A Primal-Dual Weak Galerkin Finite Element Method for Second Order - - PowerPoint PPT Presentation

▶
a primal dual weak galerkin finite element method for
SMART_READER_LITE
LIVE PREVIEW

A Primal-Dual Weak Galerkin Finite Element Method for Second Order - - PowerPoint PPT Presentation

Aug 28, 2023 280 likes •594 views

A Primal-Dual Weak Galerkin Finite Element Method for Second Order Elliptic Equations in Non-Divergence Form Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang (NSF)

slide-1
SLIDE 1

A Primal-Dual Weak Galerkin Finite Element Method for Second Order Elliptic Equations in Non-Divergence Form

Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang (NSF) Supported by NSF Grant DMS-1522586 Jiangsu Provincial Foundation Award BK20050538 October 26, 2015

Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Second

slide-2
SLIDE 2

Talk Outline

Model Problem and Background Primal-Dual Weak Galerkin Finite Element Method Error Estimates Numerical Tests

Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Secon

slide-3
SLIDE 3

Model Problem—Second Order Elliptic Equations in Non-Divergence Form

Find u = u(x) satisfying u|∂Ω = 0, such that

d

  • i,j=1

aij∂2

iju = f ,

in Ω. Assume a(x) = (aij(x))d×d ∈ L∞(Ω).

Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Secon

slide-4
SLIDE 4

Background of Second Order Elliptic Equations in Non-Divergence Form

probability and stochastic processes nonlinear PDEs in conjunction with linearization techniques such as the Newton’s iterative method a(x) is hardly smooth nor even continuous.

a(x) is merely essentially bounded in the application to Hamilton-Jacobi-Bellman equations. For nonlinear PDEs discretized by discontinuous finite elements, their linearization involves at most piecewise smooth coefficients.

Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Secon

slide-5
SLIDE 5

Variational Equation

Assume the model problem has a unique strong solution in X = H2(Ω) ∩ H1

0(Ω) satisfying

u2 ≤ Cf 0. Variational Equation: Find u ∈ X such that b(u, w) = (f , w) ∀w ∈ Y = L2(Ω).

b(u, w) = (Lu, w) Lu = d

i,j=1 aij∂2 iju

b(·, ·) satisfies the inf-sup condition sup

v∈X,v=0

b(v, σ) vX ≥ ΛσY , ∀σ ∈ Y .

Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Secon

slide-6
SLIDE 6

Weak Hessian

W (K) = {v = {v0, vb, vg} : v0 ∈ L2(K), vb ∈ L2(∂K), vg ∈ [L2(∂K)]d}. Weak Second Order Partial Derivative The weak second order partial derivative of v ∈ W (K) is defined as a bounded linear functional ∂2

ij,wv on H2(K) so that its action

  • n each ϕ ∈ H2(K) is given by

∂2

ij,wv, ϕK := (v0, ∂2 jiϕ)K − vbni, ∂jϕ∂K + vgi, ϕnj∂K.

weak Hessian of v ∈ W (K): ∇2

w,Kv =

  • ∂2

ij,wv

  • d×d

Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Second

slide-7
SLIDE 7

Discrete Weak Hessian

Discrete Weak Second Order Partial Derivative A discrete weak second order partial derivative of v ∈ W (K), denoted by ∂2

ij,w,r,Kv, is defined as the unique polynomial satisfying

(∂2

ij,w,r,Kv, ϕ)K = (v0, ∂2 jiϕ)K−vbni, ∂jϕ∂K+vgi, ϕnj∂K, ∀ϕ ∈ Pr(K).

discrete weak Hessian of v ∈ W (K): ∇2

w,r,Kv = {∂2 ij,w,r,Kv}d×d

Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Second

slide-8
SLIDE 8

Weak Finite Element Spaces

Wk(T) = {{v0, vb, vg} ∈ Pk(T) × Pk(e) × [Pk−1(e)]d} Wh,k =

  • {v0, vb, vg} : {v0, vb, vg}|T ∈ Wk(T), T ∈ Th
  • W 0

h,k = {{v0, vb, vg} ∈ Wh,k, vb|e = 0, e ⊂ ∂Ω}

Sh,k =

  • σ : σ|T ∈ Sk(T), T ∈ Th
  • Pk−2(T) ⊆ Sk(T) ⊆ Pk−1(T)

The choice of Sk(T) = Pk−2(T) has the least degrees of freedom, but the resulting numerical solution may not be as accurate as the case of Sk(T) = Pk−1(T). Figure: WG element

Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Secon

slide-9
SLIDE 9

Primal-Dual Weak Galerkin Finite Element Algorithms

Find uh ∈ W 0

h,k satisfying

bh(v, σ) = (f , σ), ∀σ ∈ Sh,k. bh(v, σ) =

T∈Th

d

i,j=1(aij∂2 ij,wv, σ)T.

The problem is not well-posed unless an inf-sup condition is satisfied. Constrained optimization problem: Find uh ∈ W 0

h,k such that

uh = argminv∈W 0

h,k,bh(v,σ)=(f ,σ), ∀σ∈Sh,k

1 2sh(v, v)

  • .

Stablizer sh(v, v) =

  • T

h−3

T v0−vb, v0−vb∂T+h−1 T ∇v0−vg, ∇v0−vg∂T

measures the “continuity” of v ∈ W 0

h,k in the sense that v is a

classical conforming element if and only if sh(v, v) = 0.

Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Secon

slide-10
SLIDE 10

Primal-Dual Weak Galerkin Finite Element Algorithms

Primal-Dual Weak Galerkin Algorithm Find (uh; λh) ∈ W 0

h,k × Sh,k satisfying

sh(uh, v) + bh(v, λh) = 0, ∀v ∈ W 0

h,k,

bh(uh, σ) = (f , σ), ∀σ ∈ Sh,k.

Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Second

slide-11
SLIDE 11

L2 Projections

Q0: L2 projection onto Pk(T) Qb: L2 projection onto Pk(e) Qg = (Qg1, Qg2, . . . , Qgd): L2 projection onto [Pk−1(e)]d Qh: L2 projection onto Wh,k, such that on each element T, Qhw = {Q0w, Qbw, Qg(∇w)} Qh: L2 projection onto Sh,k

Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Secon

slide-12
SLIDE 12

Inf-Sup condition ( Brezzi and Babuska )

Inf-Sup Condition Assume that the coefficient matrix a = {aij}d×d is uniformly piecewise continuous in Ω with respect to the finite element partition Th. For any σ ∈ Sh,k, there exists vσ ∈ W 0

h,k satisfying

bh(vσ, σ) ≥1 2σ2

0,

vσ2

2,h ≤Cσ2 0,

provided that the meshsize h < h0 for a sufficiently small, but fixed parameter h0 > 0.

Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Second

slide-13
SLIDE 13

Error Estimates

Error Estimates in a Discrete H2 Norm Assume that the coefficient functions aij are uniformly piecewise continuous in Ω. Assume that the exact solution u is sufficiently regular such that u ∈ Hk+1(Ω). Let (uh; ρh) ∈ W 0

h,k × Sh,k be

primal-dual WG solution. There exists a constant C such that uh − Qhu2,h + λh − Qhλ0 ≤ Chk−1uk+1, provided that the meshsize h < h0 holds true for a sufficiently small, but fixed h0 > 0. v2

2,h = T∈Th d i,j=1 Qh(aij∂2 ijv0)2 T + sh(v, v)

Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Second

slide-14
SLIDE 14

Error Estimates

Error Estimates in H1 Let uh = {u0, ub, ug} ∈ W 0

h,k be the primal-dual WG solution.

Assume that aij ∈ C 1(Ω), and the exact solution u ∈ Hk+1(Ω). There exists a constant C such that  

T∈Th

∇u0 − ∇u2

T

 

1 2

≤ Chkuk+1, provided that the meshsize h is sufficiently small and the dual problem has the H1-regularity with the a priori estimate w1 ≤ Cθ−1.

Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Second

slide-15
SLIDE 15

Error Estimates

Error Estimates in L2 Assume aij ∈ C 1(Ω) ∩

  • ΠT∈ThW 2,∞(T)
  • . In addition, assume that

the dual problem has H2-regularity with the a priori estimate w2 ≤ Cθ0, and P1(T) ⊂ Sk(T) for all T ∈ Th. Then, we have u0 − u0 ≤Chk+1uk+1, ub − QbuL2 ≤Chk+1uk+1, ug − Qb∇uL2 ≤Chkuk+1. provided that the meshsize h is sufficiently small. ebL2 =

T∈Th

hTeb2

∂T

1

2

egL2 =

T∈Th

hTeg2

∂T

1

2 Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Second

slide-16
SLIDE 16

Error Estimates

Remark In the case of P1(T) S2(T), we have the following sub-optimal

  • rder error estimate

u0 − u0 ≤ Chkuk+1 provided that (1) aij ∈ C 1(Ω), (2) the meshsize h is sufficiently small, and (3) the dual problem has the H1-regularity with the a priori estimate w1 ≤ Cθ−1.

Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Second

slide-17
SLIDE 17

Numerical Tests

the exact solution u = sin(x1) sin(x2) a11 = 3, a12 = a21 = 1, a22 = 2 λh is piecewise linear

Table: numerical error and convergence order for domain Ω = (0, 1)2

1/h e00

  • rder

egL2

  • rder

λh0

  • rder

1 0.00624 0.126 0.0335 2 0.00147 2.09 0.0448 1.50 0.0650

  • 0.955

4 1.39e-004 3.40 0.0116 1.95 0.0284 1.20 8 1.03e-005 3.75 0.00284 2.03 0.0132 1.10 16 6.95e-007 3.89 7.02e-004 2.02 0.00643 1.04 32 4.52e-008 3.94 1.75e-004 2.01 0.00317 1.02

Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Secon

slide-18
SLIDE 18

Numerical Tests

Table: numerical error and convergence order for L-shaped domain with vertexes (0,0), (2,0), (1,1), (1,2), and (0,2).

2/h e00

  • rder

egL2

  • rder

λh0

  • rder

1 0.0168 0.481 0.448 2 0.00248 2.76 0.125 1.95 0.195 1.20 4 2.30e-004 3.43 0.0310 2.01 0.0875 1.16 8 1.93e-005 3.57 0.00767 2.01 0.0413 1.08 16 1.61e-006 3.59 0.00191 2.01 0.0202 1.03 32 1.37e-007 3.56 4.75e-004 2.00 0.00999 1.01

Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Secon

slide-19
SLIDE 19

Numerical Tests

the exact solution u = sin(x1)sin(x2) Ω = (−1, 1)2 a11 = 1 + |x1|, a12 = a21 = 0.5|x1x2|

1 3 , a22 = 1 + |x2|

Table: numerical error and convergence order ( λh is piecewise linear)

2/h e00

  • rder

egL2

  • rder

λh0

  • rder

1 0.177

  • 1.25
  • 0.00390
  • 2

0.0357 2.30 0.486 1.36 0.00820

  • 1.07

4 0.00360 3.31 0.130 1.90 0.00324 1.34 8 2.78e-004 3.70 0.0318 2.03 0.00151 1.10 16 2.02e-005 3.78 0.00783 2.02 7.42e-004 1.03 32 2.37e-006 3.09 0.00194 2.01 3.68e-004 1.01

Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Secon

slide-20
SLIDE 20

Numerical Tests

Table: numerical error and convergence order (λh is piecewise constant)

2/h e00

  • rder

egL2

  • rder

λh0

  • rder

1 2.80e-006

  • 1.76
  • 2.10e-006
  • 2

0.176

  • 16.0

0.676 1.38 0.0895

  • 15.4

4 0.0395 2.15 0.164 2.04 0.0518 0.790 8 0.00896 2.14 0.0386 2.08 0.0190 1.45 16 0.00217 2.05 0.00938 2.04 0.00685 1.47 32 5.37e -004 2.01 0.00231 2.02 0.00288 1.25

Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Secon

slide-21
SLIDE 21

Numerical Tests

Consider

2

  • i,j=1

(1 + δij) xi |xi| xj |xj|∂2

iju = f ,

in Ω, u = 0,

  • n

∂Ω. Ω = (−1, 1)2 the exact solution u(x1, x2) = (x1e1−|x1| − x1)(x2e1−|x2| − x2).

Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Secon

slide-22
SLIDE 22

Numerical Tests

Table: Numerical error and convergence order (λh is piecewise linear).

2/h e00

  • rder

egL2

  • rder

λh0

  • rder

1 0.0940

  • 0.766
  • 0.338
  • 2

0.249

  • 1.40

1.35

  • 0.815

0.642

  • 0.927

4 0.106 1.23 0.538 1.32 1.28

  • 1.00

8 0.0306 1.80 0.137 1.97 0.537 1.26 16 0.00749 2.03 0.0327 2.07 0.212 1.34 32 0.00174 2.11 0.00785 2.06 0.0923 1.20

Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Secon

slide-23
SLIDE 23

Numerical Tests

−1 −0.5 0.5 1 −1 −0.5 0.5 1 −0.2 −0.15 −0.1 −0.05 0.05 0.1 0.15 0.2 figure for u0

Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Secon

slide-24
SLIDE 24

Numerical Tests

−1 −0.5 0.5 1 −1 −0.5 0.5 1 −0.2 −0.1 0.1 0.2 0.3 figure for rhoh

Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Secon

slide-25
SLIDE 25

Numerical Tests

Table: Numerical error and convergence order (λh is piecewise constant).

2/h e00

  • rder

egL2

  • rder

λh0

  • rder

1 0.0393

  • 0.672
  • 0.137
  • 2

0.0322 0.284 0.322 1.06 0.104 0.396 4 0.00750 2.10 0.0791 2.03 0.0532 0.963 8 0.00161 2.22 0.0180 2.13 0.0204 1.39 16 3.85e-004 2.07 0.00427 2.08 0.00818 1.32 32 9.52e-005 2.02 0.00104 2.04 0.00371 1.14

Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Secon

slide-26
SLIDE 26

Numerical Tests

−1 −0.5 0.5 1 −1 −0.5 0.5 1 −0.1 −0.05 0.05 0.1 0.15 figure for rhoh

Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Secon

slide-27
SLIDE 27

Numerical Tests

Consider

2

  • i,j=1

(δij + xixj |x|2 )∂2

iju = (2α2 − α)|x|α−2,

in Ω. the exact solution u = |x|α(α > 1) Ω = (0, 1)2 α = 1.6

Table: numerical error and convergence order(λh is piecewise linear)

1/h e00

  • rder

egL2

  • rder

λh0

  • rder

1 0.020

  • 0.315
  • 0.304
  • 2

0.00629 1.68 0.126 1.32 0.248 0.296 4 0.00174 1.86 0.0446 1.50 0.182 0.445 8 4.43e-004 1.97 0.0152 1.56 0.126 0.537 16 1.08e-004 2.03 0.00508 1.58 0.0846 0.570 32 2.60e-005 2.05 0.00169 1.59 0.0564 0.584

Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Secon

slide-28
SLIDE 28

Numerical Tests

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 figure for u0

Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Secon

slide-29
SLIDE 29

Numerical Tests

Table: numerical error and convergence order (λh is piecewise constant)

1/h e00

  • rder

egL2

  • rder

λh0

  • rder

1 0.00405

  • 0.489
  • 0.0623
  • 2

0.00803

  • 0.988

0.177 1.46 0.0616 0.0156 4 0.00263 1.61 0.0616 1.53 0.0476 0.372 8 7.90e-004 1.74 0.0210 1.55 0.0327 0.544 16 2.20e-004 1.85 0.00705 1.57 0.0218 0.582 32 5.85e-005 1.91 0.00235 1.59 0.0145 0.593

Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Secon

slide-30
SLIDE 30

Thank you very much for your attention!

Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology email: cwang462@math.gatech.edu Homepage: http://people.math.gatech.edu/ ∼ cwang462/

Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Secon