Stabilized Control Volume Finite Element Method for Drift-Diffusion · 2017. 8. 1. · Stabilized Control Volume Finite Element Method for Drift-Diffusion Kara Peterson Pavel Bochev

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


∂n

∂t−∇ · J +R = 0

J = µEn+D∇n


∑j

nj(t)φj(x)



2/6/2017 5

CVFEM Formulation


∂n

∂t−∇ · J +R = 0

J = µEn+D∇n

Ci

vi


∑j

nj(t)φj(x)



2/6/2017 6

CVFEM Formulation


∂n

∂t−∇ · J +R = 0

J = µEn+D∇n

Ci

vi


∑j

nj(t)φj(x)



∫Ci

∂n

∂tdV −

∫∂Ci

J(n)·~ndS+

∫Ci

R(n), dV = 0

2/6/2017 7

CVFEM Formulation


∂n

∂t−∇ · J +R = 0

J = µEn+D∇n

Ci

vi


∑j

nj(t)φj(x)



∫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


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



nj(t)

∫∂Ci


Kij =

∫∂Ci


Ci

vi




2/6/2017 10



nj(t)

∫∂Ci


Kij =

∫∂Ci


Ci

vi




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



dJij


dn(s)

ds


Jij =aijDnhij


where aij =hijEijµ


hij

Ci

vi

vj

Jij

∫∂Ci

Jn · ~n dS ≈∑

∂Cij∈∂Ci

Jij |∂Cij |


2/6/2017 13



dJij


dn(s)

ds


Jij =aijDnhij


where aij =hijEijµ


hij

Ci

∂Cij

vi

vj

Jij

∫∂Ci

Jn · ~n dS ≈∑

∂Cij∈∂Ci

Jij |∂Cij |


2/6/2017 14



dJij


dn(s)

ds


Jij =aijDnhij


where aij =hijEijµ


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 |


2/6/2017 15



dJij


dn(s)

ds


Jij =aijDnhij


where aij =hijEijµ


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 |


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


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


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



J · ~ndS ≈∑

∂Cij∈∂Ci

Jij |∂Cij |

Ci

∂Cik

vi vkJik


J · ~ndS ≈∑

∂CKij∈∂Ci

JE · ~n |∂CKij |

Ci

vi

JE

JE

2/6/2017 20



J · ~ndS ≈∑

∂Cij∈∂Ci

Jij |∂Cij |

Ci

∂Cil

vi

vl

Jil


J · ~ndS ≈∑

∂CKij∈∂Ci

JE · ~n |∂CKij |

Ci

vi

JE

JE

2/6/2017 21



J · ~ndS ≈∑

∂Cij∈∂Ci

Jij |∂Cij |

Ci

∂Cim

vivm Jim


J · ~ndS ≈∑

∂CKij∈∂Ci

JE · ~n |∂CKij |

Ci

vi

JE

JE

2/6/2017 22



J · ~ndS ≈∑

∂Cij∈∂Ci

Jij |∂Cij |

Ci

∂Cim

vivm Jim


J · ~ndS ≈∑

∂CKij∈∂Ci

JE · ~n |∂CKij |

Ci

vi

JE

JE

2/6/2017 23



J · ~ndS ≈∑

∂Cij∈∂Ci

Jij |∂Cij |

Ci

∂Cim

vivm Jim


J · ~ndS ≈∑

∂CKij∈∂Ci

JE · ~n |∂CKij |

Ci

vi

JE

JE

2/6/2017 24



J · ~ndS ≈∑

∂Cij∈∂Ci

Jij |∂Cij |

Ci

∂Cim

vivm Jim


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


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


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


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)


∂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


∂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)


∂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|


∂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



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



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


n = g on Γ

D = 0.00001

g =


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


n = g on Γ

D = 0.00001

g =


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






Edge flux




sub-elements

ni

nj

nk

K

Jik

JkjK1

K2K3

K4


JE(nh) =∑

eij∈E(Ω)

Jij−→W ij


2/6/2017 43






Edge flux




sub-elements

ni

nj

nk

K

Jik

JkjK1

K2K3

K4


JE(nh) =∑

eij∈E(Ω)

Jij−→W ij


2/6/2017 44






Edge flux




sub-elements

ni

nj

nk

K

Jik

JkjK1

K2K3

K4


JE(nh) =∑

eij∈E(Ω)

Jij−→W ij


2/6/2017 45


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


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


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


n = g on Γ

g =


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


n = g on Γ

D = 1.0× 10−5

g =


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

Documents

Stabilized Control Volume Finite Element Method for Drift-Diffusion · 2017. 8. 1. · Stabilized Control Volume Finite Element Method for Drift-Diffusion Kara Peterson Pavel Bochev