Exercises

Problem 1: Linear algebra refresher

Look up the topic of vector space in your favorite linear algebra book or search for the term at Wikipedia.

a) Prove that vectors in the plane $(a,b)$ form a vector space by showing that all the axioms of a vector space are satisfied.

b) Prove that all linear functions of the form $ax+b$ constitute a vector space, $a,b\in\Real$.

c) Show that all quadratic functions of the form $1 + ax^2 + bx$ do not constitute a vector space.

d) Check out the topic of inner product spaces. Suggest a possible inner product for the space of all linear functions of the form $ax+b$, $a,b\in\Real$, defined on some interval $\Omega=[A,B]$. Show that this particular inner product satisfies the general requirements of an inner product in a vector space.

Filename: linalg1.

Problem 2: Approximate a three-dimensional vector in a plane

Given $\f = (1,1,1)$ in $\Real^3$, find the best approximation vector $\u$ in the plane spanned by the unit vectors $(1,0)$ and $(0,1)$. Repeat the calculations using the vectors $(2,1)$ and $(1,2)$.

Filename: vec111_approx.

Problem 3: Approximate a parabola by a sine

Given the function $f(x)=1 + 2x(1-x)$ on $\Omega=[0,1]$, we want to find an approximation in the function space $$V = \hbox{span}\{1, \sin(\pi x)\}\tp$$

a) Sketch or plot $f(x)$. Think intuitively how an expansion in terms of the basis functions of $V$, $\baspsi_0(x)=1$, $\baspsi_1(x)=\sin(\pi x)$, can be construction to yield a best approximation to $f$. Or phrased differently, see if you can guess the coefficients $c_0$ and $c_1$ in the expansion $$u(x) = c_0\baspsi_0 + c_1\baspsi_1 = c_0 + c_1\sin (\pi x)\tp$$ Compute the $L^2$ error $||f-u||_{L^2}=(\int_0^1(f-u)^2\dx)^{1/2}$.

Hint.

b) Perform the hand calculations for a least squares approximation.

Filename: parabola_sin.

Problem 4: Approximate the exponential function by power functions

Let $V$ be a function space with basis functions $x^i$, $i=0,1,\ldots,N$. Find the best approximation to $f(x)=\exp(-x)$ on $\Omega =[0,8]$ among all functions in $V$ for $N=2,4,6$. Illustrate the three approximations in three separate plots.

Hint.

Filename: exp_powers.

Problem 5: Approximate the sine function by power functions

In this exercise we want to approximate the sine function by polynomials of order $N+1$. Consider two bases: $$\begin{align*} V_1 &= \{x, x^3, x^5, \ldots, x^{N-2}, x^N \}, \\ V_2 &= \{1,x,x^2,x^3,\ldots, x^N\}\tp \end{align*}$$ The basis $V_1$ is motivated by the fact that the Taylor polynomial approximation to the sine function has only odd powers, while $V_2$ is motivated by the assumption that also the even powers could improve the approximation in a least-squares setting.

Compute the best approximation to $f(x)=\sin(x)$ among all functions in $V_1$ and $V_2$ on two domains of increasing sizes: $\Omega_{1,k} = [0, k\pi]$, $k=2,3\ldots,6$ and $\Omega_{2,k} = [-k\pi /2, k\pi/2]$, $k=2,3,4,5$. Make plots for all combinations of $V_1$, $V_2$, $\Omega_1$, $\Omega_2$, $k=2,3,\ldots,6$.

Add a plot of the $N$-th degree Taylor polynomial approximation of $\sin(x)$ around $x=0$.

Hint.

Filename: sin_powers.

Problem 6: Approximate a steep function by sines

Find the best approximation of $f(x) = \tanh (s(x-\pi))$ on $[0, 2\pi]$ in the space $V$ with basis $\baspsi_i(x) = \sin((2i+1)x)$, $i\in\If = \{0,\ldots,N\}$. Make a movie showing how $u=\sum_{j\in\If}c_j\baspsi_j(x)$ approximates $f(x)$ as $N$ grows. Choose $s$ such that $f$ is steep ($s=20$ is appropriate).

Hint 1.

Hint 2.

Filename: tanh_sines.

Remarks

Approximation of a discontinuous (or steep) $f(x)$ by sines, results in slow convergence and oscillatory behavior of the approximation close to the abrupt changes in $f$. This is known as the Gibb's phenomenon.

Problem 7: Approximate a steep function by sines with boundary adjustment

We study the same approximation problem as in Problem 6: Approximate a steep function by sines. Since $\baspsi_i(0)=\baspsi_i(2\pi)=0$ for all $i$, $u=0$ at the boundary points $x=0$ and $x=2\pi$, while $f(0)=-1$ and $f(2\pi)=1$. This discrepancy at the boundary can be removed by adding a boundary function $B(x)$: $$u(x) = B(x) + \sum_{j\in\If} c_j\baspsi_j(x),$$ where $B(x)$ has the right boundary values: $B(x_L)=f(x_L)$ and $B(x_R)=f(x_R)$, with $x_L=0$ and $x_R=2\pi$ as the boundary points. A linear choice of $B(x)$ is $$B(x) = \frac{(x_R-x)f(x_L) + (x-x_L)f(x_R)}{x_R-x_L}\tp$$

a) Use the basis $\baspsi_i(x) = \sin((i+1)x)$, $i\in\If = \{0,\ldots,N\}$ and plot $u$ and $f$ for $N=16$. (It suffices to make plots for even $i$.)

b) Use the basis from Problem 6: Approximate a steep function by sines, $\baspsi_i(x) = \sin((2i+1)x)$, $i\in\If = \{0,\ldots,N\}$. (It suffices to make plots for even $i$.) Observe that the approximation converges to a piecewise linear function!

c) Use the basis $\baspsi_i(x) = \sin(2(i+1)x)$, $i\in\If = \{0,\ldots,N\}$, and observe that the approximation converges to a piecewise constant function.

Filename: tanh_sines_boundary_term.

Remarks

The strange results in b) and c) are due to the choice of basis. In b), $\basphi_i(x)$ is an odd function around $x=\pi/2$ and $x=3\pi/2$. No combination of basis functions is able to approximate the flat regions of $f$. All basis functions in c) are even around $x=\pi/2$ and $x=3\pi/2$, but odd at $x=0,\pi,2\pi$. With all the sines represented, as in a), the approximation is not constrained with a particular symmetry behavior.

Exercise 8: Fourier series as a least squares approximation

a) Given a function $f(x)$ on an interval $[0,L]$, look up the formula for the coefficients $a_j$ and $b_j$ in the Fourier series of $f$: $$\begin{equation*} f(x) = \frac{1}{2}a_0 + \sum_{j=1}^\infty a_j\cos \left(j\frac{2\pi x}{L}\right) + \sum_{j=1}^\infty b_j\sin \left(j\frac{2\pi x}{L}\right)\tp \end{equation*}$$

b) Let an infinite-dimensional vector space $V$ have the basis functions $\cos j\frac{2\pi x}{L}$ for $j=0,1,\dots,\infty$ and $\sin j\frac{2\pi x}{L}$ for $j=1,\dots,\infty$. Show that the least squares approximation method from the section Approximation of functions leads to a linear system whose solution coincides with the standard formulas for the coefficients in a Fourier series of $f(x)$ (see also the section Fourier series).

Hint.

c) Choose $f(x) = H(x-\half)$ on $\Omega=[0,1]$, where $H$ is the Heaviside function: $H(x)=0$ for $x < 0$, $H(x)=1$ for $x>0$ and $H(0)=\half$. Find the coefficients $a_j$ and $b_j$ in the Fourier series for $f(x)$. Plot the sum for $j=2N+1$, where $N=5$ and $N=100$.

Filename: Fourier_ls.

Problem 9: Approximate a steep function by Lagrange polynomials

Use interpolation with uniformly distributed points and Chebychev nodes to approximate $$\begin{equation*} f(x) = -\tanh(s(x-\half)),\quad x\in [0,1],\end{equation*}$$ by Lagrange polynomials for $s=5$ and $s=20$, and $N=3,7,11,15$. Combine $2\times 2$ plots of the approximation for the four $N$ values, and create such figures for the four combinations of $s$ values and point types.

Filename: tanh_Lagrange.

Problem 10: Approximate a steep function by Lagrange polynomials and regression

Redo Problem 9: Approximate a steep function by Lagrange polynomials, but apply a regression method with $N$-degree Lagrange polynomials and $2N+1$ data points. Recall that Problem 9: Approximate a steep function by Lagrange polynomials applies $N+1$ points and the resulting approximation interpolates $f$ at these points, while a regression method with more points does not interpolate $f$ at the data points. Do more points and a regression method help reduce the oscillatory behavior of Lagrange polynomial approximations?

Filename: tanh_Lagrange_regression.

Problem 11: Define nodes and elements

Consider a domain $\Omega =[0,2]$ divided into the three elements $[0,1]$, $[1,1.2]$, and $[1.2,2]$.

For P1 and P2 elements, set up the list of coordinates and nodes (nodes) and the numbers of the nodes that belong to each element (elements) in two cases: 1) nodes and elements numbered from left to right, and 2) nodes and elements numbered from right to left.

Filename: fe_numberings1.

Problem 12: Define vertices, cells, and dof maps

Repeat Problem 11: Define nodes and elements, but define the data structures vertices, cells, and dof_map instead of nodes and elements.

Filename: fe_numberings2.

Problem 13: Construct matrix sparsity patterns

Problem 11: Define nodes and elements describes a element mesh with a total of five elements, but with two different element and node orderings. For each of the two orderings, make a $5\times 5$ matrix and fill in the entries that will be nonzero.

Hint.

Filename: fe_sparsity_pattern.

Problem 14: Perform symbolic finite element computations

Perform symbolic calculations to find formulas for the coefficient matrix and right-hand side when approximating $f(x) = \sin (x)$ on $\Omega=[0, \pi]$ by two P1 elements of size $\pi/2$. Solve the system and compare $u(\pi/2)$ with the exact value 1.

Filename: fe_sin_P1.

Problem 15: Approximate a steep function by P1 and P2 elements

Given $$\begin{equation*} f(x) = \tanh(s(x-\half))\end{equation*}$$ use the Galerkin or least squares method with finite elements to find an approximate function $u(x)$. Choose $s=20$ and try $N_e=4,8,16$ P1 elements and $N_e=2,4,8$ P2 elements. Integrate $f\basphi_i$ numerically.

Hint.

Filename: fe_tanh_P1P2.

Problem 16: Approximate a steep function by P3 and P4 elements

a) Solve Problem 15: Approximate a steep function by P1 and P2 elements using $N_e=1,2,4$ P3 and P4 elements.

b) How will an interpolation method work in this case with the same number of nodes?

Filename: fe_tanh_P3P4.

Exercise 17: Investigate the approximation error in finite elements

The theory (101) from the section Computing the error of the approximation predicts that the error in the Pd approximation of a function should behave as $h^{d+1}$, where $h$ is the length of the element. Use experiments to verify this asymptotic behavior (i.e., for small enough $h$). Choose three examples: $f(x)=Ae^{-\omega x}$ on $[0,3/\omega]$, $f(x) = A\sin (\omega x)$ on $\Omega=[0, 2\pi/\omega]$ for constant $A$ and $\omega$, and $f(x)=\sqrt{x}$ on $[0,1]$.

Hint 1.

Hint 2.

xc, u = u_glob(c, vertices, cells, dof_map)
e = f_func(xc) - u
L2_error = 0
e2 = e**2
for i in range(len(xc)-1):
    L2_error += 0.5*(e2[i+1] + e2[i])*(xc[i+1] - xc[i])
L2_error = np.sqrt(L2_error)

Filename: Pd_approx_error.

Problem 18: Approximate a step function by finite elements

Approximate the step function $$\begin{equation*} f(x) = \left\lbrace\begin{array}{ll} 0 & 0\leq x < \halfi,\\ 1 & \halfi \leq x \geq \halfi \end{array}\right. \end{equation*}$$ by 2, 4, 8, and 16 P1, P2, P3, and P4. Compare approximations visually.

Hint.

f = sym.Heaviside(x - sym.Rational(1,2))

Filename: fe_Heaviside_P1P2.

Exercise 19: 2D approximation with orthogonal functions

a) Assume we have basis functions $\basphi_i(x,y)$ in 2D that are orthogonal such that $(\basphi_i,\basphi_j)=0$ when $i\neq j$. The function least_squares in the file approx2D.py will then spend much time on computing off-diagonal terms in the coefficient matrix that we know are zero. To speed up the computations, make a version least_squares_orth that utilizes the orthogonality among the basis functions.

b) Apply the function to approximate $$f(x,y) = x(1-x)y(1-y)e^{-x-y}$$ on $\Omega = [0,1]\times [0,1]$ via basis functions $$\basphi_i(x,y) = \sin ((p+1)\pi x)\sin((q+1)\pi y),\quad i=q(N_x+1) + p,$$ where $p=0,\ldots,N_x$ and $q=0,\ldots,N_y$.

Hint.

c) Make a unit test for the least_squares_orth function.

Filename: approx2D_ls_orth.

Exercise 20: Use the Trapezoidal rule and P1 elements

Consider approximation of some $f(x)$ on an interval $\Omega$ using the least squares or Galerkin methods with P1 elements. Derive the element matrix and vector using the Trapezoidal rule (109) for calculating integrals on the reference element. Assemble the contributions, assuming a uniform cell partitioning, and show that the resulting linear system has the form $c_i=f(\xno{i})$ for $i\in\If$.

Filename: fe_P1_trapez.

Exercise 21: Compare P1 elements and interpolation

We shall approximate the function $$f(x) = 1 + \epsilon\sin (2\pi nx),\quad x\in \Omega = [0,1],$$ where $n\in\Integer$ and $\epsilon \geq 0$.

a) Plot $f(x)$ for $n=1,2,3$ and find the wave length of the function.

b) We want to use $N_P$ elements per wave length. Show that the number of elements is then $nN_P$.

c) The critical quantity for accuracy is the number of elements per wave length, not the element size in itself. It therefore suffices to study an $f$ with just one wave length in $\Omega = [0,1]$. Set $\epsilon = 0.5$.

Run the least squares or projection/Galerkin method for $N_P=2,4,8,16,32$. Compute the error $E=||u-f||_{L^2}$.

Hint.

d) Repeat the set of experiments in the above point, but use interpolation/collocation based on the node points to compute $u(x)$ (recall that $c_i$ is now simply $f(\xno{i})$). Compute the error $E=||u-f||_{L^2}$. Which method seems to be most accurate?

Filename: fe_P1_vs_interp.

Exercise 22: Implement 3D computations with global basis functions

Extend the approx2D.py code to 3D applying ideas from the section Extension to 3D. Construct some 3D problem to make a test function for the implementation.

Hint.

Filename: approx3D.

Exercise 23: Use Simpson's rule and P2 elements

Redo Exercise 20: Use the Trapezoidal rule and P1 elements, but use P2 elements and Simpson's rule based on sampling the integrands at the nodes in the reference cell.

Filename: fe_P2_simpson.

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