A Quartic Finite Element Method for a 4th Order
Nonlinear PDE
Nick Bird
University of Reading
May 2012
Supervisors: Mike Baines, Steve Langdon
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Contents
1
Introduction
2
The Finite Element Method
3
Numerical Results
4
Conclusions and Future Work
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A 4th Order Nonlinear Diffusion Equation
We consider the 1D 4th order nonlinear partial differential equation
3 ∂u
∂
∂ u
=−
u 3 ,
∂t
∂x
∂x
x ∈ Ω(t) ≡ (a(t), b(t)),
subject to initial condition u 0 (x) and boundary conditions
u = 0,
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∂u
= 0,
∂x
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u
∂3u
= 0.
∂x 3
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A 4th Order Nonlinear Diffusion Equation
2
Putting q = − ∂∂xu2 , we rewrite the 4th order equation as a pair of coupled
equations
∂
∂q
∂2u
∂u
=
u
,
q = − 2,
∂t
∂x
∂x
∂x
x ∈ Ω(t) ≡ (a(t), b(t)),
subject to initial condition u 0 (x) and boundary conditions
u = 0,
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∂u
= 0,
∂x
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u
∂q
= 0.
∂x
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Similarity Solutions
It is known that for problems of the form
2m+1 u
∂u
m ∂
n∂
= (−1)
u
,
∂t
∂x
∂x 2m+1
with n = 1 and m = 0, 1, 2, . . ., there exist source-type similarity solutions
(Smyth and Hill (1988)). (Note: In the case m = 0, similarity solutions
exist for all n).
Since the 4th order problem presented here is an example of such a
problem, we can use a self-similar solution as a means of testing the
accuracy of our numerical results.
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Similarity Solutions
Figure: Plot of u S (x, t) =
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1
120t 1/5
h
4−
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x2
t 2/5
i2
at time t = 1.
+
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Velocity
As the domain Ω(t) is allowed to vary in time, we seek a velocity v (x, t) in
addition to the solution u(x, t).
This velocity dictates the evolution of Ω(t).
Through use of a local conservation principle
Z
u(x, t) dx = ρ(Ω0 ),
ρ(Ω0 ) constant,
Ω0 (t)
where Ω0 (t) ⊂ Ω(t), we are able to show that the velocity is given by
v =−
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∂q
.
∂x
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Weak Forms
Before we outline the Finite Element Method, we firstly construct weak
forms of the various equations that are required. These weak forms are
Z
b(t)
Z
b(t)
w (x, t)q dx =
a(t)
Z
a(t)
b(t)
Z
b(t)
w (x, t)v dx = −
a(t)
∂w ∂u
dx,
∂x ∂x
w (x, t)
a(t)
∂q
dx,
∂x
∀w ∈ W ,
∀w ∈ W ,
for test functions w lying in suitable test spaces W .
We also have a weighted conservation of mass principle for use in solution
recovery
Z
b(t)
w (x, t)u dx = ρ(w ),
ρ(w ) constant in time.
a(t)
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1
Introduction
2
The Finite Element Method
3
Numerical Results
4
Conclusions and Future Work
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Finite Element Method Algorithm
The algorithm in the Finite Element Method is:
Calculate Q from a given U, where Q(x, t) ≈ q(x, t) and
U(x, t) ≈ u(x, t).
Calculate V using Q, where V (x, t) ≈ v (x, t).
Update x using a time-stepping scheme and V .
Recover the solution U on the updated domain.
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Choice of Test Functions
We use a combination of piecewise linear, quadratic and quartic basis
functions in the Finite Element Method,
Figure: Linear basis
functions φ(x, t).
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Figure: Quadratic
Figure: Quartic basis
basis functions ϕ(x, t). functions χ(x, t).
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Function Approximations and Linear Systems
Using our choice of basis functions, we approximate the functions in the
method by the linear combinations
X
u(x, t) ≈ U(x, t) =
χj (x, t)Uj ,
j
q(x, t) ≈ Q(x, t) =
X
ϕj (x, t)Qj ,
j
v (x, t) ≈ V (x, t) =
X
φj (x, t)Vj .
j
We obtain the various coefficients required in vector form through the
solution of linear systems
AQ = K U,
MV = −BQ
Aik =< ϕi , ϕk >, Kik =< (ϕi )x , (χk )x >, Mik =< φi , φk >,
Bik =< φi , (ϕk )x >.
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Time-stepping
Using the velocity, the domain is moved using a scale-invariant Forward
Euler time-stepping scheme (Budd and Piggott (2001),Baines et al.
(2006)). This requires the introduction of a scaled time variable s = t 1/5 .
The time-stepping scheme then becomes
x new = x + ∆sV,
where V is the velocity written in terms of the scaled time variable s, with
truncation error
∆s d 2 x ∆x
−V =
.
∆s
2 ds 2 ∗
s ∈(s,s+∆s)
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Solution Recovery
The solution is recovered on the updated domain via the weighted
conservation of mass principle
Z
b(s)
w (x, s)u dx = ρ(w ),
a(s)
where ρ(w ) is constant in time.
We obtain coefficients Uj by solving the linear system
C U = ρ,
where Cik =< χi , χk > and ρj = ρ(χj ).
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First Time-Step
If the initial condition u 0 (x) is taken to be equal to the similarity
solution at time s0 , then our initial approximation U 0 (x) coincides
with the similarity solution to within rounding error.
Using this U 0 (x), the approximation Q(x, s0 ) is a quadratic and
matches the exact quadratic q(x, s0 ) to rounding error.
Similarly, using Q(x, s0 ), the approximation V (x, s0 ) is linear and
matches the exact linear velocity v (x, s0 ) to rounding error.
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Further Time-Steps
Under invariant time-stepping, any consistent time-stepping scheme
has a truncation error of zero under a quartic similarity solution. This
results in the error in x being zero to within rounding error at the
next time step s1 = s0 + ∆s.
The error in the solution U(x, s0 ) recovered from the algebraic
equation MU = ρ is therefore zero to within rounding error.
The same arguments used for the first time step can be used to show
that all quantities are then exact to within rounding error for each
timestep.
We find that in practice we experience a buildup of rounding error in
the timestepping scheme.
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1
Introduction
2
The Finite Element Method
3
Numerical Results
4
Conclusions and Future Work
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Outline
We now present some numerical results generated using the Quartic Finite
Element method.
A mesh of 21 Nodes is used, with results generated in a time window of
s ∈ [1, 3] over 1 million time steps of size ∆s = 2 × 10−6 .
The similarity solution
2
x2
1
u (x, s) =
,
4− 2
120s
s +
S
is used to calculate the L2 error in the solution at each time step, as well
as setting the initial conditions.
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Solution Profiles
Figure: Profiles of the approximate solution U(x, s) over the time window
s ∈ [1, 3].
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Error in Solution
Figure: L2 Error in the Quartic Finite Element Method over the time window
s ∈ [1, 3]. Scale on y-axis is O(10−12 ).
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Conclusions
We have developed a moving mesh Finite Element method for the solution
of 4th Order nonlinear diffusion equations using a combination of quartic,
quadratic and linear basis functions.
Under initial conditions taken from the similarity solution, the method
calculates all quantities exactly to within rounding error for all time.
A Finite Difference method has also been developed for the same problem
which is exact at grid points to within rounding error. This is similar to a
method developed by Parker (2010) for a second order problem.
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Future Work
Work is currently ongoing on the effect of taking initial conditions which
do not correspond to the similarity solution, and the resulting L2 error in
the solution incurred by the method.
We are also looking to extend the method into 2D, using both Finite
Element and Finite Differences.
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Thank you for your attention!
For further reading, please see:
M.J. Baines, M.E. Hubbard, P.K. Jimack, and A.C. Jones. Scale-invariant
moving finite elements for nonlinear partial differential equations in two
dimensions. Applied Numerical Mathematics, 56:230–252, 2006.
C.J. Budd and M.D. Piggott. The geometric integration of scale-invariant
ordinary and partial differential equations. Journal of Computational
and Applied Mathematics, 128:399–422, 2001.
J. Parker. An invariant approach to moving-mesh methods for PDEs.
Master’s thesis, University of Oxford, 2010.
N.F. Smyth and J.M Hill. High-Order Nonlinear Diffusion. IMA Journal of
Applied Mathematics, 40:73–86, 1988.
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