Classify each term in the following equations as linear or nonlinear. Assume that \( a \) and \( b \) are unknown numbers and that \( u \) and \( v \) are unknown functions. All other symbols are known quantities.
Consider a nonlinear vibration problem
$$ \begin{equation} mu'' + bu'|u'| + s(u) = F(t), \end{equation} $$ where \( m>0 \) is a constant, \( b\geq 0 \) is a constant, \( s(u) \) a possibly nonlinear function of \( u \), and \( F(t) \) is a prescribed function. Such models arise from Newton's second law of motion in mechanical vibration problems where \( s(u) \) is a spring or restoring force, \( mu'' \) is mass times acceleration, and \( bu'|u'| \) models water or air drag.
Approximate \( u'' \) by a centered finite difference \( D_tD_t u \), and use a centered difference \( D_t u \) for \( u' \) as well. Observe then that \( s(u) \) does not contribute to making the resulting algebraic equation at a time level nonlinear. Use a geometric mean to linearize the quadratic nonlinearity arising from the term \( bu'|u'| \).
Consider a typical nonlinear Laplace term like \( \nabla\cdot\dfc(u)\nabla u \) discretized by centered finite differences. Explain why the Jacobian corresponding to this term has the same sparsity pattern as the matrix associated with the corresponding linear term \( \dfc\nabla^2 u \).
Hint. Set up the unknowns that enter the difference stencil and find the sparsity of the Jacobian that arise from the stencil.
Filename: nonlin_sparsity_Jacobian.pdf.
Suppose we have a linear system \( F(u) = Au- b=0 \). Apply Newton's method
to this system, and show that the method converges in one iteration.
Filename: nonlin_Newton_linear.pdf.
The operator \( \nabla\cdot(\dfc(u)\nabla u) \) with \( \dfc(u) = ||\nabla u||^q \) appears in several physical problems, especially flow of Non-Newtonian fluids. In a Newton method one has to carry out the differentiation \( \partial\dfc(u)/\partial c_j \), for \( u=\sum_kc_k\baspsi_k \). Show that
$$ {\partial\over\partial u_j} ||\nabla u||^q =
q||\nabla u||^{q-2}\nabla u\cdot
\nabla\baspsi_j\tp $$
Filename: nonlin_differentiate.pdf.
We consider the problem
$$ \begin{equation} ((1 + u^2)u')' = 1,\quad x\in (0,1),\quad u(0)=u(1)=0\tp \tag{50} \end{equation} $$
a) Discretize (50) by a centered finite difference method on a uniform mesh.
b) Discretize (50) by a finite element method with P1 of equal length. Use the Trapezoidal method to compute all integrals. Set up the resulting matrix system.
Filename: nonlin_1D_coeff_discretize.pdf.
We have a two-point boundary value problem
$$ \begin{equation} ((1 + u^2)u')' = 1,\quad x\in (0,1),\quad u(0)=u(1)=0\tp \tag{51} \end{equation} $$
a) Construct a Picard iteration method for (51) without discretizing in space.
b) Apply Newton's method to (51) without discretizing in space.
c) Discretize (51) by a centered finite difference scheme. Construct a Picard method for the resulting system of nonlinear algebraic equations.
d) Discretize (51) by a centered finite difference scheme. Define the system of nonlinear algebraic equations, calculate the Jacobian, and set up Newton's method for solving the system.
Filename: nonlin_1D_coeff_linearize.pdf.
We address the so-called Bratu problem
$$ \begin{equation} u'' + \lambda e^u=0,\quad x\in (0,1),\quad u(0)=u(1)=0, \tag{52} \end{equation} $$ where \( \lambda \) is a given parameter and \( u \) is a function of \( x \). This is a widely used model problem for studying numerical methods for nonlinear differential equations. The problem (52) has an exact solution
$$ u(x) = -2\ln\left(\frac{\cosh((x-\half)\theta/2)}{\cosh(\theta/4)}\right),$$ where \( \theta \) solves
$$ \theta = \sqrt{2\lambda}\cosh(\theta/4)\tp$$ There are two solutions of (52) for \( 0<\lambda <\lambda_c \) and no solution for \( \lambda >\lambda_c \). For \( \lambda = \lambda_c \) there is one unique solution. The critical value \( \lambda_c \) solves
$$ 1 = \sqrt{2\lambda_c}\frac{1}{4}\sinh(\theta(\lambda_c)/4)\tp$$ A numerical value is \( \lambda_c = 3.513830719 \).
a) Discretize (52) by a centered finite difference method.
b) Set up the nonlinear equations \( F_i(u_0,u_1,\ldots,u_{N_x})=0 \) from a). Calculate the associated Jacobian.
Filename: nonlin_1D_Bratu_fd.pdf.
We shall investigate integrals on the form
$$
\begin{equation}
\int_0^L f(\sum_ku_k\basphi_k(x))\basphi_i(x)\dx,
\tag{53}
\end{equation}
$$
where \( \basphi_i(x) \) are P1 finite element basis functions and \( u_k \)
are unknown coefficients, more precisely the values of the unknown
function \( u \) at nodes \( \xno{k} \). We introduce a node numbering that
goes from left to right and also that all cells have
the same length \( h \). Given \( i \), the integral
only gets contributions from \( [\xno{i-1},\xno{i+1}] \). On this
interval \( \basphi_k(x)=0 \) for \( k
$$
\sum_k u_k\basphi_k(x) = u_{i-1}\basphi_{i-1}(x) +
u_{i}\basphi_{i}(x) + u_{i+1}\basphi_{i+1}(x)\tp
$$
The integral (53) now takes the
simplified form
$$
\int_{\xno{i-1}}^{\xno{i+1}}
f(u_{i-1}\basphi_{i-1}(x) +
u_{i}\basphi_{i}(x) + u_{i+1}\basphi_{i+1}(x))\basphi_i(x)\dx\tp
$$
Split this integral in two integrals over cell L (left),
\( [\xno{i-1},\xno{i}] \), and cell R (right), \( [\xno{i},\xno{i+1}] \). Over
cell L, \( u \) simplifies to \( u_{i-1}\basphi_{i-1} + u_{i}\basphi_{i} \)
(since \( \basphi_{i+1}=0 \) on this cell), and over cell R, \( u \)
simplifies to \( u_{i}\basphi_{i} + u_{i+1}\basphi_{i+1} \). Make a
Hint.
Introduce symbols
The formula is displayed by loading
Filename:
We address the same 1D Bratu problem as described in
Problem 8: Finite differences for the 1D Bratu problem.
a)
Discretize (Problem 10: Finite elements for the 1D Bratu problem) by a finite element
method using a uniform mesh with P1 elements. Use a group
finite element method for the \( e^u \) term.
b)
Set up the nonlinear equations \( F_i(u_0,u_1,\ldots,u_{N_x})=0 \)
from a). Calculate the associated Jacobian.
Filename:
We study the multi-dimensional heat conduction PDE
$$ \varrho c(T) T_t = \nabla\cdot (k(T)\nabla T)$$
in a spatial domain \( \Omega \), with a nonlinear Robin boundary condition
$$ -k(T)\frac{\partial T}{\partial n} = h(T)(T-T_s(t)),$$
at the boundary \( \partial\Omega \).
The primary unknown is the temperature \( T \), \( \varrho \) is the density
of the solid material, \( c(T) \) is the heat capacity, \( k(T) \) is
the heat conduction, \( h(T) \) is a heat transfer coefficient, and
\( T_s(T) \) is a possibly time-dependent temperature of the surroundings.
a)
Use a Backward Euler or Crank-Nicolson time discretization and
derive the variational form for the spatial problem to be solved
at each time level.
b)
Define a Picard iteration method from the variational form at
a time level.
c)
Derive expressions for the matrix and the right-hand side of the
equation system that arises from applying Newton's method to
the variational form at a time level.
d)
Apply the Backward Euler or Crank-Nicolson scheme in time first.
Derive a Newton method at the PDE level. Make a variational
form of the resulting PDE at a time level.
Filename:
Consider a 1D heat conduction PDE
$$ \varrho c(T) T_t = (k(T)T_x)_x,$$
where \( \varrho \) is the density of the solid material, \( c(T) \) is
the heat capacity, \( T \) is the temperature, and \( k(T) \) is the
heat conduction coefficient.
Use a uniform finite element mesh, P1 elements, and the group finite
element method to derive the algebraic equations arising from the
heat conduction PDE
a)
Discretize the PDE by a finite difference method. Use either a
Backward Euler or Crank-Nicolson scheme in time.
b)
Derive the matrix and right-hand side of a Newton method applied
to the discretized PDE.
Filename:
Flow of a pseudo-plastic power-law fluid between two flat plates can be
modeled by
$$ \frac{d}{dx}\left(\mu_0\left\vert\frac{du}{dx}\right\vert^{n-1}
\frac{du}{dx}\right) = -\beta,\quad u'(0)=0,\ u(H) = 0,$$
where \( \beta>0 \) and \( \mu_0>0 \) are constants.
A target value of \( n \) may be \( n=0.2 \).
a)
Formulate a Picard iteration method directly for the differential
equation problem.
b)
Perform a finite difference discretization of the problem in
each Picard iteration. Implement a solver that can compute \( u \)
on a mesh. Verify that the solver gives an exact solution for \( n=1 \)
on a uniform mesh regardless of the cell size.
c)
Given a sequence of decreasing \( n \) values, solve the problem for each
\( n \) using the solution for the previous \( n \) as initial guess for
the Picard iteration. This is called a continuation method.
Experiment with \( n=(1,0.6,0.2) \) and \( n=(1,0.9,0.8,\ldots,0.2) \)
and make a table of the number of Picard iterations versus \( n \).
d)
Derive a Newton method at the differential equation level and
discretize the resulting linear equations in each Newton iteration
with the finite difference method.
e)
Investigate if Newton's method has better convergence properties than
Picard iteration, both in combination with a continuation method.
sympy program that can compute the integral and write it out as a
difference equation. Give the \( f(u) \) formula on the command line.
Try out \( f(u)=u^2, \sin u, \exp u \).
u_i, u_im1, and u_ip1 for \( u_i \), \( u_{i-1} \),
and \( u_{i+1} \), respectively, and similar symbols for \( x_i \), \( x_{i-1} \),
and \( x_{i+1} \). Find formulas for the basis functions on each of the
two cells, make expressions for \( u \) on the two cells, integrate over
each cell, expand the answer and simplify. You can make
LaTeX code and render it via
http://latex.codecogs.com. Here are some appropriate Python statements
for the latter purpose:
from sympy import *
...
# expr_i holdes the integral as a sympy expression
latex_code = latex(expr_i, mode='plain')
# Replace u_im1 sympy symbol name by latex symbol u_{i-1}
latex_code = latex_code.replace('im1', '{i-1}')
# Replace u_ip1 sympy symbol name by latex symbol u_{i+1}
latex_code = latex_code.replace('ip1', '{i+1}')
# Escape (quote) latex_code so it can be sent as HTML text
import cgi
html_code = cgi.escape(latex_code)
# Make a file with HTML code for displaying the LaTeX formula
f = open('tmp.html', 'w')
# Include an image that can be clicked on to yield a new
# page with an interactive editor and display area where the
# formula can be further edited
text = """
<a href="http://www.codecogs.com/eqnedit.php?latex=%(html_code)s"
target="_blank">
<img src="http://latex.codecogs.com/gif.latex?%(html_code)s"
title="%(latex_code)s"/>
</a>
""" % vars()
f.write(text)
f.close()
tmp.html into a web browser.
fu_fem_int.py.
Problem 10: Finite elements for the 1D Bratu problem
nonlin_1D_Bratu_fe.pdf.
Problem 11: Derive the Newton system from a variational form
nonlin_heat_Newton.pdf.
Problem 12: Derive algebraic equations for nonlinear 1D heat conduction
nonlin_1D_heat_PDE.pdf.
Problem 13: Investigate a 1D problem with a continuation method
Bibliography