[PPT] - A Multiscale Discontinuous Galerkin Method P. Bochev Computational PowerPoint Presentation

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Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.

A Multiscale Discontinuous Galerkin Method

LSSC 05 Sozopol

P. Bochev

Computational Mathematics and Algorithms Sandia National Laboratories

T.J.R Hughes

Institute for Computational Engineering and Science, The University of Texas at Austin

G. Scovazzi

Computational Physics Department Sandia National Laboratories

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Computational mathematics and algorithms

Nomenclature

T h

( )

→ Partition of Ω into finite elements K h K → A generic finite element: T or Q ˆ K → A generic reference element

Q2 Q3 Q1 T1 T2

h V() → The set of all vertices of an element or a collection of elements E() → The set of all edges of an element or a collection of elements → The set of all edges = E(T h()) → The set of internal edges = E(T h(/))

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Computational mathematics and algorithms

The Discontinuous Space

S p(K ) K

( ) image of the reference space

S p( ˆ

K ) ˆ

K

( ) polynomials of degree ≤ p( )

ˆ K Local element space: Reference element space h() = h L2()|h| K S p(K )(K) K T h

{ }

Discontinuous space:

h = vvVv x

( )

v

+

eeE x

( )

e

+

bkBk x

( )

k

Hierarchical basis

Formal union of the local spaces, polynomial degrees not constrained by continuity across element boundaries.

Q3 p=3 Q2 p=1 Q1 p=1 T2 p=2 T1 p=1

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Computational mathematics and algorithms

The (Minimal) Continuous Space

Q2 p=1 Q1 p=1 T1 p=1 T2 p=2 p=3 Q3

– Required for the additive decomposition step in VMS – Defined with respect to the same partition of Ω into finite elements – Completely unrelated to the discontinuous space The continuous space

h() =

h H1()| h| K S1(K) K T h

{ }

The minimal continuous space – Used here for simplicity – Spanned by vertex shape functions – H1 conforming

N

i(v j) = ij

N

v|K = Vv

Basis

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Computational mathematics and algorithms

Orientations, Jumps and Averages

Orientation

+ – + – + + + – – – e1 e2 e3 e4 e5

– Elements oriented as sources – Edges oriented by choosing a normal Upwind (-) & downwind (+) elements

K– K+

– Determined by shared edge orientation Inflow & outflow element boundary

∂K– ∂K+

– Determined by edge orientations = 1 2 + +

( )

[ ] = +n+ + n

Jumps and averages u = 1 2 u+ + u

( )

u

[ ] = u+ n+ + u n

+

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Computational mathematics and algorithms

Variational Multiscale DG approach

L(x,D) = f in and R(x,D) = g on BDG h,h

( ) = FDG h ( )

h h

( )

Model BVP

T :

h

( ) a h ( )

Interscale

perator

Donor DG formulation

BDG T

h,T h

( ) = FDG T

h

( )

h

h

( )

VMS-DG formulation

h

( )

h

( )

DG space C0 space

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Computational mathematics and algorithms

Interscale operator

– T must reflect PDE’s structure – T must be locally defined (for efficiency)

BDG

h, h

( ) + BDG

h,

h

( ) = FDG

h

( )

h

h

( )

BDG

h,
h

( ) + BDG

h,

h

( ) = FDG

h

( )

h h

( )

VMS equations The key to VMS-DG is the definition of T

h

( )

h

( )

Coarse scale Fine scale

h

( ) a = +

Additive decomposition

← Coarse scale equation ← Fine scale equation

We use VMS to derive the local problems

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Computational mathematics and algorithms

The local problem

Assume:

BDG h,h

( ) =

BK h,h

( )

K

+

B h,h

( )

eh

+

Be h

,h +

{ }, h

,h +

{ }

( )

eh

← element form

← boundary form ← edge form Element equations coupled through the edge form

BDG

h,
h

( ) = FDG

h

( ) BDG

h,

h

( ) RK (

h)

h S p(K ) K

( )

To derive the local problem 1. Treat coarse scale function as data for the fine scale equation 2. Restrict fine scale equation to an element K 3. Uncouple fine scale equation from adjacent elements

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Computational mathematics and algorithms

Restriction to an element

h

( )

+ =

h for e K

0 for e K +

h

( )

= 0 for e K

h for e K +
h

( )

+ = K

( )

h
h

( )

= K +

( )

h
h = 0
h = 0
h = 0
h = 0

∂K– ∂K+

h 0

K=K– K=K+

States of the restricted weight function

h C0
h

( )

+ = h

( )

= h

States of the coarse scale solution

RK

h

( ) = FDG

h

( ) BK

h,

h

( ) B

h,

h

( )

Be

h,

h

{ }, (K +)

h ,(K)
h

{ }

( )

eE(K )

Coarse scale residual

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Computational mathematics and algorithms

Uncoupling from adjacent elements

Coupled fine scale equation on K

BK

h,
h

( ) + B

h,
h

( ) +

Be

h

( )

+,

h

( )

{

}, (K +)

h ,(K)
h

{ }

eE(K )
= RK

h

( )

Be

h

( )

+,

h

( )

{

}, (K +)

h ,(K)
h

{ }

B

e

,
{

}, (K +)

h ,(K)
h

{ }

( )

Redefine edge form in terms of the element fine scale function only

B

K

h,
h

( ) BK

h,
h

( ) + B

h,
h

( ) +

B

e

h

{ },

h

{ }

( )

eE(K )

= RK

h

( )

Uncoupled fine scale equation on K

B

K

h,
h

( ) = RK

h

( )

Interscale operator

TK :

h

( ) a S p(K ) K ( )

T :

h

( ) a h ( )

T|K = TK

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Computational mathematics and algorithms

Application to scalar advection-diffusion

Fa + Fd

( ) = f in

= g on D Fa + Fd

( ) n = h on N

Fd

( ) n = h+ on N

+

(N

)Fa + Fd

( ) n = h on N

N

N

+

Edge orientation Boundary value problem

Fa = a Fd =

advective diffusive Flux By advective direction ⇒ Upwind & downwind elements consistent with physical meaning

a +

N

D D

D

+

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Computational mathematics and algorithms

A donor DG formulation

0 =

F K + fK

( )dx

K

+

(N

+)Fa n h

( )d

N

+

F n

( )Kd

D

K
+

g

( )W ()d

D

+

Fb +;

( ) + ( )d

e

eh
+

Fc +;

( ) + ( )d

e

eh
+

[ ] [ ]d

e

eh
Residual formulation

← Weak Dirichlet condition ← Flux balance ← Weak continuity ← Least-squares stabilization Numerical fluxes

Fb +;

( ) = Fb

h +;

( ) n = s

11Fa h + s 12Fd h

( ) n

Fc +;

( ) = Fc

h +;

( ) n = s21Fa

h + s22Fd h

( ) n

coupling

terms ← Approximate true flux

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Computational mathematics and algorithms

Be +,

{ }, +, { }

( ) =

Fb +;

( )

[ ] + Fc +;

( )

[ ] + [ ] [ ]d

e

eE(K )
Weak form components

Fb +;

( ) + ( )d

e

=

Fb

h +;

( )

[ ]d

e

Fc +;

( ) + ( )d

e

=

Fc +;

( )

[ ]d

e

Equivalent form of flux terms

BK ,

( ) =

F K dx

K

DG bilinear form components

B ,

( ) =

(N

+)Fa n

( )d

N

+

F n

( )Kd

D

+

W ()d

D

← volume

← boundary ← edge

Coupling terms

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Computational mathematics and algorithms

Numerical flux & weight functions

DG-B DG-A Flux

Fa

( ) + Fd

( )

Fa

( ) + Fd ( )

s Fd

( )

sFd

( )

+ sFd

( ) n

Fc +;

( )

Fb

h +;

( )

W

( )

Fa

( ) + Fd ( )

← Total flux is upwinded

s=1 skew-symmetric excellent stability properties s=0 neutral requires LS stabilization s=-1 symmetric adjoint consistent, optimal L2 rates

DG formulation (see D. Arnold at al, SINUM 2000)

Fa

( ) + Fd

( )

← Advective flux is upwinded, diffusive is averaged DG-A DG-B

Hughes, Scovazzi, Bochev, Buffa, submitted to CMAME (2005)

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Computational mathematics and algorithms

Specialization of the edge form

DG-A DG-B

Be +,

{ }, +, { }

( ) =

[ ] [ ]d

e

eE(K )
+

a + +

( )/2

( )

Fb

h

1 2 4 4 4 3 4 4 4

[ ] s + +

( )/2

( )

Fc

h

1 2 4 4 3 4 4

[ ]d

e

Be

+,

{ }, +, { }

( ) =

[ ] [ ]d

e

eE(K )
+

a

( )

Fb

h

1 2 4 4 3 4 4

[ ] s

( )

Fc

h

1 2 4 3 4

[ ]d

e

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Computational mathematics and algorithms

T+ T— e1 e3 e2 a

Upwinding of total flux in DG-B

Local conservation properties of DG-B

T+ T— T—

Consider an internal element T and

h = 1 on K = T 0 on K T

f
+

ah

h

(

) n

e

eT
=

h

[ ] n

e

f
+

ah

h

(

) n

e

eT
= 0

Locally conservative without LS Approximately conservative with LS

ah

h = ah C h C

n e1 e3

Inflow flux

n contiguous elements

ah

h = ah T h T

n e2

Outflow flux

n the given element

Total Inflow = Total outflow

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Computational mathematics and algorithms

BK

h,
h

( ) + B

h,
h

( ) +

Fb

h

( )

+;

h

( )

(

)nK

h
h

[ ]

1 2 3 + Fc((K)

h
h

( )

+

1 2 4 3 4 ;(K +)

h
h

( )

1 2

4 3 4 )

h

[ ] +

h

[ ]nK

h
h

[ ]

1 2 3 d

e

eE(K )
= RK

h

( )

The interscale operator

To obtain T we start with the coupled element equation and redefine the states, jump and average of the fine scale as follows:

( )

+ = (K)

( )

= (K +)

=
[ ] = nK
S p(K )(K)

To compute states and jump extend by zero:

To compute edge averages extend by continuity:
= 1

2

+
(

) =

[

] = n+(K)

0 + n(K +)
(

)

+ = (K)

(

)

= (K +)

∂K–

∂K+

= 0
= 0
= 0
= 0

SLIDE 18

Computational mathematics and algorithms

Redefined edge form

DG-B DG-A Flux

Fa (K +)

( ) + Fd

( )

Fa (K +)

( ) + Fd (K +) ( )

sFd

( )

sFd (K +)

( )

Fc

( )

Fb

h

( )

B

e

{ },
{ }

( ) =

a(K +)

(

)

Fb

h

1 2 4 4 4 3 4 4 4 nK s

Fc

h

1 2 3 nK +

LS

1 2 3 d

e

eE(K )
B

e

{ },
{ }

( ) =

(K +) a

(

)

Fb

h

1 2 4 4 4 3 4 4 4 nK s(K +)

Fc

h

1 2 4 4 3 4 4 nK +

LS

1 2 3 d

e

eE(K )
DG-A

DG-B The new edge forms involve values of the fine scale φ’ from one element only

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Computational mathematics and algorithms

The local problems

DG-A DG-B

BK

,
(

) + B

,
(

) +

a(K +)

(

) nK

s nK + d

e

eE(K )
= RK

( )

BK

,
(

) + B

,
(

) +

(K +) a

(

) nK

s(K +) nK + d

e

eE(K )
= RK

( )

F

( ) = f F

( ) in K

PDE form of the local problem

= 0 on K
= 0 on K

advective ← limit → diffusive Residual driven!

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Computational mathematics and algorithms

VMS-DG Revisited

B

K

h,
h

( ) = RK

h

( )

Interscale operator

TK :

h

( ) a S p(K ) K ( )

T :

h

( ) a h ( )

T|K = TK BDG T

h,T h

( ) = FDG T

h

( )

h

h

( )

VMS-DG formulation Algebraic form

BDG h,h

( ) = FDG h ( )

Ax = f A Rnn; n = dim h

( )

BDG T

h,T h

( ) = FDG T

h

( )

TtATx = Ttf T Rnm; m = dim

h

( ) A

T

Tt

SLIDE 21

Computational mathematics and algorithms

Numerical examples 1D

Red = exact Blue = MDG (continuous) L-blue = MDG (discontinuous) Magenta= Donor DG

f=0 VMS-MDG is very close to the DG solution, while costs much less to compute. f=1 Solution plots 2, 8 and 32 elements

SLIDE 22

Computational mathematics and algorithms

Convergence rates

neutral s=0 symmetric s=-1 Skew similar to neutral For some donor DG formulations VMS-MDG converges better than the donor DG itself.

SLIDE 23

Computational mathematics and algorithms

Numerical examples in 2D

Advection skew to the mesh

h

h =

h +

h

h

DG

SLIDE 24

Computational mathematics and algorithms

Numerical examples 2D

Rotating cone

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Computational mathematics and algorithms

Conclusions

 Solves a long-standing problem with proliferation of DoFs in DG methods  Provides a template for “clever”, problem-dependent local condensation  Local problems are residual-driven  Attains and even somewhat improves upon both DG and continuous Galerkin We proposed a new VMS-DG method that combines advantages of DG with the more efficient computational structure of the continuous Galerkin method. VMS-DG  The local problem – Adding discontinuity capturing operators – Connection of with Riemann solvers  The donor DG method Future research directions