MATH 676
–
Finite element methods in
scientific computing
Wolfgang Bangerth, Texas A&M University
http://www.dealii.org/
Wolfgang Bangerth
Lecture 31.6:
Nonlinear problems
Part 3: Newton's method for the
minimal surface equation, step-15
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Wolfgang Bangerth
The minimal surface equation
Consider the minimal surface equation:
(
−∇⋅
A
∇u = f
2
√ 1+|∇ u|
u = g
)
in Ω
on ∂Ω
where we choose
Ω=B1 (0)⊂ℝ
2,
f =0,
g=sin(2 π( x + y))
Goal: Solve this numerically with Newton's method.
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Wolfgang Bangerth
Approach
Newton's method requires us to iterate these two
steps (in weak form):
(
A (∇ uk⋅∇ δ uk )
A
∇φ,
∇ δ uk − ∇ φ ,
∇ uk
2
2 3 /2
√ 1+|∇ uk|
( 1+|∇ u k| )
)(
(
= − ∇ φ,
)
A
∇ uk
2
√ 1+|∇ uk|
)
∀ φ∈H 10
uk + 1 = u k +δ uk
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Wolfgang Bangerth
Approach
Discrete form:
∑j
[(
)]
A (∇ uk , h⋅∇ φ j )
A
∇ φi ,
∇ φ j − ∇ φi ,
∇ uk ,h δ U k , j
2
2 3/ 2
√ 1+|∇ uk , h|
( 1+|∇ u k , h| )
)(
A
= − ∇ φi ,
∇ uk , h
2
√ 1+|∇ uk , h|
(
)
∀ i=1 ... N
U k +1 = U k +δ U k
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Wolfgang Bangerth
Approach
Discrete form (in matrix form):
Ak δ U k = Fk
U k +1
= U k +δ U k
where
A k , ij
A ( ∇ uk ,h⋅∇ φ j )
A
= ∇ φi ,
∇ φ j − ∇ φi ,
∇ uk , h
2
2 3 /2
√1+|∇ uk ,h|
( 1+|∇ uk , h| )
F k ,i
A
= − ∇ φi ,
∇ u k ,h
2
√ 1+|∇ uk , h|
)(
(
(
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)
)
Wolfgang Bangerth
Practical considerations
Consideration 1: To solve
Ak δ U k = Fk
with
A k , ij
A (∇ uk , h⋅∇ φ j )
A
= ∇ φi ,
∇ φj − ∇ φ,
∇ uk
2
2 3/ 2
√ 1+|∇ uk ,h|
( 1+|∇ u k , h| )
(
)(
)
what solver is appropriate?
Answer: The matrix is symmetric and positive definite, so
CG is ok.
Note: A is SPD because the minimal surface equation
corresponds to minimizing a convex energy functional E(u).
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Wolfgang Bangerth
Practical considerations
Consideration 2: What boundary values does δuk have?
Answer: Let us choose u0 so that it already has the
correct boundary values:
u 0 |∂ Ω = g
Then all of the δuk have zero boundary values because we
want that
(uk +α k δ u k )|∂ Ω = g
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Wolfgang Bangerth
Practical considerations
Consideration 3: Newton's method does not always
converge if we choose
U k +1 = U k +δ U k
But, we can often make it converge by relaxing the
iteration:
U k +1 = U k +α k δ U k
Algorithms for choosing αk are typically called “line search”.
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Wolfgang Bangerth
Practical considerations
Idea of line search: Using the damped iteration:
U k +1 = U k +α k δ U k
●
We can prove quadratic convergence if
●
We get slower convergence if
●
We may not converge if
●
We frequently converge if
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αk=1
αk<1
αk=1
αk<1
Wolfgang Bangerth
Practical considerations
Idea of line search: We want to achieve that
R (u) = L(u)−f = 0
So try this in iteration k:
●
Set αk=1
●
If
‖R (u k +αk δ uk )‖ ≤ c‖R (uk )‖
then use this αk; otherwise set αk:=αk/2 and try again
●
Compute
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U k +1 = U k +α k δ U k
Wolfgang Bangerth
Practical considerations
Practice of line search:
●
We need to say which norm we mean in
‖R (u k )‖
●
There are other criteria in addition to
‖R (u k +αk δ uk )‖ ≤ c‖R (uk )‖
These additional criteria are often called Wolfe and
Armijo-Goldstein conditions
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Wolfgang Bangerth
Practical considerations
In step-15:
●
For simplicity, just always choose
α k =0.1
●
The “Results” section of step-15 has more details on how
to implement an actual line search
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Wolfgang Bangerth
Practical considerations
Consideration 4: We want to use a sequence of meshes.
Practical implementation:
1. Start with a coarse mesh
2. Do 5 Newton iterations
3. If
‖R (u k )‖≤tol
then stop
4. Save the solution on the current mesh
5. Refine the mesh
6. Go to step 2
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Wolfgang Bangerth
The minimal surface equation
Let's look at all of this in real code!
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Wolfgang Bangerth
MATH 676
–
Finite element methods in
scientific computing
Wolfgang Bangerth, Texas A&M University
http://www.dealii.org/
Wolfgang Bangerth