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Multiscale Stabilized Control Volume Finite ElementMethod for Advection-Diffusion
Kara Peterson Pavel Bochev Mauro PeregoSuzey Gao
Sandia National Laboratories
FEF 2017
CCRCenter for Computing Research
Sandia National Laboratories is a multi-mission laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary ofLockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.
SAND 2017-3512C 1
Application Driver
Semiconductor electrical transport simulation
Scaled semiconductor drift-diffusion equations
Poisson equation ∇ · (λ2∇φ) + (p− n + C) = 0
Electron continuity equation∂n
∂t−∇ · Jn + R(φ, n, p) = 0
Hole continuity equation∂p
∂t+∇ · Jp + R(φ, n, p) = 0
Electric field E = −∇φElectron current density Jn = µnEn +Dn∇n
Hole current density Jp = µpEn−Dp∇p
source drain
gate
substrate
Nd Nd
Na
Ωox
Ωsi
Vg = 2 V
Vd = 1 V
Vs = 0
Vsub = 0
↑ ↑drift diffusion
Require a numerical scheme that is accurate and stable in the strong drift regime(Dn µnE, Dp µpE)
SAND 2017-3512C 2
Application Driver
Semiconductor electrical transport simulation
Scaled semiconductor drift-diffusion equations
Poisson equation ∇ · (λ2∇φ) + (p− n + C) = 0
Electron continuity equation∂n
∂t−∇ · Jn + R(φ, n, p) = 0
Hole continuity equation∂p
∂t+∇ · Jp + R(φ, n, p) = 0
Electric field E = −∇φElectron current density Jn = µnEn +Dn∇n
Hole current density Jp = µpEn−Dp∇p
source drain
gate
substrate
Nd Nd
Na
Ωox
Ωsi
Vg = 2 V
Vd = 1 V
Vs = 0
Vsub = 0
↑ ↑drift diffusion
Require a numerical scheme that is accurate and stable in the strong drift regime(Dn µnE, Dp µpE)
SAND 2017-3512C 3
Application Driver
Semiconductor electrical transport simulation
Scaled semiconductor drift-diffusion equations
Poisson equation ∇ · (λ2∇φ) + (p− n + C) = 0
Electron continuity equation∂n
∂t−∇ · Jn + R(φ, n, p) = 0
Hole continuity equation∂p
∂t+∇ · Jp + R(φ, n, p) = 0
Electric field E = −∇φElectron current density Jn = µnEn +Dn∇n
Hole current density Jp = µpEn−Dp∇p
source drain
gate
substrate
Nd Nd
Na
Ωox
Ωsi
Vg = 2 V
Vd = 1 V
Vs = 0
Vsub = 0
↑ ↑drift diffusion
Require a numerical scheme that is accurate and stable in the strong drift regime(Dn µnE, Dp µpE)
SAND 2017-3512C 4
Numerical DiscretizationControl Volume Finite Element Method (CVFEM)
Electron continuity equation
∂n
∂t−∇ · J + R = 0
J = un +D∇nu = µE
Finite element approximation of theelectron densitynh(x, t) =
∑j
nj(t)Nj(x)
Integrate over control volume and applythe divergence theorem
SAND 2017-3512C 5
Numerical DiscretizationControl Volume Finite Element Method (CVFEM)
Electron continuity equation
∂n
∂t−∇ · J + R = 0
J = un +D∇nu = µE
Ci
vi
Finite element approximation of theelectron densitynh(x, t) =
∑j
nj(t)Nj(x)
Integrate over control volume and applythe divergence theorem
∫Ci
∂nh
∂tdV−
∫∂Ci
J(nh)·~n dS+
∫Ci
R(nh) dV = 0
SAND 2017-3512C 6
Numerical DiscretizationControl Volume Finite Element Method (CVFEM)
Electron continuity equation
∂n
∂t−∇ · J + R = 0
J = un +D∇nu = µE
Ci
vi
Finite element approximation of theelectron densitynh(x, t) =
∑j
nj(t)Nj(x)
Integrate over control volume and applythe divergence theorem
∫Ci
∂nh
∂tdV−
∫∂Ci
J(nh)·~n dS+
∫Ci
R(nh) dV = 0
J(nh) =∑j
nj(t) (µENj +D∇Nj)
Nodal approximation for J(nh) can lead toinstabilities in strong drift regime.
SAND 2017-3512C 7
Numerical DiscretizationControl Volume Finite Element Method (CVFEM)
Electron continuity equation
∂n
∂t−∇ · J + R = 0
J = un +D∇nu = µE
Ci
vi
Finite element approximation of theelectron densitynh(x, t) =
∑j
nj(t)Nj(x)
Integrate over control volume and applythe divergence theorem
∫Ci
∂nh
∂tdV−
∫∂Ci
J(nh)·~n dS+
∫Ci
R(nh) dV = 0
J(nh) =∑j
nj(t) (µENj +D∇Nj)
Want a stabilized approximation for J thatincludes information on drift.
SAND 2017-3512C 8
Scharfetter-Gummel Upwinding
On edge eij solve 1-d boundary value problem forconstant Jij
dJij
ds= 0; Jij = µEijn(s) +D
dn(s)
ds
n(0) = ni and n(hij) = nj
Jij =aijD
hij(nj (coth(aij) + 1)− ni (coth(aij)− 1))
where aij =hijEijµ
2D, Eij = − (ψj−ψi)
hij
Ci
vi
vj
Jij
On structured grids, Jij is a goodestimate 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.
SAND 2017-3512C 9
Scharfetter-Gummel Upwinding
On edge eij solve 1-d boundary value problem forconstant Jij
dJij
ds= 0; Jij = µEijn(s) +D
dn(s)
ds
n(0) = ni and n(hij) = nj
Jij =aijD
hij(nj (coth(aij) + 1)− ni (coth(aij)− 1))
where aij =hijEijµ
2D, Eij = − (ψj−ψi)
hij
Ci
vi
vj
Jij
On structured grids, Jij is a goodestimate 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.
SAND 2017-3512C 10
Scharfetter-Gummel Upwinding
On edge eij solve 1-d boundary value problem forconstant Jij
dJij
ds= 0; Jij = µEijn(s) +D
dn(s)
ds
n(0) = ni and n(hij) = nj
Jij =aijD
hij(nj (coth(aij) + 1)− ni (coth(aij)− 1))
where aij =hijEijµ
2D, Eij = − (ψj−ψi)
hij
Ci
∂Cij
vi
vj
Jij
On structured grids, Jij is a goodestimate 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.
SAND 2017-3512C 11
Scharfetter-Gummel Upwinding
On edge eij solve 1-d boundary value problem forconstant Jij
dJij
ds= 0; Jij = µEijn(s) +D
dn(s)
ds
n(0) = ni and n(hij) = nj
Jij =aijD
hij(nj (coth(aij) + 1)− ni (coth(aij)− 1))
where aij =hijEijµ
2D, Eij = − (ψj−ψi)
hij
Ci
∂Cij
∂Cik
∂Cil
∂Cim
vi
vj
vk
vl
vm
Jij
Jik
Jil
Jim
On structured grids, Jij is a goodestimate 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.
SAND 2017-3512C 12
Scharfetter-Gummel Upwinding
On edge eij solve 1-d boundary value problem forconstant Jij
dJij
ds= 0; Jij = µEijn(s) +D
dn(s)
ds
n(0) = ni and n(hij) = nj
Jij =aijD
hij(nj (coth(aij) + 1)− ni (coth(aij)− 1))
where aij =hijEijµ
2D, Eij = − (ψj−ψi)
hij
Ci
vi
vj
Jijn n
On structured grids, Jij is a goodestimate 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.
SAND 2017-3512C 13
Scharfetter-Gummel Upwinding
On edge eij solve 1-d boundary value problem forconstant Jij
dJij
ds= 0; Jij = µEijn(s) +D
dn(s)
ds
n(0) = ni and n(hij) = nj
Jij =aijD
hij(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 no longer agood 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.
SAND 2017-3512C 14
Scharfetter-Gummel Upwinding
On edge eij solve 1-d boundary value problem forconstant Jij
dJij
ds= 0; Jij = µEijn(s) +D
dn(s)
ds
n(0) = ni and n(hij) = nj
Jij =aijD
hij(nj (coth(aij) + 1)− ni (coth(aij)− 1))
where aij =hijEijµ
2D, Eij = − (ψj−ψi)
hij
Ci
vi
vj
vm
vpJij
Jmi
Jpm
Jjp
JE
JE
On unstructured grids, Jij is no longer agood 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.
SAND 2017-3512C 15
Multi-dimensional S-G Upwinding
Idea: Use H(curl)-conforming finite elements to expand edge current density intoprimary cell
Nodal space, GhD(Ω), and edge element space, ChD(Ω), belong toan exact sequence
given Ni ∈ GhD(Ω), then∇Ni ∈ C
hD(Ω)
∇Ni =∑
eij∈E(vi)
σij−→W ij , σij = ±1
In the limit of carrier drift velocity µE = 0,
limµE→0
Jij =D(nj − ni)
hij
JE =∑
eij∈E(Ω)
D(nj−ni)−→W ij =
∑vi∈V (Ω)
Dnj∇Nj = J(nh
)
Exponentially fitted current density
JE(x) =∑eij
hijJij−→W ij(x)
JE
JE
vi
vj
vm
vp
Jij
Jmi
Jpm
Jjp
Ci
∫eij
−→W ij · trsdl = δ
rsij
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.
SAND 2017-3512C 16
Convergence on Structured Grids
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
SAND 2017-3512C 17
Robust on Unstructured Grids
Manufactured linear solution
Grid CVFEM-SG FVM-SG 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
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
SAND 2017-3512C 18
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)
Developers: Suzey Gao, Gary Hennigan, Larry Musson, Andy Huang
SAND 2017-3512C 19
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 SUPG Stabilized FEM∫Ω
(µE +D∇n) · ∇ψdV +
∫ΩRψdV +
∫Ωτ(µE · ∇n)(µE · ∇ψ)dV = 0 ∀ψ
τ =1
√uTGu
(cothα−
1
α
)α =
√uTGu
D‖G‖u = µE
SAND 2017-3512C 20
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 ]
SAND 2017-3512C 21
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.
SAND 2017-3512C 22
Multi-scale Stabilized CVFEM
Divide each element into 4 bilinear (Q1)sub-elements
Define control volumes around each sub-cellnode
Compute 2nd order Jij at each macroelement edge
Use 2nd order H(curl) basis to evaluate JEat control volume integration points
JE(nh) =∑
eij∈E(Ω)
hijJij−→W ij
K
K1
K2
K3
K4
Bochev, Peterson, Perego "A multi-scale control-volume finite element method for advection-diffusionequations",IJNMF Vol. 77, Issue 11, pp. 641-667 (2015).
SAND 2017-3512C 23
Multi-scale Stabilized CVFEM
Divide each element into 4 bilinear (Q1)sub-elements
Define control volumes around each sub-cellnode
Compute 2nd order Jij at each macroelement edge
Use 2nd order H(curl) basis to evaluate JEat control volume integration points
JE(nh) =∑
eij∈E(Ω)
hijJij−→W ij
K
Bochev, Peterson, Perego "A multi-scale control-volume finite element method for advection-diffusionequations",IJNMF Vol. 77, Issue 11, pp. 641-667 (2015).
SAND 2017-3512C 24
Multi-scale Stabilized CVFEM
Divide each element into 4 bilinear (Q1)sub-elements
Define control volumes around each sub-cellnode
Compute 2nd order Jij at each macroelement edge
Use 2nd order H(curl) basis to evaluate JEat control volume integration points
JE(nh) =∑
eij∈E(Ω)
hijJij−→W ij
K
J04
J41
J78
J85
J36 J62
J07
J73
J48
J86
J15
J52JE
Bochev, Peterson, Perego "A multi-scale control-volume finite element method for advection-diffusionequations",IJNMF Vol. 77, Issue 11, pp. 641-667 (2015).
SAND 2017-3512C 25
Multi-scale Stabilized CVFEM2nd Order Edge Current Density
Solve 1-d boundary value problem along acompound 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)
Jkj
Jik
ni
nk
nj
Edge current density
Jik = Φ(ni, nk) + γ(ni, nj , nk)
Jkj = Φ(nk, nj) + γ(ni, nj , nk)
Φ(ni, nk) = aDh
(nk (coth(a) + 1)− ni (coth(a)− 1))
γ(ni, nj , nk) = Dh
(a coth(a)− 1)(ni(coth(a)− 1)− 2nk coth(a) + nj(coth(a) + 1)
)
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Multi-scale Stabilized CVFEM2nd Order Edge Current Density
Solve 1-d boundary value problem along acompound 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)
Jkj
Jik
ni
nk
nj
Edge current density
Jik = Φ(ni, nk) + γ(ni, nj , nk)
Jkj = Φ(nk, nj) + γ(ni, nj , nk)
Φ(ni, nk) = aDh
(nk (coth(a) + 1)− ni (coth(a)− 1))
γ(ni, nj , nk) = Dh
(a coth(a)− 1)(ni(coth(a)− 1)− 2nk coth(a) + nj(coth(a) + 1)
)
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Multi-scale Stabilized CVFEM2nd Order Edge Current Density
Solve 1-d boundary value problem along acompound 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)
Jkj
Jik
ni
nk
nj
Edge current density
Jik = Φ(ni, nk) + γ(ni, nj , nk)
Jkj = Φ(nk, nj) + γ(ni, nj , nk)
Φ(ni, nk) = aDh
(nk (coth(a) + 1)− ni (coth(a)− 1))
γ(ni, nj , nk) = Dh
(a coth(a)− 1)(ni(coth(a)− 1)− 2nk coth(a) + nj(coth(a) + 1)
)
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Multi-scale Stabilized CVFEM
Subedge fluxes are a sum ofScharfetter-Gummel fluxes and ahigher-order correction term
High-order correction contributes ananti-diffusive flux
In pure advection limit (D → 0) methodonly modifies downstream segment
Jik = Φ(ni, nk) + γ(ni, nj , nk)
Jkj
Jik
ni
nk
nj
K
J04
J41
J78
J85
J36 J62
J07
J73
J48
J86
J15
J52JE
For assembly, edge basisfunctions are used only locally sono global edge data structuresare required
JE(nh) =∑
eij∈E(Ω)
hijJij−→W ij
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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.
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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
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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
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Conclusions
Stabilization using an edge-element lifting of edge current densities 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
Can achieve 2nd-order convergence with multi-scale approach
Future workInvestigate modifications to achieve a monotone schemeImplement 2nd-order scheme in CharonMore detailed comparison of methods for full drift-diffusion equations
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