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\noindent{\bf HW \#2. Math 269C (due on Monday, April 29)} 

%\noindent{\bf [1]} Show that if $w$ is continuous on $[0,1]$, and 
%$$\int_{0}^1wvdx=0,\mbox{ for all }v\in V,$$
%with 
%$V=\{v:[0,1]\rightarrow R,\ continuous,\ v(0)=v(1)=0,\ v' \ piecewise-continuous\ and\ bounded\}$, 
%then $w(x)=0$ for $x\in[0,1]$. 

\noindent{\bf [1]} Let $V$ be the space considered for the 1D model problem derived from: 
$$\mbox{(D)} \ \ \ \ \ -u''=f\mbox{ in }(0,1),\ \ u(0)=u(1)=0.$$
Construct a finite-dimensional subspace $V_h$ of $V$ consisting of 
functions which are quadratic on each subinterval $I_j$ of a partition of $I=(0,1)$. How can one choose the parameters to describe such functions ? Find the 
corresponding basis functions. Then formulate a finite element method for 
$(D)$ using the space $V_h$ and write down the corresponding linear system of 
equations in the case of a uniform partition. 

\noindent{\bf [2]} Consider the BVP 
$$\frac{d^4u}{dx^4}=f,\ \ 0<x<1, \ \ u(0)=u'(0)=u(1)=u'(1)=0.$$

\noindent{\bf (a)} Show that this problem can be given the following variational formulation: Find
$u\in W$ such that 
$$(u'',v'')=(f,v), \ \ \forall v\in W,$$
where $W=H^2_0(0,1)$ (thus functions in $H^2(0,1)$ with $v(0)=v'(0)=v(1)=v'(1)=0$). 

\noindent{\bf (b)} For $I=[a,b]$ an interval, define 
$$P_3(I)=\{v:\ v \mbox{ is a polynomial of degree } \leq 3 \mbox{ on } I\},$$
i.e. $v$ has the form $v(x)=a_3x^3+a_2x^2+a_1x+a_0$ for $x\in I$, and $a_i\in R$. 

Show that $v\in P_3(I)$ is uniquely determined by the values $v(a)$, $v'(a)$, 
$v(b)$, $v'(b)$. Find the corresponding (local) basis functions (the basis function corresponding to the value $v(a)$ is the cubic polynomial $v$ such that 
$v(a)=1$, $v'(a)=0$, $v(b)=0$, $v'(b)=0$, etc.) 

\noindent{\bf (c)} Starting from (b) construct a finite-dimensional subspace $W_h$ of $W$ 
consisting of piecewise-cubic functions. Specify suitable parameters to describe the functions in $W_h$ and determine the corresponding (global) basis functions. 

\noindent{\bf (d)} Formulate a FEM for the problem based on the space $W_h$. Find the 
corresponding linear system of equations in the case of a uniform partition. 

$\bullet$ Note: for the question ``Find the corresponding basis functions.'' in problems [1] and [2], you can use some software that can evaluate the integrals (since the evaluation by hand would be tedious). Or it is fine to leave out this part of the question, or just give the formulas without completely evaluating the integrals. 

\noindent{\bf [3]} This question corresponds to the 2D model problem. Find the linear basis functions for the triangle $K$ with vertices at 
$(0,0)$, $(h,0)$ and $(0,h)$. Show that the corresponding element stiffness
matrix is given by 
$$
\left[
\begin{array}{ccc}
1 & -\frac{1}{2} & -\frac{1}{2} \\
-\frac{1}{2} & \frac{1}{2} & 0\\
-\frac{1}{2} & 0 & \frac{1}{2}
\end{array}
\right]
$$
Using this result, show that the linear system (1.25) of Example 1.1. (textbook) has the
stated form (pages 30-31) (there is a typo in the textbook regarding the matrix). 

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