Numerical Upscaling and Preconditioning of Flows in Highly - - PowerPoint PPT Presentation

Introduction Discretization Preconditioning of the Fine Grid Systems

Numerical Upscaling and Preconditioning of Flows in Highly Heterogeneous Porous Media

• R. Lazarov, TAMU,
• Y. Efendiev, J. Galvis, and J. Willems

WS#4: Numerical Analysis

• f Multiscale Problems & Stochastic Modelling

RICAM, Linz, Dec. 12-16, 2011 Thanks: NSF, KAUST

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Introduction Discretization Preconditioning of the Fine Grid Systems

Outline

1

Introduction Motivation and Problem Formulation

2

Discretization Single Grid Approximation Subgrid Approximation and Its Performance

3

Preconditioning of the Fine Grid Systems Overlapping DD method Some Numerical Examples When one can use this method ? Examples

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Introduction Discretization Preconditioning of the Fine Grid Systems Motivation and Problem Formulation

Outline

1

Introduction Motivation and Problem Formulation

2

Discretization Single Grid Approximation Subgrid Approximation and Its Performance

3

Preconditioning of the Fine Grid Systems Overlapping DD method Some Numerical Examples When one can use this method ? Examples

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Introduction Discretization Preconditioning of the Fine Grid Systems Motivation and Problem Formulation

Motivation: media at multiple scales Figure: Porous media: real-life scale and macro scale

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Introduction Discretization Preconditioning of the Fine Grid Systems Motivation and Problem Formulation

Motivation: industrial foams – media of low solid fraction Figure: Industrial foams on micro-scale; porosity over 93% Figure: Trabecular bone: micro- and macro-scales

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Introduction Discretization Preconditioning of the Fine Grid Systems Motivation and Problem Formulation

Modeling of flow in porous media

(1) Flows in porous media are modeled by linear Darcy law that relates the macroscopic pressure p and velocity u: ∇p = −µκ−1u, κ − permeability, µ − viscosity (1) (2) Another venue for a two-phase flow is a Richards model: ∇p = −µκ−1u, where κ = k(x)λ(x, p) (2) with k(x) heterogeneous function is the intrinsic permeability, while λ(x, p) is a smooth function that varies moderately in both x and p, related to the relative permeability. (3) For flows in highly porous media Brinkman (1947) enhanced Darcy’s law by adding dissipative term scaled by viscosity: ∇p = −µκ−1u + µ∆u. (3)

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Introduction Discretization Preconditioning of the Fine Grid Systems Motivation and Problem Formulation

Computer Generated Heterogeneities Distributions

Allaire in 1991 studied homogenization of slow viscous fluid flows (with negligible no-slip effects on interface between the fluid and the solid

• bstacles) for periodic arrangements.

Computer generated permeabilty fields are shown below.

Figure: periodic + rand; rand + rand, SPE10 slice, fractured

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Introduction Discretization Preconditioning of the Fine Grid Systems Motivation and Problem Formulation

The Goals of Our Research

Our aim is development and study of numerical method that address the following main issues of the above classes of problem: Works well in both limits, Darcy and Brinkman; Applicable to linear and nonlinear highly heterogeneous problems; Could be used as a stand alone numerical upscaling procedure; Is robust with respect to high variations of the permeability field.

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Introduction Discretization Preconditioning of the Fine Grid Systems Motivation and Problem Formulation

State of the Art: Available Numerical Methods and Tools

Numerical upscaling as subgrid approximation methods: (1) standard FEM for Darcy: Hou & Wu, 1997, Wu, Efendiev, & Hou, 2002, comprehensive exposition in the book of Efendiev & Hou, 2009, (2) mixed methods for Darcy: Aarnes, 2004, Arbogast, 2002, Chen & Hou, 2002, (3) DG FEM for Brinkman equation: Willems, 2009, Iliev, Lazarov & Willems, 2010, Juntunen & Stenberg, 2010, Könö & Stenberg, 2011 (4) Galerkin FEM augmented with multiscale finite element functions: Pechstein & Scheichl, 2008, 2011, Chu, Graham, & Hou, 2010.

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Introduction Discretization Preconditioning of the Fine Grid Systems Motivation and Problem Formulation

State of the Art: Available Numerical Methods and Tools

Preconditioning techniques based on coarse grid space with multiscale finite element functions: (1) FETI and other DD methods: Graham, Klie, Lechner, Pechstein, & Scheichl, 2007, 2008, 2009, 2011 (2) Using energy-minimizing coarse spaces: Xu & Zikatanov, 2004 (3) Spectral Element Agglomerate Algebraic Multigrid Methods, Efendiev, Galvis, & Vassielvski, 2011 (4) Use of multiscale basis in DD, Aarnes and Hou (2002)

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Introduction Discretization Preconditioning of the Fine Grid Systems Motivation and Problem Formulation

Objectives:

1

Derive, study, implement, and test a numerical upscaling procedure for highly porous media that works well in both limits, Brinkman and Darcy, so it could cover both, natural porous media and man-made materials;

2

Design and study of preconditioning techniques for such problems for porous media of high contrast with targeted applications to oil/water reservoirs, bones, filters, insulators, etc

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Introduction Discretization Preconditioning of the Fine Grid Systems Motivation and Problem Formulation

Strategy:

1

Multiscale finite element method

2

Mixed and Galerkin formulations

3

Coarse-grid finite element spaces augmented with fine-scale functions based on local weighted spectral problems (after Efendiev Galvis)

4

Robust with respect to the contrast iterative techniques for solving large fine grid systems;

5

Experimentation

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Introduction Discretization Preconditioning of the Fine Grid Systems Single Grid Approximation Subgrid Approximation and Its Performance

Outline

1

Introduction Motivation and Problem Formulation

2

Discretization Single Grid Approximation Subgrid Approximation and Its Performance

3

Preconditioning of the Fine Grid Systems Overlapping DD method Some Numerical Examples When one can use this method ? Examples

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Introduction Discretization Preconditioning of the Fine Grid Systems Single Grid Approximation Subgrid Approximation and Its Performance

Weak formulation

Find u ∈ H1

0(Ω)n := V and p ∈ L2 0(Ω) := W such that

• Ω

(˜ µ∇u : ∇v + µ κu · v)dx +

• Ω

p∇ · vdx =

• Ω

f · vdx ∀v ∈ V

• Ω

q∇ · udx = 0 ∀q ∈ L2

0(Ω)

The solution of this problem has unique solution (u, p) ∈ V × W .

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Introduction Discretization Preconditioning of the Fine Grid Systems Single Grid Approximation Subgrid Approximation and Its Performance

The finite element of Brezzi, Douglas, and Marini of degree 1

For this finite element we have: On a rectangle T the polynomial space is characterized by {v = P2

1 + span{curl(x2 1 x2), curl(x1x2 2}};

with dof normal velocity pressure H (VH, WH) ⊂ (H0(div; Ω), L2

0(Ω)) := V × W ;

Has a natural variant for n = 3.

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Introduction Discretization Preconditioning of the Fine Grid Systems Single Grid Approximation Subgrid Approximation and Its Performance

DG FEM, Wang and Ye (2007) on a Single Grid

Since the tangential derivative along the internal edges will be in general discontinuous, i.e. VH H1

0(Ω); therefore we will apply the discontinuous

Galerkin method: Find (uH, pH) ∈ (VH, WH) such that for all (v H, qH) ∈ (VH, WH) a (uH, vH) + b (v H, pH) = F(v H) b (uH, qH) = 0. (4) Because of the nonconformity of the FE spaces, the bilinear form a (uH, vH) has a special form given below.

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Introduction Discretization Preconditioning of the Fine Grid Systems Single Grid Approximation Subgrid Approximation and Its Performance

Discretization of Wang and Yang 2007 on Single Grid

b (vH, pH) :=

• Ω

pH∇ · vHdx F(v H) :=

• Ω

f · v Hdx τ +

e

n+

e

n−

e

T + τ −

e

T − a (uH, v H) :=

• T∈TH
• T

(˜ µ∇uH : ∇v H + µ κuH · v H)dx −

• e∈EH
• e

˜ µ

• {

{uH} } v H + { {vH} } uH

• symmetrization

− α |e| uH v H

• stabilization
• ds

{ {v} }|e := 1

2(n+ e · ∇(v · τ + e )|e+ + n− e · ∇(v · τ − e )|e−)

v |e := v|e+ · τ +

e + v|e− · τ − e

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Introduction Discretization Preconditioning of the Fine Grid Systems Single Grid Approximation Subgrid Approximation and Its Performance

Fine and Coarse Grid Spaces: Arbogast (2004)

Fine and coarse triangulation Th and TH. FE spaces with (Wh consist of bubbles) WH,h = Wh ⊕ WH ⊂ L2

0, VH,h = Vh ⊕ VH ⊂ H0(div)

Crucial properties:

1

∇ · Vh = Wh and ∇ · VH = WH

2

v h · n = 0 on ∂T, ∀v h ∈ Vh and ∂T ∈ TH

3

WH ⊥ Wh

Ω

TH Th

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Introduction Discretization Preconditioning of the Fine Grid Systems Single Grid Approximation Subgrid Approximation and Its Performance

Splitting of the Spaces

Unique decomposition in coarse and fine-grid components yields: find uH + uh ∈ Wh ⊕ WH, and pH + ph ∈ Vh ⊕ VH such that decomposed solution a (uH + uh, vH + v h) + b (v H + v h, pH + ph) = F(v H + v h), b (uH + uh, qH + qh) = 0, for all v H + v h ∈ Wh ⊕ WH and qH + qh ∈ Vh ⊕ VH.

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Introduction Discretization Preconditioning of the Fine Grid Systems Single Grid Approximation Subgrid Approximation and Its Performance

Splitting of the Solution

Equivalently, testing separately for vH, qH and v h, qh, we get (∗) a (uH + uh, v H) + b (v H, pH +✚

✚

ph) = F(vH) ∀vH ∈ VH b (uH +✚

✚

uh, qH) = ∀qH ∈ WH (∗∗) a (uH + uh, v h) + b (v h,✚

✚

pH + ph) = F(vh) ∀vh ∈ Vh b (✟

✟

uH + uh, qh) = ∀qh ∈ Wh Since b (v, q) = (∇ · v, q), ∇ · Vh = Wh, ∇ · VH = WH, WH ⊥ Wh.

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Introduction Discretization Preconditioning of the Fine Grid Systems Single Grid Approximation Subgrid Approximation and Its Performance

Decompose (∗∗) further into (∗∗) a (δu(uH), v h) + b (v h, δp(uH)) = −a (uH, vh) ∀vh ∈ Vh b (δu(uH), qh) = ∀qh ∈ Wh a

• δu, v h
• + b
• v h, δp
• =

F(v h) ∀vh ∈ Vh b

• δu, qh
• =

∀qh ∈ Wh Note that (δu(uH), δp(uH)) is linear in uH (δu, δp) and (δu(uH), δp(uH)) can be computed locally

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Introduction Discretization Preconditioning of the Fine Grid Systems Single Grid Approximation Subgrid Approximation and Its Performance

The Upscaled Equation

We have (uh, ph) = (δu + δu(uH), δp + δp(uH)). Plugging this into the coarse equation (∗) yields: Putting all these together we obtain:

Symmetric Form of the Upscaled Equation

Thus we get a (uH + δu(uH), vH + δu(v H)) + b (v H, pH) =F(vH) − a

• δu, v H
• ,

b (uH, qH) =0, which involves only coarse-grid degrees of freedom.

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Introduction Discretization Preconditioning of the Fine Grid Systems Single Grid Approximation Subgrid Approximation and Its Performance

Sub-grid Algorithm, Willems, 2009

Sub-grid Algorithm: Willems, 2009

1

Solve for the fine responses (δu, δp) and (δu(ϕH), δp(ϕH)) for coarse basis functions ϕH and each coarse cell.

2

Solve the upscaled equation for (uH, pH).

3

Piece together the solutions to get (uH,h, pH,h) = (uH, pH) + (δu(uH), δp(uH)) + (δu, δp). For pure Darcy this reduces to the method of Arbogast, 2002, for BDM FEs.

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Introduction Discretization Preconditioning of the Fine Grid Systems Single Grid Approximation Subgrid Approximation and Its Performance

Vuggy media - Subgrid Brinkman

u = [1, 0] on ∂Ω, f = 0, µ = 1e − 2, K −1 = 1e3 Th : 1282.

(a) Ref. solution (b) H = 1/16. (c) H = 1/8. (d) H = 1/4.

Figure: Velocity component, u1, for vuggy geometry.

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Introduction Discretization Preconditioning of the Fine Grid Systems Single Grid Approximation Subgrid Approximation and Its Performance

SPE10 - Subgrid for Brinkman

u = [1, 0] on ∂Ω, f = 0, µ = 1e − 2, K −1 : ranging from 1e5 in blue to 1e2 in red. Th : 1282.

(a) Ref. solution (b) H = 1/16. (c) H = 1/8. (d) H = 1/4.

Figure: Velocity component, u1, for SPE10 on 3 coarse grids.

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Introduction Discretization Preconditioning of the Fine Grid Systems Single Grid Approximation Subgrid Approximation and Its Performance

SPE10 benchmark

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Introduction Discretization Preconditioning of the Fine Grid Systems Overlapping DD method Some Numerical Examples When one can use this method ? Examples

Outline

1

Introduction Motivation and Problem Formulation

2

Discretization Single Grid Approximation Subgrid Approximation and Its Performance

3

Preconditioning of the Fine Grid Systems Overlapping DD method Some Numerical Examples When one can use this method ? Examples

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Introduction Discretization Preconditioning of the Fine Grid Systems Overlapping DD method Some Numerical Examples When one can use this method ? Examples

Preconditioning of the fine grid system

The algebraic system is an ill-conditioned system due to two main factors: (1) the permeability K (orders of magnitude) and (2) the mesh size h. The discussed numerical upscaling of Brinkman equations leads to a saddle-point system; The matrix corresponding to the form a(·, ·) is very ill-conditioned due to the large heterogeneous variation of K and very small h; The known iterative methods, e.g. Uzawa, Bramble-Pasciak, etc either converge slow or practically do not converge for media with very high contrast.

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Introduction Discretization Preconditioning of the Fine Grid Systems Overlapping DD method Some Numerical Examples When one can use this method ? Examples

Abstract form of DD for s.p.d. forms a(u, v) (Efendiev,Galvis,L.,Willems,2011)

Here and below all spaces are finite dimensional defined on the fine grid of mesh-size h. To avoid too many indexes we have omitted h in the notations for these spaces. Global problem: find u ∈ V0 such that a(u, v) = f(v), ∀v ∈ V0: a(u, v) is a symmetric positive definite form; Ωi, i = 1, . . . , N is an overlapping cover of the domain Ω; V(Ωi) are the subspaces corresponding to Ωi and aΩi (φ, ψ) = a(φ|Ωi , ψ|Ωi ), with φ|Ωi , ψ|Ωi ∈ V(Ωi); VH(Ω) are the subspace based on the coarse mesh TH φ = φH + N

i=1 φi,

φH ∈ VH(Ω), φi ∈ V0(Ωi); DD preconditioner is based on local solutions based on V0(Ωi) and coarse-grid solution based on VH(Ω)

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Introduction Discretization Preconditioning of the Fine Grid Systems Overlapping DD method Some Numerical Examples When one can use this method ? Examples

Abstract form of DD for s.p.d. forms a(u, v) (Efendiev,Galvis,L.,Willems,2011)

Ω xj Ωj

Ωj Ωs

j,i

Ωp

j

Ωs

j

Left: Vertex xi and domain Ωi; Right: Subdomain with 7 connected components.

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Introduction Discretization Preconditioning of the Fine Grid Systems Overlapping DD method Some Numerical Examples When one can use this method ? Examples

Robust DD Preconditioners for the Global Fine Grid System

GOAL is to construct a coarse grid space VH(Ω) s.t. for φ ∈ V0, φ = φH +

j φj, φH ∈ VH(Ω),

φj ∈ V(Ωj) a(φH, φH) +

N

• i=1

a(φi, φi) ≤ Ca(φ, φ), where in the “ideal case” the constant C does not depend on the contrast and h; the coarse VH(Ω) space is “small” (e.g. ≈ #TH). Possibilities for construction of a coarse space based on:

1

Multiscale coarse grid functions

2

Energy minimizing functions

3

Functions with local spectral information on the problem

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Introduction Discretization Preconditioning of the Fine Grid Systems Overlapping DD method Some Numerical Examples When one can use this method ? Examples

Abstract Spectral Problems

Let ξj: Ω → [0, 1] be a partition of unity subordinated to the partition Ωj, so that supp(ξj) = ¯ Ωj. For any φ ∈ V0 the function (ξjφ)|Ωj ∈ V0(Ωj) and define mΩj (φ, ψ) :=

• i

aΩj (ξiξjφ, ξiξjψ), where the summation is over all i s.t. Ωj ∩ Ωi = ∅. Consider the spectral problem: find (λj

i, φj i) ∈ (R, V(Ωi)), s.t.

aΩj (ψ, φj

i) = λj imΩj (ψ, φj i)

∀ψ ∈ V(Ωi) and order the eigenvalues 0 ≤ λ1

i ≤ · · · ≤ λLi i ≤ . . . .

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Introduction Discretization Preconditioning of the Fine Grid Systems Overlapping DD method Some Numerical Examples When one can use this method ? Examples

Coarse Space Based on Spectral Problems - Efendiev&Galvis, 2009

Main assumptions: The forms aΩj (·, ·) are positive definite on V0(Ωj) and positive semi-definite on V(Ωj); {ξi} is a partition of unity, i.e. ξi(x) = 1; For a small threshold τ −1 there is Lj so that λLj+1 ≥ τ −1. Construction of the coarse space VH = span{P(ξjφj

i) :

i = 1, . . . , Lj, j = 1, . . . , N}, Where P projection back to the V0 (e.g. a-projection or interpolation). Note, that dim(VH) has increased. For φ ∈ V define φH =

j P(ξjφj 0), where

mΩj (φ − φj

0, φj i) = 0 for all i = 1, . . . , Lj. Then the desired decomposition is:

φj = P(ξjφ −

• i≥1

ξjξiφi

0) supported in Ωj and

φ = φH +

• j

φj.

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Introduction Discretization Preconditioning of the Fine Grid Systems Overlapping DD method Some Numerical Examples When one can use this method ? Examples

The First Main Result

For any φ ∈ V0 the decomposition φ = φH +

• j

φj, φH ∈ VH(Ω), φj ∈ V0(Ωj) satisfies a(φH, φH) +

N

• i=1

a(φi, φi) ≤ Cτa(φ, φ) : Good news: If τ is chosen properly so it takes care of the contrast, then the constant C depends on the max number of overlaps in the partition Ωj. Bad news: The dim(VH) depends on the topology of high contrast inclusions and could be large. More precisely: One asymptotically (with the contrast) small eigenvalue for each highly conductive connected component in each subdomain.

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Introduction Discretization Preconditioning of the Fine Grid Systems Overlapping DD method Some Numerical Examples When one can use this method ? Examples

Some examples Figure: Left: Geometry 1; Middle: Geometry 2; Right: periodic plus randomly distributed larger inclusions; all resolved by a fine mesh 256 × 256

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Introduction Discretization Preconditioning of the Fine Grid Systems Overlapping DD method Some Numerical Examples When one can use this method ? Examples

Illustration for the good and bad news

Standard coarse space VH

st

Spectral coarse space VH η # iter. dim VH

st

• cond. num.

# iter. dim VH

• cond. num.

1e2 29 49 2.29e1 25 76 15.59 1e4 55 49 1.79e3 18 162 6.20 1e6 66 49 1.77e5 19 162 6.19

Table: Geometry 1: Elliptic Problem, η is the contrast

Standard coarse space VH

st

Spectral coarse space VH η # iter. dim VH

st

• cond. num.

# iter. dim VH

• cond. num.

1e2 29 49 2.29e1 22 163 12.15 1e4 55 49 1.79e3 15 838 4.92 1e6 66 49 1.77e5 16 838 4.92

Table: Geometry 2: Elliptic Problem, η is the contrast

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Introduction Discretization Preconditioning of the Fine Grid Systems Overlapping DD method Some Numerical Examples When one can use this method ? Examples

Geometry 1: Brinkman

We note that Brinkman system is a saddle point problem. Using stream function in 2-D we were able to reduce it to the abstract form of a s.p.d. bilinear form. Details are in the paper Efendiev, Galvis, L., Willems, 2011

Standard coarse space VH

st

spectral coarse space VH η # iter. dim VH

st

• cond. num.

# iter. dim VH

• cond. num.

1e2 27 49 2.13e1 25 60 14.69 1e4 70 49 2.25e3 29 106 21.83 1e6 113 49 1.24e5 22 164 13.82

Table: Numerical results for Brinkman’s equation using standard coarse space VH st and spectral coarse spaces VH.

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Introduction Discretization Preconditioning of the Fine Grid Systems Overlapping DD method Some Numerical Examples When one can use this method ? Examples

Reducing the Dimension of the Coarse Space: Efendiev&Galvis, 2010

Following Efendiev & Galvis, 2010, instead of the standard partition of unity ξj we shall use multiscale partition of unity ξj that has restriction on each FE T ⊂ Ωj that solves the local problem (understood in the sense of fine-grid approximation) aT( ξj, ψ) = 0,

• ξj = ξj on ∂T.

Then we define the multiscale spectral coarse space

• VH = span{P(

ξjφj

i) :

i = 1, . . . , Lj ≤ Lj, j = 1, . . . , N}. Under some conditions, we have proved Lj ≤ Lj so that dim( VH) has

• decreased. By how much ?

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Introduction Discretization Preconditioning of the Fine Grid Systems Overlapping DD method Some Numerical Examples When one can use this method ? Examples

Reduction of the dimention of VH (Efendiev,Galvis,L.,Willems,2011)

Ωp

j

Ωs

j

• Ωj

Ωj Ωs

j,k,

k = 1, . . . , Lj Ωs

j,k,

k = Lj + 1, . . . , Lj Ωj Ωs

j,i

Ωp

j

Ωs

j

Reduction is equal to the # of inclusions that are entirely in one FE. Coarse spaces used: V st

H : Standard coarse space, e.g. piecewise bilinears

V ms

H

: Multiscale space by Graham, Lechner, & Scheichl (2007) VH: Spectral coarse space

• VH: Multiscale spectral coarse space

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Introduction Discretization Preconditioning of the Fine Grid Systems Overlapping DD method Some Numerical Examples When one can use this method ? Examples

Geometry 2: Second Order Elliptic Problem For comparison we define also the multiscale coarse space VH ms of Graham, Lechner, & Scheichl, 2007 V ms

H

= span{ ξj : ξj|∂Ωj = 0 j = 1, . . . , N}.

VH

st

VH

ms

η iter. dim VH

st

cond. iter. dim VH

st

cond. 1e2 27 49 2.1e1 19 49 8.7 1e4 70 49 2.2e3 44 49 5.6e2 1e6 113 49 1.2e5 66 49 5.6e4 VH

• VH

η iter. dim VH cond. iter. dim VH cond. 1e2 22 163 12.2 21 44 10.8 1e4 15 838 4.92 22 60 10.9 1e6 17 838 4.92 22 60 11.0

Table: Scalar elliptic equation: results for spectral VH, multiscale spectral VH, and multiscale VH ms coarse spaces.

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Introduction Discretization Preconditioning of the Fine Grid Systems Overlapping DD method Some Numerical Examples When one can use this method ? Examples

What about the computational complexity of this method ? In one word it is expensive ! Then the question is When one can see some advantages in using this method ? There is a number of situations when this method will be useful, these are mostly cases when the precomputed coarse grid space can be use multiple times:

1

In nonlinear problems when the heterogeneity is separated from the nonlinearity, e.g. Richards equation;

2

In computations of various scenarios of the boundary and source data;

3

In stochastic environment. Also, due it the inherently parallel nature of the constructions, these could be very useful in parallel computations.

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Introduction Discretization Preconditioning of the Fine Grid Systems Overlapping DD method Some Numerical Examples When one can use this method ? Examples

Richards Equation

we assume that the coefficient could be represented in the form k(x, p) = k(x)λ(x, p), where k(x) is highly heterogeneous. We have tested two models: Haverkamp λ(x, p) = A A + (|p|/B)γ , and Van Genuchten λ(x, p) = {1 − (α|p|/B)n−1[1 + (α|p|)n]−m}2 [1 + (α|p|)n]

m 2

. Here the coefficients A, B, γ, α, m, n are fitted to the experimental data.

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Introduction Discretization Preconditioning of the Fine Grid Systems Overlapping DD method Some Numerical Examples When one can use this method ? Examples

Richards Equation

Then for the nonlinear Richards equation we run a simple Picard iteration −div(k(x)λ(x, pn)∇pn+1) = f with pn being the previous iterate and apply our preconditioning technique for this linear equation. Details about the theory, the conditions it is valid and more computations we refer to Efendiev, Galvis, Ki Kang, L. (2011)

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Introduction Discretization Preconditioning of the Fine Grid Systems Overlapping DD method Some Numerical Examples When one can use this method ? Examples

Richards Equation (Left) Field 1: In blue are the regions where the coefficient is 1 and other in colors the regions where the coefficient is a random number between η and 10 ∗ η. (Right) Field 2: In blue are the regions where the coefficient is 1 and in red the regions where the coefficient is η, representing the contrast.

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Introduction Discretization Preconditioning of the Fine Grid Systems Overlapping DD method Some Numerical Examples When one can use this method ? Examples

Numerics for Haverkamp Model: A = 1, B = 1, γ = 1

Permeability field 1 Permeablity field 2 η R-iter CG-iter Max Cond R-iter CG-iter Max Cond 103 4 113 2.6e + 2 3 83 1.3e + 2 104 4 165 2.5e + 3 3 88 2.5e + 2 105 4 228 2.5e + 4 3 89 3.0e + 2 106 4 292 2.5e + 5 3 101 3.1e + 2

Table: Above: based on preconditioner using VH ms, dim(VH ms) = 81; Below: for preconditioner based on VH, dim( VH) = 166.

Permeability field 1 Permeability field 2 η R-iter CG-iter Max Cond R-iter CG-iter Max Cond 103 3 31 6.2 3 31 6.2 104 3 35 7.0 3 33 6.3 105 3 35 7.0 3 33 6.3 106 3 36 7.0 3 34 6.3

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Introduction Discretization Preconditioning of the Fine Grid Systems Overlapping DD method Some Numerical Examples When one can use this method ? Examples

Numerics for Haverkamp Model: A = 1, B = 0.01, γ = 0.5

Permeability field 1 Permeability field 2 η R-iter CG-iter Max Cond R-iter CG-iter Max Cond 103 11 120 3.9e + 2 8 98 1.9e + 2 104 11 190 3.6e + 3 8 104 3.9e + 3 105 11 250 3.6e + 4 8 107 4.6e + 4 106 11 315 3.6e + 5 8 111 4.6e + 5

Table: Above: based on preconditioner using VH ms, dim(VH ms) = 81; Below: for preconditioner based on VH, dim( VH) = 166.

Permeability field 1 Permeability field 2 η R-iter CG-iter Max Cond R-iter CG-iter Max Cond 103 8 38 9.6 8 38 9.6 104 8 41 9.7 8 40 9.7 105 8 41 9.7 8 41 9.7 106 8 43 9.7 8 42 9.7

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Introduction Discretization Preconditioning of the Fine Grid Systems Overlapping DD method Some Numerical Examples When one can use this method ? Examples

Numerics for van Genuhten Model

α = 0.005, B = 1, n = 2, m = 0.5 α = 0.01, B = 1, n = 4, m = 0.75 η R-iter CG-iter Max Cond R-iter CG-iter Max Cond 103 2 116 1.1e + 3 2 115 1.1e + 3 104 2 168 1.1e + 4 2 174 1.1e + 4 105 2 219 1.1e + 5 2 219 1.1e + 5 106 2 280 1.1e + 6 2 271 1.1e + 6

Table: Numerics for VH ms with k(x) taken as field 1, dim(VH ms) = 81.

α = 0.005, B = 1, n = 2, m = 0.5 α = 0.01, B = 1, n = 4, m = 0.75 η R-iter CG-iter Max Cond R-iter CG-iter Max Cond 103 2 33 6.8 1 33 6.8 104 2 34 6.8 1 34 6.8 105 2 35 6.8 1 35 6.8 106 2 36 6.8 1 36 6.8

Table: Numerics for VH with k(x) taken as field 1, dim( VH) = 166.

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Introduction Discretization Preconditioning of the Fine Grid Systems Overlapping DD method Some Numerical Examples When one can use this method ? Examples

Conclusions

In summary, multiscale computations involving media with heterogeneity on various spacial scales is a very challenging problem. For a class of problems modeling flows in porous media, we have developed: (1) methods for numerical upscaling of flows in porous media that works well in both limits, Darcy and Brinkman. (2) robust DD preconditioners that use coarse spaces augmented with functions (that take care of the high contrast on fine level) obtained by solving some local spectral problems.

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Introduction Discretization Preconditioning of the Fine Grid Systems Overlapping DD method Some Numerical Examples When one can use this method ? Examples

Thank you for your attention !!!

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Introduction Discretization Preconditioning of the Fine Grid Systems Overlapping DD method Some Numerical Examples When one can use this method ? Examples

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Introduction Discretization Preconditioning of the Fine Grid Systems Overlapping DD method Some Numerical Examples When one can use this method ? Examples

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