basic principles of weak galerkin finite element methods for pdes

Basic Principles of Weak Galerkin FiniteElement Methods for PDEs

Junping WangComputational Mathematics

Division of Mathematical SciencesNational Science Foundation

Arlington, VA 22230

Polytopal Element Methods in Mathematics and Engineering

Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt]Basic Principles of Weak Galerkin Finite Element Methods for PDEs

References in Weak Galerkin (WG)

1 Search “weak Galerkin” or “Junping Wang” on arXiV.org2 Partial List of Contributors:

Xiu Ye, University of ArkansasChunmei Wang, Georgia Institute of TechnologyLin Mu, Michigan State UniversityGuowei Wei, Michigan State UniversityYanqiu Wang, Oklahoma State UniversityLong Chen, University of California, IrvineShan Zhao, University of AlabamaRan Zhang, Jilin University, ChinaRuishu Wang, Jilin University, ChinaQilong Zhai, Jilin University, China


Talk Outline

1 Basics of Weak Galerkin Finite Element Methods (WG-FEM)

weak gradientstabilization (weak continuity)implementation and error analysis

2 An Abstract Framework3 WG-FEM for Model PDEs

mixed formulationhybridized WGlinear elasticity

4 Primal-Dual Weak Galerkin – What is it briefly?

The Fokker-Planck equationThe Cauchy problem for elliptic equationsAn abstract framework


Related Numerical Methods

1 FEM

2 Stabilized FEMs

3 MFD

4 DG, HDG

5 VEM


Second Order Elliptic Problems

Find u ∈ H10 (Ω) such that

(a∇u,∇v) = (f , v), ∀v ∈ H10 (Ω).

Procedures in the standard Galerkin finite element method:

1 Partition Ω into triangles or tetrahedra.

2 Construct a subspace, denoted by Sh ⊂ H10 (Ω), using

piecewise polynomials.

3 Seek for a finite element solution uh from Sh such that

(a∇uh,∇v) = (f , v) ∀v ∈ Sh.


An Out-of-Box Thinking

Replace uh and v by any distribution, and ∇uh and ∇v by anotherdistribution, say ∇wv as the generalized derivative, and seek for adistribution uh such that

(a∇wuh,∇wv) = (f , v), ∀v ∈ Sh.

Main Issues:

1 Functions in Sh are to be more general (as distributions orgeneralized functions) — a good feature

2 The gradient ∇v is computed weakly or as distributions —Questionable and fixable?

3 The numerical approximations are stable and convergent —questionable, how to fix?

4 The schemes are easy to implement and broadly applicable —Ideal, and can be achieved.


Motivation for WG

The classical gradient ∇u for u ∈ C 1(K ) can be computed as∫K∇u · φ = −

∫K

u ∇ · φ+

∫∂K

u(φ · n)

for all φ ∈ [C 1(K )]2.

The integrals on the right-hand side requires only u0 = u inthe interior of K , plus ub = u (trace) on the boundary ∂K .We symbolically have∫

K∇wu · φ = −

∫K

u0 ∇ · φ+

∫∂K

ubφ · n

Thus, u can be extended to u0, ub with ∇u being extendedto ∇wu.


Generalized Weak Derivatives: Foundation of WG

Weak Derivative

For any u = u0; ub with u0 ∈ L2(K ) and ub ∈ L2(∂K ), thegeneralized weak derivative of u in the direction ν is the followinglinear functional on H1(K ):

〈∂νu, φ〉 = −∫

Ku0∂νφ+

∫∂K

(n · ν)ubφ.

for all φ ∈ H1(K ).

The generalized weak derivative shall be called weak derivative.


Weak Functions

Weak Functions

A weak function on the region K refers to a generalized functionv = v0, vb such that v0 ∈ L2(K ) and vb ∈ L2(∂K ).

The first component v0 represents the value of v in theinterior of K , and the second component vb represents v onthe boundary of K .

vb may or may not be related to the trace of v0 on ∂K .

The space of weak functions:

W (K ) = v = v0, vb : v0 ∈ L2(K ), vb ∈ L2(∂K ).


Weak Gradient

For any v ∈ W (K ), the weak gradient of v is defined as abounded linear functional ∇wv in H1(K ) whose action oneach q ∈ H1(K ) is given by

〈∇wv , q〉K := −∫

Kv0∇ · qdK +

∫∂K

vbq · nds,

where n is the outward normal direction on ∂K .

The weak gradient is identical with the strong gradient forsmooth weak functions (e.g., as restriction of smoothfunctions).


Discrete Weak Gradients

For computational purpose, the weak gradient needs to beapproximated, which leads to discrete weak gradients, ∇w ,r , givenby ∫

K∇w ,rv · qdK = −

∫K

v0∇ · qdK +

∫∂K

vbq · nds,

for all q ∈ V (K , r). Here

V (K , r) j [Pr (K )]2 is a subspace.

Pr (K ) is the set of polynomials on K with degree r ≥ 0.

V (K , r) does not enter into the degrees of freedom indiscretization.


Weak Finite Element Spaces

Th: polygonal/polytopal partition of the domain Ω, shaperegular

construct local discrete elements

Wk(T ) := v = v0, vb : v0 ∈ Pk(T ), vb ∈ Pk−1(∂T ) .

patch local elements together to get a global space

Sh := v = v0, vb : v0, vb|T ∈ Wk(T ),∀T ∈ Th .

Weak FE Space

Weak finite element space with homogeneous boundary value:

S0h := v = v0, vb ∈ Sh, vb|∂T∩∂Ω = 0, ∀T ∈ Th .


Weak Finite Element Functions

Element Shape Functions for P1(K )/P0(∂K ):

φi = λi , 0, i = 1, 2, 3,

φ3+j = 0, τj, j = 1, 2, · · · ,N,

where N is the number of sides.

Figure: WG element


Shape Regularity for Polytopal Elements

xe

AeA

B

CD

EF

ne

Why Shape Regularity?The shape regularity is needed for (1) trace inequality, (2) inverseinequality, and (3) domain inverse inequality.


Weak Galerkin Finite Element Formulation

WG-FEM

Find uh = u0; ub ∈ S0h such that

(a∇wuh, ∇wv) + s(uh, v) = (f , v0), ∀v = v0; vb ∈ S0h ,

where

1 ∇wv ∈ Pk−1(T ) is the discrete weak gradient computedlocally on each element,

2 s(·, ·) is a stabilizer enforcing a weak continuity,

3 the stabilizer s(·, ·) measures the discontinuity of the finiteelement solution.


Depiction of the General WG Element

Pj−1(e) Pj−1(e)

Pj−1(e)

Pj(T )

The polynomial spaces Pj−1(T ) or Pj(T ) can be used for thecomputation of the weak gradient ∇w .


Stabilizer

Commonly used stabilizer:

s(w , v) = ρ∑T∈Th

h−1T 〈Qbw0 − wb,Qbv0 − vb〉∂T ,

where Qb is the L2 projection onto Pk−1(e), e ⊂ ∂T .

Discrete and computation-friendly stabilizer:

s(w , v) = ρ∑T∈Th

∑xj

(Qbw0 − wb)(xj) (Qbv0 − vb)(xj),

where xj is a set of (nodal) points on ∂T .


WG from the Minimization Perspective

The original problem can be characterized as

u = arg minv∈H1

0 (Ω)

(1

2(a∇v ,∇v)− (f , v)

)Weak Galerkin finite element scheme

uh = arg minv∈S0

h

(1

2(a∇wv ,∇wv)− (f , v0) +

1

2s(v , v)

).


L2 Projections and Weak Gradients

The following is a commutative diagram:

H1(T ) [L2(T )]d

Sh(T ) V (r ,T ) 0

∇

Qh Qh∇w

or equivalently

∇w (Qhu) = Qh(∇u), ∀u ∈ H1(T ).

Implication:

The discrete weak gradient is a good approximation of thetrue gradient operator.


Error Estimate

With the correct regularity assumptions, one has the followingoptimal order error estimate

‖Qhu − u0‖0 + h‖Qhu − uh‖1,h ≤ Chk+1‖u‖k+1.

Error estimates in negative norms hold true as well. Thus, theWG-FEM solutions are of superconvergent at certain places.


Computational Implementation

1 The stiffness matrix can be assembled as the sum of elementstiffness matrices

2 WG preserves physical quantities of importance: massconservation, energy conservation, etc

3 Suitable for parallel computation; element degree of freedomscan be eliminated in parallel

4 Suitable for multiscale analysis

5 Ideal for problems with discontinuous solutions


An Abstract Framework

Abstract Problem

Find u ∈ V such that

a(u, v) = f (v), ∀v ∈ V .

In application to PDE, the space V has certain embedded“continuities”, such as H1, H(div), H(curl), H2, orweighted-version of them. Assume

Vh: finite dimensional spaces that approximate V

ah(·, ·): bilinear forms on Vh × Vh that approximate a(·, ·)fh: linear functionals on Vh that approximate f .

sh(·, ·): stabilizers that provide necessary “smoothness”

In WG for Poisson equation, the first bilinear form refers to

ah(u, v) = (a∇wu,∇wv),

and the second one sh(·, ·) is the stabilizer.Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 [14pt]Basic Principles of Weak Galerkin Finite Element Methods for PDEs

Abstract WG

Abstract WG

Find uh ∈ Vh such that

ah(uh, v) + sh(uh, v) = fh(v), ∀ v ∈ Vh.

Some Assumptions:

Regularity: The solution of the abstract problem lies in asubspace H ⊂ V

The (discrete) norm ‖ · ‖Vhcan be extended to H + Vh so

that the topology of H is given by the family of semi-norms‖ · ‖Vh

, h ∈ (0, h0)

Boundedness and Coercivity: The bilinear form

awh(·, ·) := ah(·, ·) + sh(·, ·)

is bounded and coercive in Vh.


Almost Consistency

The Abstract WG algorithm is said to be almost consistent if thereexists a linear projection/interpolation operator Qh : V −→ Vh

such that for each uf ∈ H of the abstract problem, one has

Interpolation approximation:

limh→0

‖uf − Qhuf ‖Vh= 0

Residual consistency:

limh→0

supv 6=0,v∈Vh

|ah(Qhuf , v)− fh(v)|‖v‖Vh

= 0

Almost smoothness:

limh→0

supv 6=0,v∈Vh

|sh(Qhuf , v)|‖v‖Vh

= 0.


Convergence

Convergence

Assume that the Abstract WG algorithm is almost consistent.Furthermore, assume that awh(·, ·) is bounded and coercive in Vh,and the solution to the abstract problem lies in the subspace H.Then, we have

limh→0

‖uf − uh‖Vh= 0.

For the second order elliptic problem, the convergence can beinterpreted as

limh→0

(‖Qh(∇uf )−∇wuh‖0 + s(uh, uh)

12

)= 0.


WG: a New Paradigm of Discretization

Primal Formulation

Find uh such that

(a∇wuh,∇wv) + s(uh, v) = (f , v), ∀ v .

Key in WG: discrete weak gradient + stabilization to ensure aweak continuity of uh in H1.

Primal-Mixed Formulation

Find uh and qh such that

(a−1qh, v) + (∇wuh, v) = 0, ∀ v

s(uh,w)− (qh,∇ww) = (f ,w), ∀ w .

Key in WG: discrete weak gradient + stabilization to provide aweak continuity for uh in H1.


WG: a New Paradigm of Discretization

Dual-Mixed Formulation

Find qh and uh such that

s(qh, vh) + (a−1qh, v)− (u,∇w · v) = 0, ∀ v

(∇w · qh,w) = (f ,w), ∀ w .

Key in WG: discrete weak divergence + stabilization in velocity toensure a weak continuity with respect to H(div).

The space Pj+1(T ) is used for the computation of ∇w · v.

Lowest order element: pw constant for flux, pw linear forpressure

The finite element partition is of general polytopal type.


WG Can Be Hybridized

Hybridized Dual-Mixed Formulation

Find qh, uh, and λh such that

s(qh, vh) + (a−1qh, v)− (u,∇w · v) +∑T

〈λh, vb · n〉∂T = 0,

(∇w · qh,w) +∑T

〈σ,qb · n〉∂T = (f ,w).

Key in HWG: variable reduction in the sense that qh and uh canbe eliminated locally on each element.

Pjn Pjn

Pjn

[Pj ]d × Pj+1


Linear Elasticity Problems

Model Problem: Find a displacement vector field u satisfying

−∇ · σ(u) = f, in Ω,

u = u, on Γ.

Stress-strain relation for linear, homogeneous, and isotropicmaterials:

σ(u) = 2µε(u) + λ(∇ · u)I,

(Primal Form) Find u ∈ [H1(Ω)]d satisfying u = u on Γ and

2(µε(u), ε(v)) + (λ∇ · u,∇ · v) = (f, v), ∀v ∈ [H10 (Ω)]d .


Linear Elasticity Problems-Mixed Formulation

Introducing a pressure variable p = λ∇ · u, the elasticity problemcan be reformulated as follows:

(Mixed Formulation) Find u ∈ [H1(Ω)]d and p ∈ L2(Ω) satisfyingu = u on Γ, the compatibility condition

∫Ω λ

−1pdx =∫Γ u · nds,

2(µε(u), ε(v)) + (∇ · v, p) = (f, v), ∀v ∈ [H10 (Ω)]d ,

(∇ · u, q)− (λ−1p, q) = 0, ∀q ∈ L20(Ω).


Weak Divergence Operator

The space of weak vector-valued functions in K

V (K ) = v = v0, vb : v0 ∈ [L2(K )]d , vb ∈ [L2(∂K )]d.

Weak Divergence

The weak divergence of v ∈ V (K ), ∇w · v, is a bounded linearfunctional on H1(K ), so that its action on any φ ∈ H1(K ) is givenby

〈∇w · v, φ〉K := −(v0,∇φ)K + 〈vb · n, φ〉∂K .

Discrete Weak Divergence

The discrete weak divergence of v ∈ V (K ), denoted by∇w ,r ,K · v, is the unique polynomial, satisfying

(∇w ,r ,K · v, φ)K = −(v0,∇φ)K + 〈vb · n, φ〉∂K , ∀φ ∈ Pr (K ).


Weak Finite Element Spaces

[Pk−1]2 + PRM [Pk−1]

2 + PRM

[Pk−1]2 + PRM

[Pk ]2

∇wv ∈ [Pk−1(T )]2×2

∇w · v ∈ Pk−1(T ).


Weak Galerkin Algorithms for Primal Formulation

WG-FEM Primal

Find uh = u0,ub ∈ Vh with ub = Qbu on Γ such that for allv = v0, vb ∈ V 0

h ,∑T∈Th

2(µεw (uh), εw (v))T + (λ∇w · uh,∇w · v)T + s(uh, v) = (f, v0).

Qb: L2 projection onto [Pk−1(e)]d + PRM(e)

εw (u) = 12(∇wu +∇wuT )

Stablizer: s(w, v) =∑

T∈Thh−1T 〈Qbw0 −wb,Qbv0 − vb〉∂T


Weak Galerkin Algorithms for Mixed Formulation

WG-FEM in Mixed Form

Find uh = u0,ub ∈ Vh and ph ∈ Wh satisfying ub = Qbu on Γ,the compatibility condition (λ−1ph, 1) =

∫Γ u · nds, and

2(µεw (uh), εw (v))h + s(uh, v) + (∇w · v, ph)h = (f, v0),∀v ∈ V 0h ,

(∇w · uh, q)h − λ−1(ph, q) = 0,∀q ∈ W 0h .

Wh = q : q|T ∈ Pk−1(T ), T ∈ Th W 0h = Wh ∩ L2

0(Ω)

WG-FEM Primal = WG-FEM Mixed

The two weak Galerkin Algorithms are equivalent in the sense thatthe solutions to the two weak Galerkin Algorithms are identical toeach other.


Error Estimate in a Discrete H1-Norm

Error Estimates and Convergence in H1

Let the exact solution be sufficiently smooth such that(u; p) ∈ [Hk+1(Ω)]d × Hk(Ω). Let (uh; ph) ∈ Vh ×Wh be theweak Galerkin finite element solution.

|||Qhu−uh|||+λ−12 ‖Qhp−ph‖+|||Qhp−ph|||0 ≤ Chk(‖u‖k+1+‖p‖k),

where C is a generic constant independent of (u; p). Consequently,

|||u− uh|||+ λ−12 ‖p − ph‖+ |||p − ph|||0 ≤ Chk(‖u‖k+1 + ‖p‖k).


Error Estimate in L2-Norm

Error Estimates and Convergence in L2

Assume that the exact solution is sufficiently smooth such that(u; p) ∈ [Hk+1(Ω)]d × Hk(Ω). Let (uh; ph) ∈ Vh ×Wh be theweak Galerkin finite element solution. Then, under the regularityassumption, there exists a constant C , such that

‖Q0u− u0‖ ≤ Chk+s(‖u‖k+1 + ‖p‖k

).

Moreover,‖u− u0‖ ≤ Chk+s

(‖u‖k+1 + ‖p‖k

).


Numerical Results

Ω = (0, 1)2

the exact solution u =

(sin(x) sin(y)

1

)

Table: WG based on P1(T )/PRM(e), λ = 1, µ = 0.5.

1/h ‖u0 − Q0u‖ order ‖ub − Qbu‖ order |||uh − Qhu||| order

2 0.0750 – 0.0424 – 0.3103 –4 0.0192 1.97 0.0115 1.88 0.1566 0.998 0.0049 1.98 0.0031 1.87 0.0787 0.9916 0.0012 1.99 0.0008 1.93 0.0394 1.0032 0.0003 2.00 0.0002 1.97 0.0197 1.00


Numerical Results

Table: WG based on P1(T )/P1(e), λ = 1, µ = 0.5.

1h ‖uh − Q0u‖ order ‖ub − Qbu‖ order |||uh − Qhu||| order

2 0.0743 – 0.0424 – 0.3082 –4 0.0190 1.96 0.0113 1.90 0.1555 0.998 0.0048 1.98 0.0031 1.88 0.0782 0.9916 0.0012 1.99 0.0008 1.93 0.0392 1.0032 0.0003 2.00 0.0002 1.97 0.0196 1.00


Numerical Results: Locking-Free Experiments

Ω = (0, 1)2

the exact solution

u =

(sin(x) sin(y)cos(x) cos(y)

)+ λ−1

(xy

)

Table: WG based on P1(T )/PRM(e), µ = 0.5, and λ = 1.


2 0.0352 – 0.0331 – 0.1544 –4 0.0097 1.86 0.0120 1.46 0.0834 0.898 0.0026 1.91 0.0037 1.68 0.0433 0.9416 0.0007 1.96 0.0010 1.87 0.0220 0.9832 0.0002 1.98 0.0003 1.96 0.0110 0.99



Table: WG based on P1(T )/PRM(e), µ = 0.5, and λ = 1, 000, 000.


2 0.0344 – 0.0290 – 0.1447 –4 0.0100 1.79 0.0113 1.36 0.0773 0.908 0.0028 1.82 0.0038 1.59 0.0403 0.9416 0.0008 1.90 0.0011 1.81 0.0205 0.9732 0.0002 1.96 0.0003 1.93 0.0103 0.99



Table: WG based on P1(T )/P1(e), µ = 0.5, and λ = 1.


2 0.0341 – 0.0313 – 0.1518 –4 0.0093 1.87 0.0115 1.45 0.0816 0.908 0.0025 1.91 0.0036 1.67 0.0424 0.9516 0.0006 1.96 0.0010 1.86 0.0215 0.9832 0.0002 1.98 0.0003 1.95 0.0108 0.99



Table: WG based on P1(T )/P1(e), µ = 0.5, and λ = 1, 000, 000.


2 0.0340 – 0.0280 – 0.1439 –4 0.0098 1.79 0.0111 1.34 0.0768 0.918 0.0028 1.82 0.0037 1.58 0.0400 0.9416 0.0007 1.90 0.0011 1.81 0.0203 0.9732 0.0002 1.96 0.0003 1.93 0.0102 0.99


The Fokker-Planck Equation

describe the time evolution of the probability density functionof the velocity of a particle under the influence of drag forcesand random forces, as in Brownian motion.

assume the stochastic differential equation:

dXt = µ(Xt , t)dt + σ(Xt , t)dWt

the probability density f (x , t) for the random vector Xt

satisfies the Fokker-Planck equation

∂f

∂t+∇ · (µf )− 1

2

N∑i ,j=1

∂2ij [Dij f ] = 0,

where µ = (µ1, · · · , µN) is the drift vector and

Dij(x , t) =M∑

k=1

σik(x , t)σjk(x , t)

is the diffusion tensor.


Model Problem

Find u = u(x) satisfying

d∑i ,j=1

∂2ij(aiju) =g , in Ω,

u =0, on ∂Ω.

assume that a(x) is non-smooth,

weak formulation is given by seeking u such that

d∑i ,j=1

(u, aij∂2ijw) = (g ,w), ∀v ∈ H2(Ω) ∩ H1

0 (Ω).


A Cauchy Problem for Elliptic Equations

The model problem seeks u such that

−∆u = f , in Ω,

u = 0, on Γ ⊂ ∂Ω,∂u

∂n= ψ, on Γ ⊂ ∂Ω.

This is usually an ill-posed problem which does not have a solutionor has many solutions. Let Γc = ∂Ω/Γ. A variational form for thisproblem seeks u ∈ H1

0,Γ(Ω) such that

(∇u,∇w) = (f ,w) + 〈ψ,w〉Γ,

for all w ∈ H10,Γc (Ω).


An Abstract Problem

Let V and W be two Hilbert spaces

b(·, ·) is a bilinear form on V ×W

The inf-sup condition of Babuska and Brezzi is satisfied.

The spaces U and V have certain embedded “continuities”,such as L2, H1, H(div), H(curl), H2, or weighted-version ofthem.

Abstract Problem

Find u ∈ V such that b(u,w) = f (w) for all w ∈ W . Here f is abounded linear functional on W .

Goal: Design finite element methods by using weak Galerkinapproach.


Specific Examples (Revisited)

For the Fokker-Planck, we have V = L2 and W = H2 ∩ H10

and

b(v ,w) :=d∑

i ,j=1

(v , aij∂2ijw).

For the Cauchy problem for Poisson equation,V ×W = H1

0,Γ(Ω)× H10,Γc (Ω) and

b(v ,w) := (∇v ,∇w).

Note that the inf-sup condition may not be satisfied.


An Abstract Primal-Dual Formulation

Primal equation in color blue,

Dual equation in color red,

They are connected by stabilizers with weak continuity.

WG-FEM

Find uh ∈ Vh and λh ∈ Wh such that

s1(uh, v)− bh(v , λh) = 0, ∀v ∈ Vh

s2(λh,w) + bh(uh,w) = fh(w), ∀ w ∈ Wh.

s1(·, ·): stabilizer/smoother in Vh

s2(·, ·): stabilizer/smoother in Wh


The Primal-Dual WG for Elliptic Cauchy Problem

the bh(·, ·)-form is given by

bh(v ,w) := (∇wv ,∇ww),

Both stabilizers are given by:

s(u, v) =∑T∈Th

h−1T 〈u0−ub, v0−vb〉∂T+hT 〈∂nu0−ugn, ∂nv0−vgn〉∂T .

Error estimates and numerical experiments are on the way.


Numerical Tests

the right-hand side f is given by2a11+2a22−10a12−10a21−50 sin(30(x−0.5)2+30(y−0.5)2)

the boundary condition u = x2 + 2y2 − 5xy on ∂Ω

Ω = (0, 1)2

Figure: WG finite element solution with coefficients a11 = 3,a12 = a21 = 1 and a22 = 2 in mesh size 1.250e-01 (ρh is piecewise linearfunction).


Numerical Tests

Figure: WG finite element solution with coefficients a11 = 10,a12 = a21 = (0.25x)2(0.25y)2, a22 = 10 in mesh size 1.250e-01 (ρh ispiecewise linear function).


Numerical Tests

Figure: WG finite element solution with coefficients a11 = 3,a12 = a21 = 1 and a22 = 2 in mesh size 1.250e-01 (ρh is a piecewiseconstant).


Numerical Tests

Figure: WG finite element solution with coefficients a11 = 10,a12 = a21 = (0.25x)2(0.25y)2, a22 = 10 in mesh size 1.250e-01 (ρh is apiecewise constant).


Current and Future Research

The current and future research projects include

1 WG on polytopal partitions with curved sides,

2 Fokker-Planck equation,

3 Nonlinear PDEs such as MHD and Cahn-Hillard equations,

4 Variational problems where the trial and test spaces aredifferent, but an inf-sup condition is satisfied,

5 Applications and efficient implementation issues.


Thanks for your attention!


basic principles of weak galerkin finite element methods for pdes

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