dune-dpg: A Library for Discontinous
Petrov Galerkin Using Optimal Test Spaces
Felix Gruber and Angela Klewinghaus
joint work with Olga Mula
IGPM, RWTH Aachen
Dune User Meeting 2015 in Heidelberg
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dune-dpg | F. Gruber, A. Klewinghaus | September 2015
Motivation:
Variational formulation:
Find u ∈ U such that
b (u , v ) = `(v )
for all v ∈ V
for Hilbert spaces U , V , continuous bilinear form b : U × V → R and ` ∈ V 0 .
Goal: Stability
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dune-dpg | F. Gruber, A. Klewinghaus | September 2015
Motivation:
Variational formulation:
Find u h ∈ U h such that
b (u h , v h ) = `(v h )
for all v h ∈ V h
for Hilbert spaces U , V , continuous bilinear form b : U × V → R and ` ∈ V 0 .
Goal: Stability
• Discrete approximation yields a best approximation (up to a constant)
ku − uh kU . inf ku − wh kU .
wh ∈Uh
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dune-dpg | F. Gruber, A. Klewinghaus | September 2015
Motivation:
Variational formulation:
Find u h ∈ U h such that
b (u h , v h ) = `(v h )
for all v h ∈ V h
for Hilbert spaces U , V , continuous bilinear form b : U × V → R and ` ∈ V 0 .
Goal: Stability
• Discrete approximation yields a best approximation (up to a constant)
ku − uh kU . inf ku − wh kU .
wh ∈Uh
• Residual-based error bound
ku − uh kU ' kb (uh , ·) − `kV 0 .
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dune-dpg | F. Gruber, A. Klewinghaus | September 2015
Ultimate Goal:
Input
bilinear form b (·, ·)
Output
dune-dpg
righthand side `(·)
???
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dune-dpg | F. Gruber, A. Klewinghaus | September 2015
near best approximation uh
rigorous error bound
Choose Uh to Ensure Good Approximation Properties, Choose Vh to
Ensure Stability
How to choose Vh for given Uh ?
Literature:
Barret/Morton, Demkowicz/Gopalakrishnan, Dahmen/Cohen/Welper,
Dahmen/Huang/Schwab/Welper, Broersen/Stevenson, Broersen/Dahmen/Stevenson
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dune-dpg | F. Gruber, A. Klewinghaus | September 2015
Choose Uh to Ensure Good Approximation Properties, Choose Vh to
Ensure Stability
How to choose Vh for given Uh ?
• Optimal test norm
k · kV ,opt := sup
u ∈U
b (u , v )
ku k U
– yields residual based error bound
kuh − u kU = ||b (uh , ·) − `||V 0 ,opt
Literature:
Barret/Morton, Demkowicz/Gopalakrishnan, Dahmen/Cohen/Welper,
Dahmen/Huang/Schwab/Welper, Broersen/Stevenson, Broersen/Dahmen/Stevenson
4 of 15
dune-dpg | F. Gruber, A. Klewinghaus | September 2015
Choose Uh to Ensure Good Approximation Properties, Choose Vh to
Ensure Stability
How to choose Vh for given Uh ?
• Optimal test norm
k · kV ,opt := sup
u ∈U
b (u , v )
ku k U
– yields residual based error bound
kuh − u kU = ||b (uh , ·) − `||V 0 ,opt
• Optimal test space Vh = B −∗ RU Uh , where RU : U → U 0 is the Rieszisomorphism,
that is Vh := T (Uh ) where
hT (uh ), v iV ,opt = b (uh , v ) ,
∀v ∈V
– yields best approximation property
ku − uh kU = inf ku − wh kU
wh ∈Uh
Literature:
Barret/Morton, Demkowicz/Gopalakrishnan, Dahmen/Cohen/Welper,
Dahmen/Huang/Schwab/Welper, Broersen/Stevenson, Broersen/Dahmen/Stevenson
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dune-dpg | F. Gruber, A. Klewinghaus | September 2015
Computation of the Optimal Test Space
We have to compute Vh by solving
hT (uh ), v iV ,opt = b (uh , v ) ,
Literature:
∀v ∈V
Barret/Morton, Demkowicz/Gopalakrishnan, Dahmen/Cohen/Welper,
Dahmen/Huang/Schwab/Welper, Broersen/Stevenson, Broersen/Dahmen/Stevenson
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dune-dpg | F. Gruber, A. Klewinghaus | September 2015
Computation of the Optimal Test Space
We have to compute Vh by solving
hT (uh ), v iV ,opt = b (uh , v ) ,
∀v ∈V
• This is infinite dimensional
Literature:
Barret/Morton, Demkowicz/Gopalakrishnan, Dahmen/Cohen/Welper,
Dahmen/Huang/Schwab/Welper, Broersen/Stevenson, Broersen/Dahmen/Stevenson
5 of 15
dune-dpg | F. Gruber, A. Klewinghaus | September 2015
Computation of the Optimal Test Space
We have to compute Vh by solving
hT (uh ), v iV ,opt = b (uh , v ) ,
∀v ∈V
• This is infinite dimensional
– Projection approach: Replace V by some good enough finite dimensional test search
space V̂ ⊂ V
Literature:
Barret/Morton, Demkowicz/Gopalakrishnan, Dahmen/Cohen/Welper,
Dahmen/Huang/Schwab/Welper, Broersen/Stevenson, Broersen/Dahmen/Stevenson
5 of 15
dune-dpg | F. Gruber, A. Klewinghaus | September 2015
Computation of the Optimal Test Space
We have to compute Vh by solving
hT (uh ), v iV ,opt = b (uh , v ) ,
∀v ∈V
• This is infinite dimensional
– Projection approach: Replace V by some good enough finite dimensional test search
space V̂ ⊂ V
• We cannot (afford to) compute the optimal test norm and solve global problems
Literature:
Barret/Morton, Demkowicz/Gopalakrishnan, Dahmen/Cohen/Welper,
Dahmen/Huang/Schwab/Welper, Broersen/Stevenson, Broersen/Dahmen/Stevenson
5 of 15
dune-dpg | F. Gruber, A. Klewinghaus | September 2015
Computation of the Optimal Test Space
We have to compute Vh by solving
hT (uh ), v iV ,opt = b (uh , v ) ,
∀v ∈V
• This is infinite dimensional
– Projection approach: Replace V by some good enough finite dimensional test search
space V̂ ⊂ V
• We cannot (afford to) compute the optimal test norm and solve global problems
– replace the optimal test norm by an equivalent but localizable and computable norm
– use an ultraweak variational formulation
– (alternatively: use a saddlepoint formulation)
Literature:
Barret/Morton, Demkowicz/Gopalakrishnan, Dahmen/Cohen/Welper,
Dahmen/Huang/Schwab/Welper, Broersen/Stevenson, Broersen/Dahmen/Stevenson
5 of 15
dune-dpg | F. Gruber, A. Klewinghaus | September 2015
Computation of the Optimal Test Space
We have to compute Vh by solving
hT (uh ), v iV ,opt = b (uh , v ) ,
∀v ∈V
• This is infinite dimensional
– Projection approach: Replace V by some good enough finite dimensional test search
space V̂ ⊂ V
• We cannot (afford to) compute the optimal test norm and solve global problems
– replace the optimal test norm by an equivalent but localizable and computable norm
– use an ultraweak variational formulation
– (alternatively: use a saddlepoint formulation)
We can compute an approximation for Vh by solving
hT̂ (uh ), v iV ,K = bK (uh , v ) ,
∀ v ∈ V̂ K ,
where K is a single cell and h·, ·iV ,K and bK (·, ·) denote restrictions to K .
Literature:
Barret/Morton, Demkowicz/Gopalakrishnan, Dahmen/Cohen/Welper,
Dahmen/Huang/Schwab/Welper, Broersen/Stevenson, Broersen/Dahmen/Stevenson
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dune-dpg | F. Gruber, A. Klewinghaus | September 2015
dune-dpg:
Input
dune-dpg
ultraweak bilinear form b (·, ·)
compute Vh
righthand side `(·)
trial space Uh
solve b (uh , vh ) = `(vh )
test search space V̂
for all vh ∈ Vh
localizable inner product h·, ·iV
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dune-dpg | F. Gruber, A. Klewinghaus | September 2015
Output
near best approximation uh
rigorous error bound
Implementation: Layout of dune-dpg
rough overview of a DPG solver:
define grid
define trial space Uh and test search space V̂
define bilinear form and inner product
assemble system matrix and rhs vector
interpolate boundary values
solve system
compute a posteriori error estimates
[ refine adaptively ]
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dune-dpg | F. Gruber, A. Klewinghaus | September 2015
Implementation: BilinearForm / InnerProduct
• In our case, a bilinear form can be seen as a sum of integral terms, e. g.
bK ((u , q ), v ) :=
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Z
cvu −
K
dune-dpg | F. Gruber, A. Klewinghaus | September 2015
Z
K
β · ∇vu +
Z
uqβ · n.
∂K
Implementation: BilinearForm / InnerProduct
• In our case, a bilinear form can be seen as a sum of integral terms, e. g.
bK ((u , q ), v ) :=
Z
cvu −
K
Z
K
β · ∇vu +
Z
• The bilinear form is used to create a local system matrix
MijK := bK ((uj , qj ), vi )
which is a sum of matrices given by the integrals above.
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dune-dpg | F. Gruber, A. Klewinghaus | September 2015
uqβ · n.
∂K
Implementation: BilinearForm / InnerProduct
• In our case, a bilinear form can be seen as a sum of integral terms, e. g.
bK ((u , q ), v ) :=
Z
cvu −
K
Z
K
β · ∇vu +
Z
uqβ · n.
∂K
• The bilinear form is used to create a local system matrix
MijK := bK ((uj , qj ), vi )
which is a sum of matrices given by the integrals above.
• Thus we introduce an IntegralTerm class to generate the local matrix for a single
integral. The BilinearForm is then parametrized over several IntegralTerms and
sums up their local matrices.
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dune-dpg | F. Gruber, A. Klewinghaus | September 2015
Implementation: BilinearForm / InnerProduct
• In our case, a bilinear form can be seen as a sum of integral terms, e. g.
bK ((u , q ), v ) :=
Z
cvu −
K
Z
K
β · ∇vu +
Z
uqβ · n.
∂K
• The bilinear form is used to create a local system matrix
MijK := bK ((uj , qj ), vi )
which is a sum of matrices given by the integrals above.
• Thus we introduce an IntegralTerm class to generate the local matrix for a single
integral. The BilinearForm is then parametrized over several IntegralTerms and
sums up their local matrices.
• Abstractly, an inner product is simply a symmetric bilinear form over a single
space. Thus, we can reuse the “sum over IntegralTerms” logic to define an
InnerProduct class.
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dune-dpg | F. Gruber, A. Klewinghaus | September 2015
Implementation: Optimal Testspace
• Localfunctions: Linear combinations of localfunctions of the test search space
– Parameter: Localfunctions Test Search Space, Coefficient Matrix
• Functions:
– Parameter: Trial Spaces, Test Search Spaces, Bilinear Form, Inner Product
– uses IndexSet of Trial Spaces
– in LocalView.bind(): Uses Trial Spaces, Test Search Spaces, Bilinear Form, Inner Product
to set up and solve
hT̂ (uh ), v iV ,K = bK (uh , v ) ,
∀ v ∈ V̂ K ,
for every (local) solution basis function uh on the current (physical) cell.
→ yields coefficient matrix for Localfunctions
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dune-dpg | F. Gruber, A. Klewinghaus | September 2015
Implementation: SystemAssembler
• Building a global system matrix and right hand side for our method is handled by
the SystemAssembler class.
• The SystemAssembler loops over all cells and uses the BilinearForm and
OptimalTestBasis to build local matrices.
• The local matrices built by the BilinearForm are then copied into a global system
matrix. (Thanks to dune-functions we have a mapping between local and global
indices.)
• Similarily the right hand side of the system is assembled cell by cell.
• SystemAssembler also later applies the boundary values to the system.
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dune-dpg | F. Gruber, A. Klewinghaus | September 2015
Implementation: Overview of dune-dpg
SystemAssembler
assembleSystem()
BilinearForm
getLocalMatrix()
OptimalTestBasis
IntegralTerm
getLocalMatrix()
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dune-dpg | F. Gruber, A. Klewinghaus | September 2015
InnerProduct
getLocalMatrix()
Example: Transport
Transport equation:
β · ∇u + cu = f
u=g
on Ω,
on Γ− .
Challenge:
• not "second order elliptic" but hyperbolic
Theoreticel foundations:
• D. Broersen, W. Dahmen, R. Stevenson
Applications:
• discrete ordinates discretization of Boltzman equation (F. Gruber, joint work with
W. Dahmen, O. Mula)
• limit case of the convection-diffusion equation (A. Klewinghaus, joint work with W.
Dahmen)
• porous media (W. Dahmen, V. König, S. Müller)
We’d like to thank Oliver Sander for his introduction into dune(-functions).
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dune-dpg | F. Gruber, A. Klewinghaus | September 2015
Results Transport
Figure: β = (1, 1), c = 1, f = 1
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dune-dpg | F. Gruber, A. Klewinghaus | September 2015
Results Transport
Figure: β = (1, 1), c = 1, f = 1, if x1 < x2 , or f = 0 else
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dune-dpg | F. Gruber, A. Klewinghaus | September 2015
Results Convection Diffusion
u
u
1.000e+00
1.000e+00
0.75
0.75
0.5
0.5
0.25
0.25
0
0
-2.031e-01
-2.031e-01
(a) e = 0.02
(b) e = 0.01
u
(c) e = 0.002
u
1.000e+00
1.000e+00
0.75
0.75
0.5
0.5
0.25
0.25
0
0
-2.031e-01
-2.031e-01
(d) e = 0.00001
Figure: Ω = (0, 1)2 , regular square grid with 50 × 50 elements (h = 0.02)
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dune-dpg | F. Gruber, A. Klewinghaus | September 2015