Discretisation
Solution of the system
Numerical experiments
Further applications
A Finite Element Method for Nonvariational
Elliptic Problems
Tristan Pryer
joint work with Omar Lakkis
May 20, 2010
Discretisation
Solution of the system
Outline
1
Discretisation
2
Solution of the system
3
Numerical experiments
4
Further applications
Numerical experiments
Further applications
Discretisation
Solution of the system
Numerical experiments
Further applications
Model problem in nonvariational form
Model problem
Given f ∈ L2 (Ω), find u ∈ H2 (Ω) ∩ H10 (Ω) such that
A:D2 u | φ = hf , φi ∀ φ ∈ H10 (Ω),
X:Y := trace (X| Y) is the Frobenius inner product of two matrices.
Discretisation
Solution of the system
Numerical experiments
Further applications
Model problem in nonvariational form
Model problem
Given f ∈ L2 (Ω), find u ∈ H2 (Ω) ∩ H10 (Ω) such that
A:D2 u | φ = hf , φi ∀ φ ∈ H10 (Ω),
X:Y := trace (X| Y) is the Frobenius inner product of two matrices.
Discretisation
Solution of the system
Numerical experiments
Further applications
Model problem in nonvariational form
Model problem
Given f ∈ L2 (Ω), find u ∈ H2 (Ω) ∩ H10 (Ω) such that
A:D2 u | φ = hf , φi ∀ φ ∈ H10 (Ω),
X:Y := trace (X| Y) is the Frobenius inner product of two matrices.
Don’t want to rewrite in divergence form!
A:D2 u | φ = hdiv (A∇u) | φi − hdiv (A) ∇u, φi .
Discretisation
Solution of the system
Numerical experiments
What should we do?
Main idea
Define appropriately the Hessian of a function who’s not twice
differentiable, i.e., as a distribution.
Further applications
Discretisation
Solution of the system
Numerical experiments
Further applications
What should we do?
Main idea
Define appropriately the Hessian of a function who’s not twice
differentiable, i.e., as a distribution.
Construct a finite element approximation of this distribution. What
is meant by the Hessian of a finite element function?
[Aguilera and Morin, 2008]
Discretisation
Solution of the system
Numerical experiments
Further applications
What should we do?
Main idea
Define appropriately the Hessian of a function who’s not twice
differentiable, i.e., as a distribution.
Construct a finite element approximation of this distribution. What
is meant by the Hessian of a finite element function?
[Aguilera and Morin, 2008]
Discretise the strong form of the PDE directly.
Discretisation
Solution of the system
Numerical experiments
Hessians as distributions
generalised Hessian
Given v ∈ H1 (Ω) it’s generalised Hessian is given by
2
D v | φ = − h∇v ⊗ ∇φi + h∇u ⊗ n φi∂Ω ∀ φ ∈ H1 (Ω),
x ⊗ y := xy| is the tensor product between two vectors.
Further applications
Discretisation
Solution of the system
Numerical experiments
Further applications
Hessians as distributions
generalised Hessian
Given v ∈ H1 (Ω) it’s generalised Hessian is given by
2
D v | φ = − h∇v ⊗ ∇φi + h∇u ⊗ n φi∂Ω ∀ φ ∈ H1 (Ω),
x ⊗ y := xy| is the tensor product between two vectors.
finite element space notation
V :=
Φ ∈ H1 (Ω) : Φ|K ∈ Pp ∀ K ∈ T
V̊ := V ∩
H10 (Ω),
,
Discretisation
Solution of the system
Numerical experiments
Further applications
Hessians as distributions
generalised Hessian
Given v ∈ H1 (Ω) it’s generalised Hessian is given by
2
D v | φ = − h∇v ⊗ ∇φi + h∇u ⊗ n φi∂Ω ∀ φ ∈ H1 (Ω),
x ⊗ y := xy| is the tensor product between two vectors.
finite element space notation
V :=
Φ ∈ H1 (Ω) : Φ|K ∈ Pp ∀ K ∈ T
V̊ := V ∩
,
H10 (Ω),
finite element Hessian
For each V ∈ V̊ there exists a unique H[V ] ∈ Vd×d such that
hH[V ], Φi = D2 V | Φ
∀ Φ ∈ V.
Discretisation
Solution of the system
Numerical experiments
Further applications
Nonvariational finite element method
Substitute the finite element Hessian directly into the model problem.
We seek U ∈ V̊ such that
D
E D
E
A:H[U], Φ̊ = f , Φ̊
∀ Φ̊ ∈ V̊.
Discretisation
Solution of the system
Numerical experiments
Further applications
Nonvariational finite element method
Substitute the finite element Hessian directly into the model problem.
We seek U ∈ V̊ such that
D
E D
E
A:H[U], Φ̊ = f , Φ̊
∀ Φ̊ ∈ V̊.
Discretisation
D
d X
d D
E X
E
Aα,β Hα,β [U], Φ̊
f , Φ̊ =
α=1 β=1
=
d X
d D
X
α=1 β=1
E
Φ̊, Aα,β Φ| hα,β .
Discretisation
Solution of the system
Numerical experiments
Further applications
Nonvariational finite element method
Substitute the finite element Hessian directly into the model problem.
We seek U ∈ V̊ such that
D
E D
E
A:H[U], Φ̊ = f , Φ̊
∀ Φ̊ ∈ V̊.
Discretisation
D
d X
d D
E X
E
Aα,β Hα,β [U], Φ̊
f , Φ̊ =
α=1 β=1
=
d X
d D
X
α=1 β=1
E
Φ̊, Aα,β Φ| hα,β .
Discretisation
Solution of the system
Numerical experiments
Further applications
Nonvariational finite element method
Substitute the finite element Hessian directly into the model problem.
We seek U ∈ V̊ such that
D
E D
E
A:H[U], Φ̊ = f , Φ̊
∀ Φ̊ ∈ V̊.
Discretisation
D
d D
d X
E X
E
f , Φ̊ =
Aα,β Hα,β [U], Φ̊
α=1 β=1
=
d X
d D
X
E
Φ̊, Aα,β Φ| hα,β .
α=1 β=1
hΦ, Φ| i hα,β = hΦ, Hα,β [U]i
D
E D
E
|
|
= − ∂β Φ, ∂α Φ̊ + Φnβ , ∂α Φ̊
∂Ω
u.
Discretisation
Solution of the system
Numerical experiments
Further applications
Nonvariational finite element method
Substitute the finite element Hessian directly into the model problem.
We seek U ∈ V̊ such that
D
E D
E
A:H[U], Φ̊ = f , Φ̊
∀ Φ̊ ∈ V̊.
Discretisation
D
d D
d X
E X
E
f , Φ̊ =
Aα,β Hα,β [U], Φ̊
α=1 β=1
=
d X
d D
X
E
Φ̊, Aα,β Φ| hα,β .
α=1 β=1
hΦ, Φ| i hα,β = hΦ, Hα,β [U]i
D
E D
E
|
|
= − ∂β Φ, ∂α Φ̊ + Φnβ , ∂α Φ̊
∂Ω
u.
Discretisation
Solution of the system
Numerical experiments
Further applications
Linear system
U = Φ̊ u, where u ∈ RN̊ is the solution to the following linear system
|
Du :=
d
d X
X
α=1 β=1
Bα,β M−1 Cα,β u = f.
Discretisation
Solution of the system
Numerical experiments
Further applications
Linear system
U = Φ̊ u, where u ∈ RN̊ is the solution to the following linear system
|
Du :=
d
d X
X
Bα,β M−1 Cα,β u = f.
α=1 β=1
Components of the linear system
D
E
Bα,β := Φ̊, Aα,β Φ| ∈ RN̊×N ,
M := hΦ, Φ| i ∈ RN×N ,
D
E D
E
|
|
Cα,β := − ∂β Φ, ∂α Φ̊ + Φnβ , ∂α Φ̊
∈ RN×N̊ ,
∂Ω
D
E
f := f , Φ̊ ∈ RN̊ .
Discretisation
Solution of the system
Numerical experiments
Further applications
The system is hard to solve
Linear system
U = Φ̊ u, where u ∈ RN̊ is the solution to the following linear system
|
Du :=
d
d X
X
α=1 β=1
Remarks
The matrix D is not sparse
Bα,β M−1 Cα,β u = f.
Discretisation
Solution of the system
Numerical experiments
Further applications
The system is hard to solve
Linear system
U = Φ̊ u, where u ∈ RN̊ is the solution to the following linear system
|
Du :=
d
d X
X
Bα,β M−1 Cα,β u = f.
α=1 β=1
Remarks
The matrix D is not sparse
Mass lumping only works for P1 elements AND in this case only
gives a reasonable solution U for very simple A
Discretisation
Solution of the system
Numerical experiments
Further applications
The system is hard to solve
Linear system
U = Φ̊ u, where u ∈ RN̊ is the solution to the following linear system
|
Du :=
d
d X
X
Bα,β M−1 Cα,β u = f.
α=1 β=1
Remarks
The matrix D is not sparse
Mass lumping only works for P1 elements AND in this case only
gives a reasonable solution U for very simple A
Notice D is the sum of Schur complements
Discretisation
Solution of the system
Numerical experiments
Further applications
The system is hard to solve
Linear system
U = Φ̊ u, where u ∈ RN̊ is the solution to the following linear system
|
Du :=
d
d X
X
Bα,β M−1 Cα,β u = f.
α=1 β=1
Remarks
The matrix D is not sparse
Mass lumping only works for P1 elements AND in this case only
gives a reasonable solution U for very simple A
Notice D is the sum of Schur complements
We can create a block matrix to exploit this
Discretisation
Solution of the system
Numerical experiments
Further applications
Block system





E=



M
0
..
.
0
M
..
.
0
0
B1,1
0
0
B1,2
···
···
..
.
0
0
..
.
···
M
...
0
d,d−1
... B
0
0
..
.
−C1,1
−C1,2
..
.
0
M
Bd,d
−Cd,d−1
−Cd,d
0
|
v = (h1,1 , h1,2 , . . . , hd,d−1 , hd,d , u) ,
|
h = (0, 0 . . . , 0, 0, f) .





,



Discretisation
Solution of the system
Numerical experiments
Further applications
Block system





E=



M
0
..
.
0
M
..
.
0
0
B1,1
0
0
B1,2
···
···
..
.
0
0
..
.
···
M
...
0
d,d−1
... B
0
0
..
.
−C1,1
−C1,2
..
.
0
M
Bd,d
−Cd,d−1
−Cd,d
0
|
v = (h1,1 , h1,2 , . . . , hd,d−1 , hd,d , u) ,
|
h = (0, 0 . . . , 0, 0, f) .
Equivalence of systems
Then solving the system Du = f is equivalent to solving
Ev = h.
for u.





,



Discretisation
Solution of the system
Numerical experiments
Further applications
structure of the block matrix
the discretisation presented nothing but:
Find U ∈ V̊ such that


hH[U], Φi = − h∇U ⊗ ∇Φi + h∇U ⊗ n Φi∂Ω




∀Φ∈V


D
E D
E


 A:H[U], Φ̊ = f , Φ̊
∀ Φ̊ ∈ V̊.
Discretisation
Solution of the system
Numerical experiments
Further applications
structure of the block matrix
the discretisation presented nothing but:
Find U ∈ V̊ such that


hH[U], Φi = − h∇U ⊗ ∇Φi + h∇U ⊗ n Φi∂Ω




∀Φ∈V


D
E D
E


 A:H[U], Φ̊ = f , Φ̊









M
0
..
.
0
M
..
.
0
0
B1,1
0
0
B1,2
···
···
..
.
0
0
..
.
···
M
...
0
. . . Bd,d−1
0
0
..
.
−C1,1
−C1,2
..
.
0
M
Bd,d
−Cd,d−1
−Cd,d
0

∀ Φ̊ ∈ V̊.
h1,1
h1,2
..
.





  hd,d−1

  hd,d
u


0
0
..
.

 

 


 
 

.
=
  0 
 

  0 
f
Discretisation
Solution of the system
Numerical experiments
A Linear PDE in NDform
Operator choice - heavily oscillating
1
0
A=
0 a(x)
Further applications
Discretisation
Solution of the system
Numerical experiments
Further applications
A Linear PDE in NDform
Operator choice - heavily oscillating
1
0
A=
0 a(x)
1
a(x) = sin
|x1 | + |x2 | + 10−15
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Discretisation
Solution of the system
Numerical experiments
Figure: Choosing f appropriately such that u(x) = exp (−10 |x|2 ).
1
10
X = L2
X = H1
10
EOC = 0.94816
EOC = 0.96833
EOC = 1.0129
−1
10 EOC = 1.6815
EOC = 1.0149
EOC = 1.0094
EOC = 1.0052
EOC
=
2.0751
−2
10
EOC = 2.0465
||u−uh||X
0
−3
EOC = 2.0345
10
EOC = 2.02
−4
10
EOC = 2.0107
−5
10
2
10
3
10
dim V
4
10
Further applications
Discretisation
Solution of the system
Numerical experiments
Another Linear PDE in NDform
Operator choice
A=
1
0
0
a(x)
Further applications
Discretisation
Solution of the system
Numerical experiments
Another Linear PDE in NDform
Operator choice
A=
1
0
0
a(x)
2
a(x) := arctan 5000(|x| − 1) + 2 .
Further applications
Discretisation
Solution of the system
Numerical experiments
Further applications
Figure: Choosing f appropriately such that u(x) = sin (πx1 ) sin (πx2 ).
1
10
||u−uh||X
X = L2
X = H1
EOC
= 0.91495
0
10
EOC = 1.0384
EOC = 1.0339
EOC
=
1.8547
EOC = 1.02
−1
10
EOC = 1.0107
EOC = 1.0055
EOC = 2.0811
−2
10
EOC = 2.069
EOC = 2.0402
−3
10
EOC = 2.0214
EOC = 2.011
−4
10
2
10
3
10
dim V
4
10
Discretisation
Solution of the system
Numerical experiments
Further applications
The same Linear PDE in NDform
Figure: On the left we present the maximum error of the standard FE-solution.
Notice the oscillations apparant on the unit circle. On the right we show the
maximum error of the NDFE-solution
Discretisation
Solution of the system
Numerical experiments
Fully nonlinear PDEs
Model problem
Given f ∈ L2 (Ω), find u ∈ H2 (Ω) ∩ H10 (Ω) such that
N [u] = F (D2 u) − f = 0
Further applications
Discretisation
Solution of the system
Numerical experiments
Fully nonlinear PDEs
Model problem
Given f ∈ L2 (Ω), find u ∈ H2 (Ω) ∩ H10 (Ω) such that
N [u] = F (D2 u) − f = 0
Newton’s method
Given u 0 for each n ∈ N find u n+1 such that
0 n
N [u ] | u n+1 − u n = −N [u n ]
Further applications
Discretisation
Solution of the system
Numerical experiments
Fully nonlinear PDEs
Model problem
Given f ∈ L2 (Ω), find u ∈ H2 (Ω) ∩ H10 (Ω) such that
N [u] = F (D2 u) − f = 0
Newton’s method
Given u 0 for each n ∈ N find u n+1 such that
0 n
N [u ] | u n+1 − u n = −N [u n ]
N [u + v ] − N [u]
F (D2 u + D2 v ) − F (D2 u)
= lim
→0
0
2
2
= lim F (D u) : D v + O().
hN 0 [u] | v i = lim
→0
→0
Further applications
Discretisation
Solution of the system
Numerical experiments
Fully nonlinear PDEs
Model problem
Given f ∈ L2 (Ω), find u ∈ H2 (Ω) ∩ H10 (Ω) such that
N [u] = F (D2 u) − f = 0
Newton’s method
Given u 0 for each n ∈ N find u n+1 such that
0 n
N [u ] | u n+1 − u n = −N [u n ]
N [u + v ] − N [u]
F (D2 u + D2 v ) − F (D2 u)
= lim
→0
= lim F 0 (D2 u) : D2 v + O().
hN 0 [u] | v i = lim
→0
→0
Further applications
Discretisation
Solution of the system
Numerical experiments
Discretisation
VERY roughly
Given U 0 = Λu 0 find U n+1 such that
F 0 (H[U n ]):H[U n+1 − U n ] = f − F (H[U n ])
Further applications
Discretisation
Solution of the system
Numerical experiments
Further applications
Discretisation
VERY roughly
Given U 0 = Λu 0 find U n+1 such that
F 0 (H[U n ]):H[U n+1 − U n ] = f − F (H[U n ])
H[U n ] is given in the solution of the previous iterate!









M
0
..
.
0
M
..
.
···
···
..
.
0
0
0
0
B1,1
n−1
B1,2
n−1
···
M
...
0
d,d−1
. . . Bn−1
0
0
..
.
0
0
..
.
−C1,1
−C1,2
..
.
0
M
d,d
Bn−1
−Cd,d−1
−Cd,d
0

hn1,1
hn1,2
..
.




 n
 h
  d,d−1
  hnd,d
un


0
0
..
.

 


 
 


 
.
=
  0 

 
  0 
f
Discretisation
Solution of the system
Numerical experiments
Further applications
Discretisation
VERY roughly
Given U 0 = Λu 0 find U n+1 such that
F 0 (H[U n ]):H[U n+1 − U n ] = f − F (H[U n ])
H[U n ] is given in the solution of the previous iterate!









M
0
..
.
0
M
..
.
···
···
..
.
0
0
0
0
B1,1
n−1
B1,2
n−1
···
M
...
0
d,d−1
. . . Bn−1
0
0
..
.
0
0
..
.
−C1,1
−C1,2
..
.
0
M
d,d
Bn−1
−Cd,d−1
−Cd,d
0

hn1,1
hn1,2
..
.




 n
 h
  d,d−1
  hnd,d
un


0
0
..
.

 


 
 

 

.
=
  0 

 
  0 
f
Discretisation
Solution of the system
Numerical experiments
Further applications
Discretisation
VERY roughly
Given U 0 = Λu 0 find U n+1 such that
F 0 (H[U n ]):H[U n+1 − U n ] = f − F (H[U n ])
H[U n ] is given in the solution of the previous iterate!









M
0
..
.
0
M
..
.
···
···
..
.
0
0
0
0
B1,1
n−1
B1,2
n−1
···
M
...
0
d,d−1
. . . Bn−1
0
0
..
.
0
0
..
.
−C1,1
−C1,2
..
.
0
M
d,d
Bn−1
−Cd,d−1
−Cd,d
0

hn1,1
hn1,2
..
.




 n
 h
  d,d−1
  hnd,d
un
Saves us postprocessing another one! [Vallet et al., 2007]
[Ovall, 2007]


0
0
..
.

 


 
 

 

.
=
  0 

 
  0 
f
Discretisation
Solution of the system
Numerical experiments
Further applications
Bibliography I
Aguilera, N. E. and Morin, P. (2008).
On convex functions and the finite element method.
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