39 we perturb tem as follows: C ( q q m Cq 1 + (λ + 2µ κ η Bq Bq1 = (λ + 2µBf, 1 + (λ + 2µ κ ( η Bq + 2 q 4 B q m = (λ + 2µBf. Taking into account tat C + 2 B = I, we encounter te discrete problem 4 and, (P tr : Au 1 + Gp 1 = 0, (41 q 1 = Bp 1, (42 q 1 + (λ + 2µκ η Bq1 = (λ + 2µBf 1, (43 D ( u 1 + κ η q1 = f 1, on ω, (44 ( q q m D Au + Gp = 0, (45 q = Bp, (46 + (λ + 2µ κ η Bq = (λ + 2µBf, (47 ( u u m + κ η q = f, on ω, (48 m = 1, 2,..., M 1, wic is te discrete problem corresponding to problem (27-30, denoted by (P tr (i.e., first transform ten discretize. Finally, we observe tat tis formulation is equivalent to te pre-transformed sceme ((P tr tr : Au 1 + Gp1 = 0, (49 Du 1 + κ η Bp Bp1 = f 1, (50 12

13 and: Du Du m + κ η Bp Au + Gp + 2 Bp m 4 m = 1, 2,..., M 1, Bp = 0, (51 = f, (52 were te initial condition Du 0 as already been applied. Tis sceme does not give oscillations. A diagram in Figure 2 summarizes te relation between te different problems considered. In te figure disc stands for discretized, stab means stabilized and indicates tat problems are equivalent. Summarizing, we ave six different problems: (P: original continuous problem, (P tr : transformed continuous problem, (P : discretization of (P (tis sceme sows oscillations, (P tr : discretization of (P tr (oscillation-free solutions, (P tr : transformation of discrete problem (P (identical to (P (i.e., oscillatory, and ((P tr tr : formulation were (P tr is transformed to an identical (stable sceme based on te original problem. Figure 2 indicates tat one can obtain ((P tr tr from (P by adding a stabilization term. We can also obtain sceme (P tr from (P tr by adding a stabilization term. (P (P tr disc (P (P tr disc stab stab (P tr ((P tr tr not stable stabilized Fig. 2. Relation between stable and unstable, continuous and discrete problem formulations. In order to illustrate wit a numerical example te effects of stabilization of term 2 B(p p m /(4 as in (52, we sow te numerical results for Terzagui problem [3], wic consists on a column of soil, bounded by a rigid, impermeable bottom and walls. A unit load is prescribed at te top wall, wic is free to drain. 13

14 Original problem Exact Solution Segregated sceme Fig. 3. Unpysical oscillations for original formulation of te incompressible poroelasticity model problem, wit central differencing wit 32 nodes, ˆt = 10 6, and stable solution from transformed system on a collocated grid. Figure 3 sows, for te initial time step, tat standard central finite differences on a grid wit 32 nodes lead to spurious oscillations in te discrete pressure (x = 0 represents te top wall. Te time discretization is te O( accurate implicit Euler sceme wit = In tis problem te boundary layer is approximated in a stable way by te discrete transformed system in O( 2 + -accuracy on a collocated grid wit 32 nodes, see also Figure Convergence of te stabilized sceme We now prove in 1D tat te stabilized discrete problem gives stable solutions. Firstly, we give an energy estimate. For simplicity, we suppose tat p 0 satisfies te boundary conditions. In tis case, te sceme writes as (51-52 even for m = 0. Oterwise, a different estimate must be done for te first step time. Proposition 4 Te solutions of te sceme (51-52 for m 0 satisfy te a priori estimates u 2 A p p 2 B 2 p0 2 B + C 2 2 B u0 2 A p0 2 B + C 1 m j=1 f j+1 f j 2 B j=1 1 + f f j 2 B 1, (53 2 B 1, (54 were C 1 and C 2 are constants independent of te discretization parameters. 14

15 respec- Proof: Multiplying scalarly (51 and (52 by (u u m tively, we get for 0 m M 1, and (D u 2 4 ( Au, u u m ( + u m (B p p m, p, p Te addition of (55 and (56 yields Gp, u + κ (Bp, p + η / and p u m = 0, (55 = ( f, p. (56 ( Au, u ( f, p u m κ + B(p, p + 2 ( B(p p m, p = η 4. (57 Applying te generalized Caucy-Scwarz inequality in te rigt and side we get 1 2 ( u 2 A 2 um 2 A + 8 ( p 2 B pm 2 B η f 2 B 4κ 1, and terefore, (53 is obtained, i.e., te solution u is stable wit respect to te initial data and rigt and side. To obtain an a priori estimate for te pressure, a splitting of te solution will be used. Let be p = p +p, were te first part p, is te solution of te problem κ η B p = f, m = 0,... M 1, (58 and te second p is solution of Au + Gp = Gp, (59 Du Du m + κ Bp + 2 η 4 B p p m = 2 4 B p p m (60 We get from (59 and (60 15

16 u u m 2 + A ( G(p p m u, u m = ( G(p p m u, u m and (D u u m p (B p m, p p m, p + κ η p m = 2 4 ( Bp, p p m p (B p m, p p m. Adding te previous equations and after simple transformations we obtain te inequality κ 2η ( p 2 B p m 2 B 2 obtaining te estimate (54. ( p G p m 2 ( + 2 p 8 A 1 p m 2. Convergence results are straigtforward using estimates (53-54 and considering te approximation errors of te sceme. 4 Multigrid Solution of Transformed System Te iterative solution metod for te transformed system of equations (Algoritm I can be interpreted as a left distributor for wit: In tat case, we obtain C L u = C f, I 0 (λ + µ ( x C = 0 I (λ + µ ( y. (61 ( x ( y (λ + 2µ µ 0 LC 1,3 C L = 0 µ LC 2,3, 0 0 ã(λ + 2µ 2 16

17 wit LC 1,3 = LC 3,1 and LC 2,3 = LC 3,2, as in (12, (13, and (λ + µ x f C f = (λ + µ y f (λ + 2µ f Te main difference wit Algoritm I is tat ere a semi-discretization in time as already taken place (as in model operator (8. Discrete operator L represents te discretization on a collocated grid. We end up wit an upper triangular system. In a first step ten, te last equation sould be updated after wic te oter two equations may be treated. For te resemblance and simplification, operator ã(λ + 2µ 2 is split into ( ã(λ + 2µ + 1q = f, p = q. Notice tat, compared to te work explained in Section 2.2, we solve te transformed system tat is almost (i.e., except for te boundary coupling decoupled. So, we do not use te reformulation system only in multigrid smooting. For te multigrid solution of (61 we can simply coose four times a igly efficient scalar multigrid algoritm tat works well for discrete Laplace type operators. All four scalar operators appearing are isotropic Laplace type operators. A multigrid metod wit igest efficiency, based on a red-black pointwise Gauss-Seidel smooter, GS-RB, and well-known coices for te remaining multigrid components [12] can be used for all coices of λ, µ, and ã. Tese include te direct coarse grid discretization of te PDE, full weigting and bilinear interpolation, as te restriction and prolongation operators, respectively. Remark 3. A similar transformation can be defined and analyzed for incompressible Stokes equation from fluid mecanics. In tat case, for te discrete Stokes operator 0 ( x L,st = 0 ( y ( x ( y 0 a corresponding distributor is given by I 0 0 C,st = 0 I 0. ( x ( y 17

18 Te transformed system ten reads 0 ( x C,st L,st = 0 ( y. 0 0 Also ere, it is not necessary to discretize on a staggered grid for a stable discretization. A generalization of tis transformation for te system of incompressible Navier-Stokes equations can be found in [12] (Section 8.8.3: Assume tat Ω is a bounded domain in IR 2 and tat velocities u, v and pressure p are sufficiently smoot. Ten te two systems u + Re(uu x + vu y + p x = 0 v + Re(uv x + vv y + p y = 0 u x + v y = 0 (Ω (Ω (Ω = Ω Ω and u + Re(vu y uv y + p x = 0 v + Re(uv x vu x + p y = 0 p + 2Re(v x u y u x v y = 0 u x + v y = 0 (Ω (Ω (Ω ( Ω are equivalent. Te result originates from [10]. Tis transformed Navier-Stokes system as recently been used in [11]. Notice te te original primary unknowns are used after te transformation. Te generalization to 3D is trivial. 5 Numerical Experiments In tis section tree numerical experiments are evaluated. Te experiments range from a model problem wit academic parameter setting to more realistic problems and parameters. We report on te accuracy of te numerical solution from te transformed system and, in particular, on te multigrid convergence. A comparison in terms of CPU time between te staggered multigrid approac and multigrid for te collocated discretization of te transformed system is made. For te coupled, staggered approac, te measure of convergence is related to te absolute value of te residual after te m-t iteration in te maximum 18

19 norm over te tree equations in te system, r m := r m 1, + r m 2, + r m 3, < T OL, (62 wit T OL = In te decoupled approac, te stopping criterion is based on te residual in eac equation separately. Convergence is acieved in eac equation s residual is less tan Multigrid Convergence for First Model Problem Some analytical reference solutions are known in te literature [1] for (7 in dimensionless form, were scaling as taken place wit respect to a caracteristic lengt of te medium l, Lamé constants λ + 2µ, time scale t 0 and a (7. By coosing a unit squared domain, a source term f = 2 δ 0.25,0.25 sin ˆt (δ x,y is te Kronecker delta function, ˆt = (λ + 2µat, te following boundary and initial conditions, at: y = {0, 1} : u = 0, v/ y = 0 at: x = {0, 1} : v = 0, u/ x = 0, and pressure p = 0 at te boundaries, we can mimic te dimensionless situation. In tis case, te solution can be written as an infinite series [1], see also [7]. An interesting feature is tat tis solution is independent of te Lamé coefficients. Te parameters in te reference experiment read µ = 1/2, λ = 0, ã = (8. Figure 4 sows for tis setting te computed displacement and pressure solution at time ˆt = π/2. Te solution resembles te exact solution in [1] very well, witout any unpysical oscillations. O( 2 + accuracy is observed for te displacements, and, asymptotically, for te pressure too (despite te occurrence of te delta function wic usually influences te numerical accuracy negatively [7]. Tis reference problem is solved wit state-of-te-art multigrid metods developed for te staggered grid (Section 2.2 and te collocated grid discretization (Section 4. Table 1 compares te CPU times, to satisfy te tolerance 10 6, of te transformed system on te collocated grid and te original system discretized on te staggered grid. Note tat tis tolerance is considered accurate w.r.t. engineering practice. Te transformed system needs, on average, for te four equations six scalar multigrid iterations per equation. Te CPU time in Table 1 for tis system on a Pentium IV, 2.6 MHz is 1 second per time step on a grid, and 4 seconds 19

20 x y 0.6 Fig. 4. Numerical solution for displacements and pressure for 2D poroelasticity model problem, grid. per time step on a grid. Tese results are more tan 10 times faster tan te CPU time results in our previous work wit multigrid for te staggered system. Te F(1,1-cycle [12] (meaning one pre- and one post-smooting iteration is used for tis problem; it is fast and sows an -independent beavior. Te average multigrid convergence factor observed on te collocated grid is 0.06, wic is in correspondence wit te multigrid teory for Laplace operators. Tese results not sensitive to variations in te Lamé coefficients. Also te multigrid convergence factor on te staggered grid is excellent, ρ We use te distributive smooter (11 in a multigrid F(1,1-cycle [12] ere. In te distributive smooter tree operators need to be smooted separately. For tis purpose, a red-black point-wise Gauss-Seidel smooter is used for te Laplace operators in te first two equations and a line-wise Gauss- Seidel relaxation metod for te tird equation. Tis multigrid metod for te staggered case was te most efficient metod in a comparison among various cycles and smooters in [7]. Te extra costs compared to te collocated version are due to te fact tat we treat a coupled system on te staggered grid. Table 1 CPU times comparison of multigrid metods for te original system on a staggered grid, and te transformed system on a collocated grid collocated decoupled staggered coupled In bot solution approaces a matrix-free version of multigrid is used; te CPU times include te time for computing te operator elements. 20

21 Fig. 5. Numerical solution for pressure for second problem, grid. 5.2 Poroelasticity Problem wit Realistic Parameters In te following experiment, te domain considered is Ω = ( 50, 50 (0, 100. As te boundary conditions zero displacements are cosen and for te pressure, 1 on Γ 1 : x 20, y = 100, p = 0 on Γ \ Γ 1. Te material properties of te porous medium are given in Table 2. Table 2 Material parameters for te second poroelastic problem. Property Value Unit Young s modulus N/m 2 Poisson s ratio Permeability 10 7 m 2 Fluid viscosity 10 3 Pas In Figure 5 te solution of te pressure is presented. In Table 3 we compare te CPU times of te two approaces: Solving te system in a coupled fasion on a staggered grid, and te solution of te decoupled transformed system on a collocated grid. Te multigrid metods for 21

22 bot discretization approaces are identical to te ones employed in te previous experiment. Te multigrid convergence factor for te transformed system is again excellent, 0.06 for eac equation, as is its staggered counterpart ρ 0.08 for te coupled system. Te CPU time used, owever, differs again substantially, as presented in Table 3. Table 3 CPU time comparison between te collocated decoupled and te staggered coupled multigrid scemes collocated decoupled staggered coupled 0.5 (5 2 (5 10 (6 40 (6 Te results wit te transformed system, comparing to te results in Table 1, confirm te independence of te multigrid convergence and CPU time wit respect to te poroelastic problem parameters. Tis is a strong robustness result for te new solver developed. 5.3 Poroelastic Footing Experiment Te tird example is a true 2d footing problem (see also [9]. Te simulation domain is a 100 by 100 meters block of porous soil, as in Figure 6. σ Fig. 6. Computational domain for te footing problem At te base of tis domain te soil is assumed to be fixed wile at some upper part of te domain a uniform load of intensity σ 0 is applied in a strip of lengt 40m. Te wole domain is assumed free to drain. Terefore, te boundary data is given as follows: 22

23 Fig. 7. Numerical solution for displacements and pressure for 2D poroelasticity reference problem, grid. p = 0, σ xy = 0, σ yy = σ 0, on Γ 1 = {(x, y Ω, / x 20, y = 50}, p = 0, σ xy = 0, σ yy = 0, on Γ 2 = {(x, y Ω, / x > 20, y = 50}. Te material properties of te porous medium are te same as in te previous problem, see Table 2 and te uniform load is taken as σ 0 = N/m 2. Notice tat te boundary condition for te footing problem involves te prescription of stress conditions. Tese conditions are applied in te discretization of te equations of te stabilized system ((P tr tr. Ten tey are transformed to problem (P tr in a similar way as it explained in Section 3.2. In Figure 7 te solution of te pressure is presented. Te unpysical oscillations for small t tat were present in te numerical results in [9], do not occur ere wit bot formulations: not wit staggered grids and not wit collocated grids adding te stabilization term. Finally, te multigrid convergence factor for te decoupled system is found to be 0.06 for te equations for p and q, wile for te oter two equations, wit te stress boundary conditions, it is found to be Te corresponding CPU times are 1 on a grid and 4 on a grid. Te convergence factor for te coupled system on te staggered grid reads ρ = 0.2. Te CPU times are ten 29 on a grid and 113 on te grid. Te improvement in CPU time wit te transformed system is ere more impressive. Tis example sows tat te transformation can easily be applied to more realistic boundary conditions, resulting in a similar performance of te solver. 23

24 6 Conclusion In tis paper we provide a fast and accurate discrete solution for te incompressible variant of te poroelasticity equations. Te system is transformed so tat a stable discretization can be obtained on a collocated grid. Tis is done by means of an (implicit, via te transformation addition of an O( 2 + stabilization term in te discretization of te transformed system. A robust and very efficient multigrid iteration as been defined based on te decoupled version of te poroelasticity system after te transformation. It is sufficient to coose a igly efficient multigrid metod for a scalar Poisson type equation for te overall solution of tis poroelasticity system. Wit standard geometric transfer operators, a direct coarse grid discretization and a point-wise red-black Gauss-Seidel smooter, an efficient multigrid metod is developed for all relevant coices of te problem parameters. According to classical multigrid teory, we observe multigrid convergence factors tat are less tan 0.1 for a variety of poroelastic problems. In te literature tis is often called textbook multigrid efficiency. Very satisfactory solution times ave been produced, tat are about 10 times faster tan te iterative solution of te coupled system, discretized on a staggered grid. Te present discretization and iterative solution metod can be seen as a basis for te generalization to more complicated problems tat are also bot porous and elastic. Here we tink of double porosity problems, or coupled problems in wic poroelasticity is coupled to Stokes flow and to termodynamical models. Te present saddle point type problem as been andled well by a transformation, tat is now well understood, at least in 1D. Te future problems offer next callenges in te treatment of coupled saddle point type problems. References [1] S. I. Barry, G.N. Mercer, Exact solutions for 2D time dependent flow and deformation witin a poroelastic medium, ASME J. Appl. Mecanics, 66: , [2] M. Benzi, G.H. Golub, J. Liesen, Numerical solution of saddle point problems. Acta Numerica 1 137, [3] M. A. Biot, General teory of tree dimensional consolidation. J. Appl. Pysics, 12: , [4] A. Brandt, Multigrid tecniques: 1984 guide wit applications to fluid dynamics. GMD-Studie Nr. 85, Sankt Augustin, Germany,

25 [5] W. Elers and J. Blum (eds., Porous media: teory, experiments and numerical applications. Springer, Berlin, Germany, [6] F. J. Gaspar, F. J. Lisbona, and P. N. Vabiscevic, A finite difference analysis of Biot s consolidation model. Appl. Num. Mat. 44: , [7] F.J. Gaspar, F.J. Lisbona, C.W. Oosterlee, and R. Wienands, A systematic comparison of coupled and distributive smooting in multigrid for te poroelasticity system. Num. Lin. Algebra Appl. 11: , [8] V. C. Mow and W. M. Lai, Recent developments in synovial joint biomecanics. SIAM Review, 22: , [9] Murad M. A., Loula A. F. D. On stability and convergence of finite element approximations of Biot s consolidation problem. Internat. J. Numer. Metods Engrg. 37:, , [10] A. Scüller, A multigrid algoritm for te incompressible Navier-Stokes equations, In: W.Hackbusc, R.Rannacer (eds., Num. treatment of Navier- Stokes equations, Notes on Numerical Fluid Mecanics, Vieweg, Braunscweig, 30: , [11] R.C. Swanson, Evaluation of a multigrid sceme for te incompressible Navier- Stokes equations, NASA/TM May [12] U. Trottenberg, C. W. Oosterlee, and A. Scüller, Multigrid. Academic Press, New York, [13] R. S. Varga, Matrix iterative analysis, Prentice-Hall. Englewood Cliffs, [14] P. Wesseling, Principles of computational fluid dynamics. Springer, Berlin, [15] R. Wienands, F.J. Gaspar, F.J. Lisbona and C.W. Oosterlee, An efficient multigrid solver based on distributive smooting for poroelasticity equations. Computing 73: ,