Interface Problem The Natural Method Poisson Interface Problem Stokes Interface Problem High-Contrast Transmission Pro Higher-0rder Finite Element Methods for Elliptic Problems with Interfaces Marcus Sarkis Mathematical Sciences Deptartment, WPI May 12, 2015. Hydraulic Fracturing IMA Workshop Joint work with Johnny Guzm´ an and Manuel Sanchez (Brown) Marcus Sarkis [email protected]Mathematical Sciences Deptartment, WPI FEM for an Interface Problem 1
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Higher-0rder Finite Element Methods for Elliptic Problems ...€¦ · Outline 1 Interface Problem 2 The Natural Method 3 Poisson Interface Problem 4 Stokes Interface Problem 5 High-Contrast
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Interface Problem The Natural Method Poisson Interface Problem Stokes Interface Problem High-Contrast Transmission Problem Concluding Remarks & Future Work References
Higher-0rder Finite Element Methods
for Elliptic Problems with Interfaces
Marcus Sarkis
Mathematical Sciences Deptartment, WPI
May 12, 2015. Hydraulic Fracturing IMA Workshop
Joint work with Johnny Guzman and Manuel Sanchez (Brown)
Marcus Sarkis [email protected] Mathematical Sciences Deptartment, WPI
FEM for an Interface Problem 1
Outline
1 Interface Problem
2 The Natural Method
3 Poisson Interface Problem
4 Stokes Interface Problem
5 High-Contrast Transmission Problem
6 Concluding Remarks & Future Work
7 References
Outline
1 Interface Problem
2 The Natural Method
3 Poisson Interface Problem
4 Stokes Interface Problem
5 High-Contrast Transmission Problem
6 Concluding Remarks & Future Work
7 References
Interface Problem
Interface Problem
−∆u± = f in Ω±
u = 0 on ∂Ω
[u] = α on Γ
[∇u · n] = β on Γ
We denote
[u] = u+ − u−
[∇u · n] = ∇u− · n− +∇u+ · n+
Illustration of interface
Illustration of Ω, Γ.
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Ω
Γ
Equivalent Formulation
For simplicity we assume here that α ≡ 0
−∆u = f + F in Ω ⊂ R2
u = 0 on ∂Ω
F (x) =
∫ A
0
β(s)δ(x−X(s))ds ∀x ∈ Ω
• X : [0, A)→ Γ is the arch-length parametrization of Γ
• δ is a two-dimensional Dirac function
• This could be thought of as Peskin’s Formulation
Previous Work
Some Finite Difference methods
• IBM Peskin (77)• IIM LeVeque, Li (94)• Beale, A. Layton (96)• Mori (98)• Marquez, Nave, Rosales (11)
Some Finite Element Methods
• Boffi, Gastaldi (03)• Gong, Li, Li (07)• He, Lin, Lin (11)• Adjerid, Ben-Romd, Lin (14)
Outline
1 Interface Problem
2 The Natural Method
3 Poisson Interface Problem
4 Stokes Interface Problem
5 High-Contrast Transmission Problem
6 Concluding Remarks & Future Work
7 References
Variational Formulation for Interface Problem
Find u ∈ H10 (Ω) such that∫
Ω
∇u · ∇vdx =
∫Ω
fvdx+
∫Γ
βvds
for all v ∈ H10 (Ω).
The Natural Method
Find uh ∈ Vh such that;∫Ω
∇uh · ∇v dx =
∫Ω
f v dx+
∫Γ
βv ds ∀v ∈ Vh
Ex: Vh is the conforming piecewise polynomials of degree k
Numerical Example
Exact solution of the interface problem in Ω = [−1, 1]2
u(x) =
1 if r ≤ R1− log( r
R) if r > R
where r = ‖x‖2 and R = 1/3
Then, the data are given by f± = 0, α = 0 and β = 1R
Semi-log plot of gradient error for the natural method with h = .0028.|∇eNh (dT )| (red) for every triangle T and curve 2h + log(1/h)(h/d)2 (blue).
The distance d in the x-axis varies from 0 to√h.
Outline
1 Interface Problem
2 The Natural Method
3 Poisson Interface Problem
4 Stokes Interface Problem
5 High-Contrast Transmission Problem
6 Concluding Remarks & Future Work
7 References
Poisson Interface Problem
Goal
Recover the high accuracy of the natural method
Vh =v ∈ H1
0 (Ω) : v|T ∈ Pk(T ) ∀T ∈ Th
The set T Γh = T ∈ Th : T ∩ Γ 6= ∅
Find uh ∈ Vh such that for all v ∈ Vh the following holds∫Ω
∇uh · ∇v dx =
∫Ω
f vdx+
∫Γ
βv ds−∑T∈T Γ
h
∫T
∇wuT · ∇v dx
Main Result
Ih : Lagrange interpolation operator onto the Vh
Theorem
If u± ∈ Ck+1(Ω±
) , f |Ω± ∈ Ck−1(Ω±
), β smooth, then
‖∇(Ihu− uh)‖L∞(Ω) ≤ C hk(‖u+‖Ck+1(Ω+) + ‖u−‖Ck+1(Ω−)
)
What do we need?
We will construct wuT , for T ∈ T Γ
h , such that satisfies
‖∇(Ihu+ wuT − u)‖L∞(T±) ≤ Chk
where T± = T ∩ Ω±
• P1(T ) conforming correction, [GSS 14]
• Pk(T ) nonconforming correction, [GSS 15a]
Interface Problem The Natural Method Poisson Interface Problem Stokes Interface Problem High-Contrast Transmission Problem Concluding Remarks & Future Work References
Construction of wuT
Consider the local space for T ∈ T Γh
Sk(T ) =w ∈ L2(T ) : w|T± ∈ Pk(T±)
For each T ∈ T Γh , let wu
T ∈ Sk(T ) such that[Dk−`η wu
T (x`,Ti )]
=[Dk−`η u(x`,Ti )
]for 0 ≤ i ≤ ` and 0 ≤ ` ≤ k
IhwuT = 0
Marcus Sarkis [email protected] Mathematical Sciences Deptartment, WPI
FEM for an Interface Problem 19
Interface Problem The Natural Method Poisson Interface Problem Stokes Interface Problem High-Contrast Transmission Problem Concluding Remarks & Future Work References
Figure: Illustration of notation. T± = T ∩ Ω±
Marcus Sarkis [email protected] Mathematical Sciences Deptartment, WPI
FEM for an Interface Problem 20
Without Knowing u
For each x`,Ti ∈ Γη = an+ bt[
Dk−`η u(x`,Ti )
]= a
[Dk−`n u(x`,Ti )
]+ b
[Dk−`t u(x`,Ti )
][D`ηu(x`,Ti )
]=∑j=0
(l
j
)ajb`−j
[DjnD
`−jt u(x`,Ti )
]The RHS obtained from normal and tangential derivatives of f andtangential derivaties of α and β. Derived from the equations
−∆u = f, [u] = α, [Dnu] = β
Existence and Uniqueness
Lemma
Given data ci,` for 0 ≤ i ≤ ` and 0 ≤ ` ≤ k. There exist aunique function in w ∈ Sk(T ) such that[
Dk−`η w(x`,Ti )
]= ci,` for 0 ≤ i ≤ ` and 0 ≤ ` ≤ k
Ihw = 0
• Note that is a square system of (k + 1)(k + 2) equations• Explict construction using
First component of the velocity (uh)1 (left) and pressure ph (right).
Outline
1 Interface Problem
2 The Natural Method
3 Poisson Interface Problem
4 Stokes Interface Problem
5 High-Contrast Transmission Problem
6 Concluding Remarks & Future Work
7 References
High-Contrast Coefficients
Interface Problem
−ρ±∆u± = f± in Ω±
u = 0 on ∂Ω
[u] = 0 on Γ
[ρ∇u · n] = 0 on Γ
Denote
[u] = u+ − u−
[ρ∇u · n] = ρ−∇u− · n− + ρ+∇u+ · n+
Discontinuous Galerkin
Find uh ∈ Vh such that
ah(uh, vh) = (f, vh) for all vh ∈ Vh,Bilinear Form
ah(w, v) :=
∫Ωρ∇hw · ∇hv −
∑e∈EΓ
h
∫e
(ρ∇hv · n
[w] +
ρ∇hw · n
[v])
+∑
e∈EΓh
(γ
|e−|
∫e−
ρ−
[w] · [v] +γ
|e+|
∫e+
ρ+
[w] · [v]
)
∑e∈EΓ
h
(|e−|
∫e−
ρ−
[∇hv · n] [∇hw] + |e+|∫e+
ρ+
[∇hv · n] [∇hw · n]
)
Here we denote by ∇hv the functions whose restriction to each T± with T ∈ Th is ∇v
Main Result
Theorem
The error estimate that we prove is of the form
‖u− uh‖V ≤ C h(√
ρ−‖u‖H2(Ω−) +√ρ+‖u‖H2(Ω+)
)
Summary & Future Work
Summarizing
Analysis of the natural method
Higher-order method for Poisson interface problem
Higher-order method for Stokes interface problem
Second-order high constrast problems
Future Work
Fracturing problems
Time-evolving problems
References
GSS 14 J. Guzman, M. Sanchez-Uribe and S. On the accuracy offinite element approximations to a class of interfaceproblems. Math. Comp. Accepted, 2014
GSS 15a J. Guzman, M. Sanchez-Uribe and S. Higher-order finiteelements methods for elliptic problems with interfaces.Submitted 2015.
GSS 15b J. Guzman, M. Sanchez-Uribe and S. A finite elementmethod for high-contrast interface problems with errorestimates independent of contrast. To be submitted 2015.