Adaptive Discontinuous Galerkin Finite Element Method

Introduction 

Mathematical Description 

Preconditioning and Iterative Solver 

Adaptive Finite Element Method 

Adaptive Discontinuous Galerkin Finite Element 

Method 

Haihang You 

April 24, 2009 

Haihang You 

Adaptive Discontinuous Galerkin Finite Element Method

Introduction 




BUILT FOR SPEED 

A FORCE 

Introduction 

◮ Finite Element Method 

was introduced in early 

1940’s to obtain the 

numerical solution of 

problems in civil and 

aeronautical engineering. 

◮ The FEM has a prominent 

position among Numerical 

Methods for solving 

PDEs(Wikipedia). 

1.5 

1 

0.5 

0 

−0.5 

−1 

−1.5 

1 

0.5 

0 

0 

0.2 

0.4 

0.6 

0.8 

1 

Haihang You 


Introduction 





A FORCE 

The North American Eagle on a 2007 test run in the desert at Black Rock, Nev., where next year Ed Shadle and 

his team will try to set a land speed record of about 800 m.p.h. 

By GUY GUGLIOTTA on New York Times 

Published: April 20, 2009 

Haihang You 


Introduction 





A FORCE 

A computational fluid dynamics study showing the flow of air around the vehicle. The orange areas indicate where 

air is approaching the speed of sound. Blue is decelerated flow. 

Haihang You 


Introduction 




Poisson’s Equation 

Variational Formulation 

Variational Formulation Cont. 

Standard Galerkin Method 

Basis Function 

Stiffness Matrix 

Discontinuous Galerkin Method 

DG Formulation 


− ∇ 2 u = f in Ω (1) 

u = g D on Γ D (2) 

∇u · n = g N on Γ N . (3) 

◮ Ω ⊂ R d , d = 2, 3 

◮ g D denotes Dirichlet boundary condition 

◮ g N denotes Neumann boundary condition 

◮ boundary ∂Ω = Γ D ∪ Γ N 

◮ n is the unit outward normal vector of ∂Ω 

Haihang You 


Introduction 













◮ V = {v ∈ H 1 (Ω)} 

◮ − ∫ Ω (∇2 u)vdx = ∫ Ω fvdx 

∀v ∈ V 

◮ − ∫ Ω (∇2 u)vdx = ∫ Ω ∇u · ∇vdx − ∫ ∂Ω 

∂u 

∂n vdx 

Haihang You 


Introduction 













Let u be the solution of Eq(1), then u is also the solution of 

following variational problem: Find u ∈ V such that 

a(u, v) = (f , v) ∀v ∈ V (4) 

where 

a(u, v) = 

(f , v) = 

∫ 

∫ 

Ω 

Ω 

∇u · ∇vdx (5) 

∫ 

fvdx + 

∂u 

vdx 

∂n 

(6) 

∂Ω 

Haihang You 


Introduction 













Find u h ∈ V r h 

such that 

a(u h , v h ) = (f , v h ) ∀v h ∈ V r h (7) 

where 

V r h = Π K∈T h 

P r−1 (K), r ≥ 2 

◮ P r−1 (K) is the space of polynomials of total degree r − 1. 

◮ T h = {K i : i = 1, 2, ...,m h }. 

◮ Ω = ∪ K∈Th K = K 1 ∪ K 2 ∪ ... ∪ K mh . 

Haihang You 


Introduction 













◮ φ j (x i ) = δ ij = 

{ 1 if i = j 

0 if i ≠ j 

0 

−1 

1 

0.8 

0.6 

0.4 

0.2 

0.8 

0.6 

0.4 

0.2 

0 

0 

1 

1 

◮ u h (x i ) = 

N∑ 

ξ i φ i (x i ) 

i=1 

1 

1 

0.5 

0.8 

0.6 

0 

1 

0.8 

0.6 

0.4 

0.2 

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0 

0 

Haihang You 



Introduction 












N∑ 

∫ 

ξ i 

i=1 

◮ Aξ = b 

Ω 

∫ 

∇φ i · ∇φ j dx = 

◮ a ji = a ij = ∫ Ω ∇φ i · ∇φ j dx 

◮ b j = ∫ Ω f φ jdx + ∫ ∂Ω 

Ω 

∂u 

∂n · ∇φ jdx 

∫ 

∂u 

f φ j dx + 

∂Ω ∂n · ∇φ jdx 

Haihang You 


Introduction 













a γ h (u,v) = ∑ K∈T h 

(∇u, ∇v) K (8) 

− ∑ e∈E I h 

[ 

〈{∂ n u} ,[v]〉 e 

+ 〈{∂ n v} ,[u]〉 e 

− γh −1 

e 〈[u],[v]〉 e 

] 

− ∑ 

e∈E D h 

Eh I set of internal edges 

Eh B set of boundary edges on Γ D 

Eh N set of boundary edges on Γ N 

[ 

〈∂ n u,v〉 e 

+ 〈∂ n v,u〉 e 

− γh −1 

e 〈u,v〉 e 

] 

Haihang You 



Introduction 












Find u h ∈ V r h 

such that 

where 

F(v h ) = (f ,v h ) − ∑ 

a γ h (uγ h ,v h) = F(v h ) ∀v h ∈ V r h (9) 

e∈E D h 

〈 

gD ,∂ n v h − γh −1 

e v h 

〉 

e + ∑ 

e∈E N h 

〈g N ,v h 〉 e 

and 

V r h = Π K∈Th P r−1 (K), r ≥ 2 

Haihang You 


Introduction 




Additive Schwarz Preconditioner 

Embeding and Projection Operators 

Performance of Conjugate Gradient Method 

Performance of Preconditioned Conjugate Gradient Method 


◮ 

Ax = b 

where A is Symmetric Positive Definite. For practical problems, the matrix is sufficiently large to require 

iterative methods such as Conjugate Gradient method(CG). However, 

κ = cond(A) ≈ O(h −2 

min ) 

where h is the diameter of the smallest triangle of the mesh. 

◮ The convergence rate for CG is close to 1 as K is big,so number of iterations grows as size of A grows. 

√ κ − 1 

‖e m ‖ A ≤ 2( √ ) m ‖e 0 ‖ A κ + 1 

Haihang You 


Introduction 








◮ T s ⊆ T H ⊆ T h , where T s is union of T H , and T H is 

union of T h . 

◮ Define additive Schwarz preconditioner as: 

P = R T 0 A −1 

0 R 0 + (R T 1 A −1 

1 R 1 + · · · R T n A −1 

n R n ) (10) 

Haihang You 


Introduction 








(EH h) dg 

V dg 

h 

↑↓ (Ph H) dg 

(EH H) cg 

V dg 

H 

↑↓ 

V cg 

H 

(PH H) cg 

◮ (Ph H) dg = (EH h) T 

dg 

◮ (PH H) cg = (EH H) T 

cg 

◮ V dg 

h 

◮ V dg 

H 

◮ V cg 

H 

represents DG function space on fine mesh. 

represents DG function space on coarse mesh. 

represents CG function space on coarse mesh. 

R 0 = (P H H ) cg (P H h ) dg 

Haihang You 


Introduction 








Performance of Conjugate Gradient Method(Time) -- Core 2 duo(2.66 GHz) 

Performance of Conjugate Gradient Method(Iters) -- Core 2 duo(2.66 GHz) 

4.1e+03 

Time 

8192 

Iters 

1.0e+03 

4096 

2.6e+02 

6.4e+01 

2048 

1.6e+01 

1024 

Time 

4.0e+00 

1.0e+00 

2.5e-01 

Iters 

512 

256 

6.2e-02 

1.6e-02 

3.9e-03 

128 

64 

9.8e-04 

1 2 3 4 5 6 7 8 

Level 

32 

1 2 3 4 5 6 7 8 

Level 

Haihang You 


Introduction 








Performance of Preconditioned Conjugate Gradient Method(Time) -- Core 2 duo(2.66 GHz) 

Performance of Preconditioned Conjugate Gradient Method(Iters) -- Core 2 duo(2.66 GHz) 

2048 

1024 

DG-cgSV 

DG-DG-pcgSV 

DG-CG-pcgSV 

8192 

4096 

DG-cgSV 

DG-DG-pcgSV 

DG-CG-pcgSV 

512 

2048 

256 

1024 

Time 

128 

Iters 

512 

64 

256 

32 

128 

16 

64 

8 

6 6.5 7 7.5 8 

32 

6 6.5 7 7.5 8 

Level 

Level 

Haihang You 


Introduction 




A Posteriori Estimator 

Adaptive Algorithm 

Refinement Algorithm 

Application 


An adaptive algorithm is an iterative method consisting of cycles of 

the form: Solve⇒Estimate⇒Refine 

1. Given a mesh T H , compute solution u H 

2. Estimate the error of u H . 

3. If: error ≤ specified tolerance, STOP. 

else: refine mesh to new mesh T h 

4. go to step 1). 

Haihang You 


Introduction 







Application 


An refinement algorithm is an method to choose certain elements 

for the refinement: 

1. Given a mesh T H , compute solution u H 

2. Estimate the error of u H . 

3. Sort each element by its error, and predict refinement levels to 

achieve the torlance. 

4. Refine chosen elements accordingly to new mesh T h . 

Haihang You 


Introduction 







Application 

Application 

{ 

−∇ 2 u = 0 

u = r 2/3 sin(2/3θ) 

in Ω 

on ∂Ω 

0.3 

Residual Error 

drastic cutting 

theta=0.6 

0.25 

0.2 

Error 

0.15 

0.1 

0.05 

0 

0 5 10 15 20 25 

aIters 

Haihang You 


Introduction 




Thank You! 

Haihang You

Adaptive Discontinuous Galerkin Finite Element Method

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