stabilized control volume finite element method for drift-diffusion · 2017. 8. 1. · stabilized...
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Stabilized Control Volume Finite Element Method forDrift-Diffusion
Kara PetersonPavel Bochev Mauro Perego Suzey Gao
Gary Hennigan Larry Musson
Sandia National Laboratories
UNM Applied Math SeminarFebruary 6, 2017
SAND2017-1293PE
CCRCenter for Computing Research
Sandia National Laboratories is a multi-mission laboratory managed and operated by Sandia Corporation,a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s
National Nuclear Security Administration under contract DE-AC04-94AL85000.
2/6/2017 1
Outline
1 Introduction to CVFEM
2 Scharfetter-Gummel Upwinding
3 Stabilization with Multi-dimensional S-G Upwinding
4 Extension to FEM
5 Multi-scale Stabilized CVFEM
6 Conclusions
2/6/2017 2
Motivation
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Scaled coupled drift-diffusion equations for semi-conductors
∇ · (λ2E)− (p− n+ C) = 0 and E = −∇ψ in Ω× [0, T ]
∂n
∂t−∇ · Jn +R(ψ, n, p) = 0 and Jn = µnEn+Dn∇n in Ω× [0, T ]
∂p
∂t+∇ · Jp +R(ψ, n, p) = 0 and Jp = µpEp−Dp∇p in Ω× [0, T ]
E - electric field n - electron densityψ - electric potential p - hole density
Want a numerical scheme that is accurate and stable in the strong drift regime
2/6/2017 3
CVFEM Formulation
Electron continuity equation
∂n
∂t−∇ · J +R = 0
J = µEn+D∇n
Finite element approximation of theelectron densitynh(t,x) =
∑j
nj(t)φj(x)
For each primary grid vertex, definea control volume
Integrate over the control volume andapply the divergence theorem
2/6/2017 4
CVFEM Formulation
Electron continuity equation
∂n
∂t−∇ · J +R = 0
J = µEn+D∇n
Finite element approximation of theelectron densitynh(t,x) =
∑j
nj(t)φj(x)
For each primary grid vertex, definea control volume
Integrate over the control volume andapply the divergence theorem
2/6/2017 5
CVFEM Formulation
Electron continuity equation
∂n
∂t−∇ · J +R = 0
J = µEn+D∇n
Ci
vi
Finite element approximation of theelectron densitynh(t,x) =
∑j
nj(t)φj(x)
For each primary grid vertex, definea control volume
Integrate over the control volume andapply the divergence theorem
2/6/2017 6
CVFEM Formulation
Electron continuity equation
∂n
∂t−∇ · J +R = 0
J = µEn+D∇n
Ci
vi
Finite element approximation of theelectron densitynh(t,x) =
∑j
nj(t)φj(x)
For each primary grid vertex, definea control volume
Integrate over the control volume andapply the divergence theorem
∫Ci
∂n
∂tdV −
∫∂Ci
J(n)·~ndS+
∫Ci
R(n), dV = 0
2/6/2017 7
CVFEM Formulation
Electron continuity equation
∂n
∂t−∇ · J +R = 0
J = µEn+D∇n
Ci
vi
Finite element approximation of theelectron densitynh(t,x) =
∑j
nj(t)φj(x)
For each primary grid vertex, definea control volume
Integrate over the control volume andapply the divergence theorem
∫Ci
∂nh
∂tdV−
∫∂Ci
J(nh)·~n dS+
∫Ci
R(nh) dV = 0
2/6/2017 8
CVFEM Formulation∫∂Ci
J(nh) · ~n dS =∑j
nj(t)
∫∂Ci
(µEφj +D∇φj) · ~n dS
Kij =
∫∂Ci
(µEφj +D∇φj) · ~n dS
Ci
vi
In strong drift regime this formulation can resultin instabilities
Unstabilized CVFEM Correct Solution
Want an approximation J(nh) · ~n that includes information about the drift
2/6/2017 9
CVFEM Formulation∫∂Ci
J(nh) · ~n dS =∑j
nj(t)
∫∂Ci
(µEφj +D∇φj) · ~n dS
Kij =
∫∂Ci
(µEφj +D∇φj) · ~n dS
Ci
vi
In strong drift regime this formulation can resultin instabilities
Unstabilized CVFEM Correct Solution
Want an approximation J(nh) · ~n that includes information about the drift
2/6/2017 10
CVFEM Formulation∫∂Ci
J(nh) · ~n dS =∑j
nj(t)
∫∂Ci
(µEφj +D∇φj) · ~n dS
Kij =
∫∂Ci
(µEφj +D∇φj) · ~n dS
Ci
vi
In strong drift regime this formulation can resultin instabilities
Unstabilized CVFEM Correct Solution
Want an approximation J(nh) · ~n that includes information about the drift
2/6/2017 11
Scharfetter-Gummel Upwinding
On edge eij solve 1-d boundary value problemalong edge for constant Jij
dJij
ds= 0; Jij = µEijn(s) +D
dn(s)
ds
n(0) = ni and n(hij) = nj
Jij =aijDnhij
(nj (coth(aij) + 1)− ni (coth(aij)− 1))
where aij =hijEijµ
2D, Eij = − (ψj−ψi)
hij
Ci
vi
vj
Jij
∫∂Ci
Jn · ~n dS ≈∑
∂Cij∈∂Ci
Jij |∂Cij |
D. L. Scharfetter and H. K Gummel, Large-signal analysis of a silicon read diode oscillator, IEEE Transactions onElectron Devices 16, 64-77, 1969.
2/6/2017 12
Scharfetter-Gummel Upwinding
On edge eij solve 1-d boundary value problemalong edge for constant Jij
dJij
ds= 0; Jij = µEijn(s) +D
dn(s)
ds
n(0) = ni and n(hij) = nj
Jij =aijDnhij
(nj (coth(aij) + 1)− ni (coth(aij)− 1))
where aij =hijEijµ
2D, Eij = − (ψj−ψi)
hij
Ci
vi
vj
Jij
∫∂Ci
Jn · ~n dS ≈∑
∂Cij∈∂Ci
Jij |∂Cij |
D. L. Scharfetter and H. K Gummel, Large-signal analysis of a silicon read diode oscillator, IEEE Transactions onElectron Devices 16, 64-77, 1969.
2/6/2017 13
Scharfetter-Gummel Upwinding
On edge eij solve 1-d boundary value problemalong edge for constant Jij
dJij
ds= 0; Jij = µEijn(s) +D
dn(s)
ds
n(0) = ni and n(hij) = nj
Jij =aijDnhij
(nj (coth(aij) + 1)− ni (coth(aij)− 1))
where aij =hijEijµ
2D, Eij = − (ψj−ψi)
hij
Ci
∂Cij
vi
vj
Jij
∫∂Ci
Jn · ~n dS ≈∑
∂Cij∈∂Ci
Jij |∂Cij |
D. L. Scharfetter and H. K Gummel, Large-signal analysis of a silicon read diode oscillator, IEEE Transactions onElectron Devices 16, 64-77, 1969.
2/6/2017 14
Scharfetter-Gummel Upwinding
On edge eij solve 1-d boundary value problemalong edge for constant Jij
dJij
ds= 0; Jij = µEijn(s) +D
dn(s)
ds
n(0) = ni and n(hij) = nj
Jij =aijDnhij
(nj (coth(aij) + 1)− ni (coth(aij)− 1))
where aij =hijEijµ
2D, Eij = − (ψj−ψi)
hij
Ci
∂Cik
vi
vj
Jijn n
On structured grids, Jij is agood estimate of J · ~n on ∂Cij∫
∂Ci
Jn · ~n dS ≈∑
∂Cij∈∂Ci
Jij |∂Cij |
D. L. Scharfetter and H. K Gummel, Large-signal analysis of a silicon read diode oscillator, IEEE Transactions onElectron Devices 16, 64-77, 1969.
2/6/2017 15
Scharfetter-Gummel Upwinding
On edge eij solve 1-d boundary value problemalong edge for constant Jij
dJij
ds= 0; Jij = µEijn(s) +D
dn(s)
ds
n(0) = ni and n(hij) = nj
Jij =aijDnhij
(nj (coth(aij) + 1)− ni (coth(aij)− 1))
where aij =hijEijµ
2D, Eij = − (ψj−ψi)
hij
Ci
vi
vj
Jijn n
On unstructured grids, Jij is nolonger a good estimate of J · ~n∫
∂Ci
Jn · ~n dS ≈∑
∂Cij∈∂Ci
Jij |∂Cij |
D. L. Scharfetter and H. K Gummel, Large-signal analysis of a silicon read diode oscillator, IEEE Transactions onElectron Devices 16, 64-77, 1969.
2/6/2017 16
Multi-dimensional S-G Upwinding
Idea: Use H(curl)-conforming finiteelements to expand edge currents into
primary cell
Exponentially fitted current density
JE(x) =∑eij
hijJij−→W ij(x)
J12
J23
J30
J01
JE
∫eij
−→W ij · trsdl = δrsij
−→W 01 =
(1−y
4, 0) −→
W 12 =(0, 1+x
4
) −→W 23 =
(− 1+y
4, 0) −→
W 30 =(0,− 1−x
4
)P. Bochev, K. Peterson, X. Gao A new control-volume finite element method for the stable and accurate solution ofthe drift-diffusion equations on general unstructured grids, CMAME, 254, pp. 126-145, 2013.
2/6/2017 17
Multi-dimensional S-G Upwinding
Exponentially fitted current density
JE(x) =∑eij
hijJij−→W ij(x)
Nodal space, GhD(Ω), and edge element space, ChD(Ω), belong to an exact sequence
given φi ∈ GhD(Ω), then ∇φi ∈ ChD(Ω)
For the lowest order case
∇φi =∑
eij∈E(vi)
σij−→W ij , σij = ±1
If the carrier drift velocity µE = 0, Jij =D(nj − ni)
hij
JE =∑
eij∈E(Ω)
D(nj − ni)−→W ij =
∑vi∈V (Ω)
Dnj∇φj = J(nh)
2/6/2017 18
Multi-dimensional S-G Upwinding
Standard S-G∫∂Ci
J · ~ndS ≈∑
∂Cij∈∂Ci
Jij |∂Cij |
Ci
∂Cij
vi
vj
Jij
Multi-dimensional S-G∫∂Ci
J · ~ndS ≈∑
∂CKij∈∂Ci
JE · ~n |∂CKij |
Ci
vi
JE
JE
2/6/2017 19
Multi-dimensional S-G Upwinding
Standard S-G∫∂Ci
J · ~ndS ≈∑
∂Cij∈∂Ci
Jij |∂Cij |
Ci
∂Cik
vi vkJik
Multi-dimensional S-G∫∂Ci
J · ~ndS ≈∑
∂CKij∈∂Ci
JE · ~n |∂CKij |
Ci
vi
JE
JE
2/6/2017 20
Multi-dimensional S-G Upwinding
Standard S-G∫∂Ci
J · ~ndS ≈∑
∂Cij∈∂Ci
Jij |∂Cij |
Ci
∂Cil
vi
vl
Jil
Multi-dimensional S-G∫∂Ci
J · ~ndS ≈∑
∂CKij∈∂Ci
JE · ~n |∂CKij |
Ci
vi
JE
JE
2/6/2017 21
Multi-dimensional S-G Upwinding
Standard S-G∫∂Ci
J · ~ndS ≈∑
∂Cij∈∂Ci
Jij |∂Cij |
Ci
∂Cim
vivm Jim
Multi-dimensional S-G∫∂Ci
J · ~ndS ≈∑
∂CKij∈∂Ci
JE · ~n |∂CKij |
Ci
vi
JE
JE
2/6/2017 22
Multi-dimensional S-G Upwinding
Standard S-G∫∂Ci
J · ~ndS ≈∑
∂Cij∈∂Ci
Jij |∂Cij |
Ci
∂Cim
vivm Jim
Multi-dimensional S-G∫∂Ci
J · ~ndS ≈∑
∂CKij∈∂Ci
JE · ~n |∂CKij |
Ci
vi
JE
JE
2/6/2017 23
Multi-dimensional S-G Upwinding
Standard S-G∫∂Ci
J · ~ndS ≈∑
∂Cij∈∂Ci
Jij |∂Cij |
Ci
∂Cim
vivm Jim
Multi-dimensional S-G∫∂Ci
J · ~ndS ≈∑
∂CKij∈∂Ci
JE · ~n |∂CKij |
Ci
vi
JE
JE
2/6/2017 24
Multi-dimensional S-G Upwinding
Standard S-G∫∂Ci
J · ~ndS ≈∑
∂Cij∈∂Ci
Jij |∂Cij |
Ci
∂Cim
vivm Jim
Multi-dimensional S-G∫∂Ci
J · ~ndS ≈∑
∂CKij∈∂Ci
JE · ~n |∂CKij |
Ci
viJE
JE
2/6/2017 25
Manufactured Solution on Structured Grid
Steady-state manufactured solution
−∇ · J +R = 0 in Ωn = g on ΓD
n(x, y) = x3 − y2
µE = (− sinπ/6, cosπ/6)
CVFEM-SG FVM-SGL2 error H1 error L2 error H1 error
Grid D = 1× 10−3
32 0.4373E-02 0.7620E-01 0.4364E-02 0.7572E-0164 0.2108E-02 0.4954E-01 0.2107E-02 0.4937E-01
128 0.9870E-03 0.3089E-01 0.9870E-03 0.3084E-01Rate 1.095 0.681 1.094 0.679Grid D = 1× 10−5
32 0.4732E-02 0.7897E-01 0.4723E-02 0.7850E-0164 0.2517E-02 0.5477E-01 0.2515E-02 0.5460E-01
128 0.1298E-02 0.3834E-01 0.1298E-02 0.3828E-01Rate 0.955 0.514 0.955 0.514
CVFEM-SG control volume finite element method with multi-dimensional S-G upwindingFVM-SG finite volume method with 1-d S-G upwinding
2/6/2017 26
Robustness on Unstructured Grids
Manufactured solution on randomlyperturbed grids
−∇ · J +R = 0 in Ωn = g on ΓD
n(x, y) = x+ yµE = (− sinπ/6, cosπ/6)
D = 1.0× 10−5
Grid Error CVFEM-SG FVM-SGO(h2) L2 0.88047E-06 0.83467E-04
H1 0.14524E-03 0.13426E-01O(h) L2 0.38594E-04 0.36687E-02
H1 0.65073E-02 0.58993E+00O(1) L2 0.21849E-02 0.24247E+00
H1 0.36473E+00 0.38469E+02
Grid CVFEM-SG FVM-SG
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2/6/2017 27
Charon
To solve coupled drift-diffusion equations CVFEM-SG has been implementedin Sandia’s Charon code
Electrical transport simulation code for semiconductor devices, solvingPDE-based nonlinear equations
Built with Trilinos libraries (https://github.com/trilinos/Trilinos)that provide
Framework and residual-based assembly (Panzer, Phalanx)Linear and Nonlinear solvers (Belos, Nox, ML, etc)Temporal and spatial discretization (Tempus, Intrepid, Shards)Automatic differentiation (Sacado)Advanced manycore performance portability (Kokkos)
2/6/2017 28
PN Diode
PN Diode coupled drift-diffusion equations
∇ · (ε0εsi∇ψ) = −q(p− n +Nd −Na) in Ω
−∇ · Jn + R(ψ, n, p) = 0 in Ω
∇ · Jp + R(ψ, n, p) = 0 in Ω
R(ψ, n, p) =np−n2
iτp(n+ni)+τn(p+ni)
+(cnn + cpp)(np− n2i )
PN Diode
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Compare with Steamlined Upwind Petrov-Galerkin(SUPG) method∫Ω
(µE +D∇n) · ∇ψdV +
∫ΩRψdV +
∫Ωτ(µE · ∇n)(µE · ∇ψ)dV = 0
τ - a parameter that depends on mesh size and velocity
2/6/2017 29
PN DiodeMesh Dependence Study
hx = 0.01µm hx = 0.02µm hx = 0.05µm
0 0.2 0.4 0.6 0.8 1
−15
−10
−5
0
5
10
15
x [µ m]
Ele
ctric
al P
oten
tial i
n un
it of
kBT
/q
SUPG−FEMCVFEM−SG
0 0.2 0.4 0.6 0.8 1
−15
−10
−5
0
5
10
15
x [µ m]
Ele
ctric
al P
oten
tial i
n un
it of
kBT
/q
SUPG−FEMCVFEM−SG
0 0.2 0.4 0.6 0.8 1
−15
−10
−5
0
5
10
15
x [µ m]
Ele
ctric
al P
oten
tial i
n un
it of
kBT
/q
SUPG−FEMCVFEM−SG
0 0.2 0.4 0.6 0.8 110
−14
10−12
10−10
10−8
10−6
10−4
10−2
100
x [µ m]
Car
rier
Den
sity
[x 1
017 c
m−
3 ]
0 0.2 0.4 0.6 0.8 110
−14
10−12
10−10
10−8
10−6
10−4
10−2
100
x [µ m]
Car
rier
Den
sity
[x 1
018 c
m−
3 ]
0 0.2 0.4 0.6 0.8 110
−14
10−12
10−10
10−8
10−6
10−4
10−2
100
x [µ m]
Car
rier
Den
sity
[x 1
017 c
m−
3 ]
2/6/2017 30
PN DiodeStrong Drift Case
Na = Nd = 1.0× 1018cm−3, Va = −1.5V
0 0.2 0.4 0.6 0.8 1−80
−70
−60
−50
−40
−30
−20
−10
0
10
20
x [µ m]
Ele
ctric
al P
oten
tial i
n un
it of
kBT
/q
SUPG−FEMCVFEM−SG
FEM-SUPG solution develops undershoots and becomes negative in junctionregion, while CVFEM-SG exhibits only minimal undershoots and values
remain positive.
2/6/2017 31
N-Channel MOSFET
source drain
gate
substrate
Nd Nd
Na
Ωox
Ωsi
Vg = 2 V
Vd = 1 V
Vs = 0
Vsub = 0
MOSFET coupled drift-diffusion equations
∇ · (ε0εox∇ψ) = 0 in Ωox
∇ · (ε0εsi∇ψ) = −q(ni exp
(−qψkBT
)− n +Nd −Na
)in Ωsi
∇ · (nµn∇ψ −Dn∇n) = 0 in Ωsi
FEM-SUPG CVFEM-SG
FEM-SUPG solution exhibits larger negative values over a wider area than theCVFEM-SG solution.
2/6/2017 32
N-Channel MOSFET
source drain
gate
substrate
Nd Nd
Na
Ωox
Ωsi
Vg = 2 V
Vd = 1 V
Vs = 0
Vsub = 0
MOSFET coupled drift-diffusion equations
∇ · (ε0εox∇ψ) = 0 in Ωox
∇ · (ε0εsi∇ψ) = −q(ni exp
(−qψkBT
)− n +Nd −Na
)in Ωsi
∇ · (nµn∇ψ −Dn∇n) = 0 in Ωsi
FEM-SUPG CVFEM-SG
FEM-SUPG solution exhibits larger negative values over a wider area than theCVFEM-SG solution.
2/6/2017 33
Stabilization for FEM
Weak form of equation∫Ω
∂n
∂tψdV +
∫ΩJ · ∇ψdV +
∫ΩRψdV = 0
Semi-discrete equation with stabilization
∫Ω
∂nh
∂tψdV+
∫ΩJE(nh)·∇ψdV+
∫ΩR(nh)ψdV = 0
where
JE(x) =∑eij
hijJij−→W ij(x)
Electron continuity equation
∂n
∂t−∇ · J +R = 0
J = µEn+D∇n
vi
vj
Jij
K
JE
P. Bochev, K. Peterson, A parameter-free stabilized finite element method for scalar advection-diffusion problems,Central European Journal of Mathematics, Vol. 11, issue 8, pp. 1458-1477 (2013)
2/6/2017 34
Symmetrized Stabilization for FEMConsider the stabilized flux
JE(x) =∑eij
hijJij−→W ij(x)
It can be divided into a projection of the nodalflux and a stabilization term
JE(x) = IE(µEnh +D∇nh
)+ Θ(nh)
where
IE(u) =∑
eij∈E(Ω)
−→W ij
∫eij
u · tijdl
Electron continuity equation
∂n
∂t−∇ · J +R = 0
J = µEn+D∇n
vi
vj
Jij
K
JE
Θ(nh) =∑eij
(nj − ni) θij−→W ij(x), θij = D (aij coth(aij)− 1) , aij =
µEhij
2D
P. Bochev, M. Perego, K. Peterson, Formulation and analysis of a parameter-free stabilized finite element method,SINUM Vol. 53, No. 5, pp. 2363-2388 (2015).
2/6/2017 35
Symmetrized Stabilization for FEMCan use the stabilization term with the originalnodal flux∫
Ω
(µEnh +D∇nh
)·∇ψdV+
∫Ω
Θ(nh)∇ψdV = 0
This form resembles a nonsymmetric artificaldiffusion, which motivates the followingsymmetrized form∫
ΩΘ(nh) · Θ(ψ)dV = 0
where
Θ(nh) =∑
eij∈E(Ω)
(nj − ni)√θij−→W ij(x)
Electron continuity equation
∂n
∂t−∇ · J +R = 0
J = µEn+D∇n
vi
vj
Jij
K
JE
Symmetrized form has same accuracy (first-order in strong drift regime), but better
stability
2/6/2017 36
Symmetrized Stabilization for FEM
Stabilizing kernel automatically adjusts strength of artificial edge diffusion
θij = D (aij coth(aij)− 1)
In the diffusion limit
limaij→0
θij = 0
In the drift limit
limaij→∞
θij =hij
2|µE|
Electron continuity equation
∂n
∂t−∇ · J +R = 0
J = µEn+D∇n
vi
vj
Jij
K
JE
2/6/2017 37
Manufactured Solution
D = 0.1Quadrilaterals Triangles
Grid ‖φ− φh‖0 ‖∇φ−∇φh‖0 ‖φ− φh‖0 ‖∇φ−∇φh‖032 0.2437E-03 0.3609E-01 0.3623E-03 0.3610E-0164 0.6099E-04 0.1804E-01 0.9093E-04 0.1804E-01
128 0.1525E-04 0.9021E-02 0.2275E-04 0.9021E-02256 0.3813E-05 0.4511E-02 0.5690E-05 0.4511E-02Rate 2.000 1.000 1.998 1.000
D = 0.001Quadrilaterals Triangles
Grid ‖φ− φh‖0 ‖∇φ−∇φh‖0 ‖φ− φh‖0 ‖∇φ−∇φh‖032 0.4262E-02 0.7536E-01 0.7703E-02 0.8802E-0164 0.2082E-02 0.4932E-01 0.3714E-02 0.5604E-01
128 0.9820E-03 0.3081E-01 0.1544E-02 0.3245E-01256 0.3910E-03 0.1505E-01 0.5566E-03 0.1594E-01Rate 1.276 0.934 1.375 0.951
D = 0.00001Quadrilaterals Triangles
Grid ‖φ− φh‖0 ‖∇φ−∇φh‖0 ‖φ− φh‖0 ‖∇φ−∇φh‖032 0.4741E-02 0.7949E-01 0.8545E-02 0.9514E-0164 0.2518E-02 0.5497E-01 0.4616E-02 0.6664E-01
128 0.1298E-02 0.3842E-01 0.2401E-02 0.4684E-01256 0.6574E-03 0.2695E-01 0.1222E-02 0.3293E-01Rate 0.964 0.517 0.952 0.510
−∇ · J(n) = fJ(n) = (D∇n− µEn)
φ(x, y) = x3 − y2µE = (− sin(π/6), cos(π/6))
2/6/2017 38
Double Glazing Test
−∇ · J(n) = R in Ω
J(n) = (D∇n+ µEn) in Ω
n = g on Γ
D = 0.00001
g =
0 on ΓL ∪ ΓT ∪ (ΓB ∩ x ≤ 0.5)1 on ΓR ∪ (ΓB ∩ x > 0.5)
µE =
(2(2y − 1)(1− (2x− 1)2)−2(2x− 1)(1− (2y − 1)2)
)Random quadrilateral grid
Mesh Symmetric Formulation Original Formulation
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
min = -0.1649 min = -0.2644
max = 1.0514 max = 1.0956
2/6/2017 39
Double Glazing Test
−∇ · J(n) = R in Ω
J(n) = (D∇n+ µEn) in Ω
n = g on Γ
D = 0.00001
g =
0 on ΓL ∪ ΓT ∪ (ΓB ∩ x ≤ 0.5)1 on ΓR ∪ (ΓB ∩ x > 0.5)
µE =
(2(2y − 1)(1− (2x− 1)2)−2(2x− 1)(1− (2y − 1)2)
)Uniform quadrilateral grid
Symmetric Formulation SUPG Artificial Diffusion
−0.2
0
0.2
0.4
0.6
0.8
1
−0.2
0
0.2
0.4
0.6
0.8
1
−0.2
0
0.2
0.4
0.6
0.8
1
min = -0.0691 min = -0.1494 min = 0.0
max = 1.0 max = 1.0 max = 1.0
2/6/2017 40
Double Glazing Test
−∇ · J(n) = R in Ω
J(n) = (D∇n+ µEn) in Ω
n = g on Γ
D = 0.00001
g =
0 on ΓL ∪ ΓT ∪ (ΓB ∩ x ≤ 0.5)1 on ΓR ∪ (ΓB ∩ x > 0.5)
µE =
(2(2y − 1)(1− (2x− 1)2)−2(2x− 1)(1− (2y − 1)2)
)
Stability estimate implies method is more diffusive on triangles
Triangle Quadrilateral
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
2/6/2017 41
Multi-scale Stabilized CVFEM
Solve 1-d boundary value problem along amacro-element edge for a linear J(s) = a+ bs
J(s) = µEsn(s) +Ddn
dsn(0) = ni, n(hs/2) = nk and n(hs) = nj
Jik = J(hs/4) Jkj = J(3hs/4)
Edge flux
Jik = J1stik (ni, nk) + γik(ni, nj , nk)
Jkj = J1stkj (nk, nj) + γkj(ni, nj , nk)
Divide each element into four
sub-elements
K
K1
K2K3
K4
Expand into primary (macro) cell using H(curl)-conforming finiteelements
JE(nh) =∑
eij∈E(Ω)
Jij−→W ij
Bochev, Peterson, Perego "A multi-scale control-volume finite element method for advection-diffusionequations",IJNMF Vol. 77, Issue 11, pp. 641-667 (2015).
2/6/2017 42
Multi-scale Stabilized CVFEM
Solve 1-d boundary value problem along amacro-element edge for a linear J(s) = a+ bs
J(s) = µEsn(s) +Ddn
dsn(0) = ni, n(hs/2) = nk and n(hs) = nj
Jik = J(hs/4) Jkj = J(3hs/4)
Edge flux
Jik = J1stik (ni, nk) + γik(ni, nj , nk)
Jkj = J1stkj (nk, nj) + γkj(ni, nj , nk)
Divide each element into four
sub-elements
ni
nj
nk
K
Jik
JkjK1
K2K3
K4
Expand into primary (macro) cell using H(curl)-conforming finiteelements
JE(nh) =∑
eij∈E(Ω)
Jij−→W ij
Bochev, Peterson, Perego "A multi-scale control-volume finite element method for advection-diffusionequations",IJNMF Vol. 77, Issue 11, pp. 641-667 (2015).
2/6/2017 43
Multi-scale Stabilized CVFEM
Solve 1-d boundary value problem along amacro-element edge for a linear J(s) = a+ bs
J(s) = µEsn(s) +Ddn
dsn(0) = ni, n(hs/2) = nk and n(hs) = nj
Jik = J(hs/4) Jkj = J(3hs/4)
Edge flux
Jik = J1stik (ni, nk) + γik(ni, nj , nk)
Jkj = J1stkj (nk, nj) + γkj(ni, nj , nk)
Divide each element into four
sub-elements
ni
nj
nk
K
Jik
JkjK1
K2K3
K4
Expand into primary (macro) cell using H(curl)-conforming finiteelements
JE(nh) =∑
eij∈E(Ω)
Jij−→W ij
Bochev, Peterson, Perego "A multi-scale control-volume finite element method for advection-diffusionequations",IJNMF Vol. 77, Issue 11, pp. 641-667 (2015).
2/6/2017 44
Multi-scale Stabilized CVFEM
Solve 1-d boundary value problem along amacro-element edge for a linear J(s) = a+ bs
J(s) = µEsn(s) +Ddn
dsn(0) = ni, n(hs/2) = nk and n(hs) = nj
Jik = J(hs/4) Jkj = J(3hs/4)
Edge flux
Jik = J1stik (ni, nk) + γik(ni, nj , nk)
Jkj = J1stkj (nk, nj) + γkj(ni, nj , nk)
Divide each element into four
sub-elements
ni
nj
nk
K
Jik
JkjK1
K2K3
K4
Expand into primary (macro) cell using H(curl)-conforming finiteelements
JE(nh) =∑
eij∈E(Ω)
Jij−→W ij
Bochev, Peterson, Perego "A multi-scale control-volume finite element method for advection-diffusionequations",IJNMF Vol. 77, Issue 11, pp. 641-667 (2015).
2/6/2017 45
Multi-scale Stabilized CVFEM
Define control volumes around each sub-cellnode
Use a second-order (bilinear) finite elementapproximation for n on each sub-elementnh(t,x) =
∑j
nj(t)φj(x)
Compute Jij at each macro element edgeand evaluate JE at control volume sideintegration points using 2nd order H(curl)basis
K
K1
K2K3
K4
2/6/2017 46
Multi-scale Stabilized CVFEM
Define control volumes around each sub-cellnode
Use a second-order (bilinear) finite elementapproximation for n on each sub-elementnh(t,x) =
∑j
nj(t)φj(x)
Compute Jij at each macro element edgeand evaluate JE at control volume sideintegration points using 2nd order H(curl)basis
K
Jik
JkjK1
K2K3
K4n
2/6/2017 47
Manufactured Solution
−∇ · J(n) = R in Ω
J(n) = (D∇n+ µEn) in Ω
n = g on Γ
n(x, y) = x3 − y2
µE = (− sinπ/6, cosπ/6)
CVFEM-MS CVFEM-SG FEM-SUPGL2 error H1 error L2 error H1 error L2 error H1 error
Grid∗ ε = 1× 10−3
32 1.57e-3 6.05e-2 4.24e-3 7.48e-2 2.09e-4 3.61e-264 3.93e-4 2.89e-2 2.07e-3 4.91e-2 4.85e-5 1.80e-2128 8.98e-5 1.24e-2 9.78e-4 3.07e-2 1.11e-5 9.02e-3Rate 2.06 1.14 1.06 0.642 2.12 1.00Grid ε = 1× 10−5
32 1.69e-3 6.60e-2 4.73e-3 7.90e-2 2.30e-4 3.61e-264 4.54e-4 3.45e-2 2.52e-3 5.48e-2 5.78e-5 1.80e-2128 1.18e-4 1.76e-2 1.30e-3 3.83e-2 1.45e-5 9.02e-3Rate 1.92 0.955 0.933 0.521 1.99 1.00
∗ For CVFEM-MS the size corresponds sub-elements rather than macro-elements.
2/6/2017 48
Skew Advection Test
−∇ · J(n) = R in Ω
J(n) = (D∇n+ µEn) in Ω
n = g on Γ
g =
0 on ΓL ∪ ΓT ∪ (ΓB ∩ x ≤ 0.5)1 on ΓR ∪ (ΓB ∩ x > 0.5)
µE = (− sinπ/6, cosπ/6) D = 1.0× 10−5
CVFEM-MS CVFEM-SG SUPG
min = -0.0445 min = 0.00 min = -0.0459
max = 1.077 max = 1.004 max = 1.251
2/6/2017 49
Double Glazing Test
−∇ · J(n) = R in Ω
J(n) = (D∇n+ µEn) in Ω
n = g on Γ
D = 1.0× 10−5
g =
0 on ΓL ∪ ΓT ∪ (ΓB ∩ x ≤ 0.5)1 on ΓR ∪ (ΓB ∩ x > 0.5)
µE =
(2(2y − 1)(1− (2x− 1)2)−2(2x− 1)(1− (2y − 1)2)
)CVFEM-MS CVFEM-SG SUPG
2/6/2017 50
Conclusions
Stabilization using an edge-element lifting of classical S-G fluxes offers astable and robust method for solving drift-diffusion equations
Works on unstructured grids
Does not require heuristic stabilization parameters
Although not provably monotone, violations of solution bounds are less than for acomparable scheme with SUPG stabilization
Future workImplementation of 2nd-order scheme in CharonMore detailed comparison of methods for full drift-diffusion equations
2/6/2017 51
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