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{\bf HW \#1. Math 269C (due on Friday, April 17)} 

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{\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 $x\in[0,1]$. 

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{\bf [2]} Construct a finite-dimensional subspace $V_h$ of $V$ (from problem [1]) 
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. 
Recall that $(D)$ is 
$$-u''=f\mbox{ in }(0,1),\ \ u(0)=u(1)=0.$$

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{\bf [3]} Consider the BVP 
$$\frac{d^4u}{dx^4}=f,\ \ 0<x<1, \ \ u(0)=u'(0)=u(1)=u'(1)=0.$$

(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 $W=\{v: v \mbox{ and } v' \mbox{ are continuous on }[0,1],\ v''$ is
piecewise-continuous and $v(0)=v'(0)=v(1)=v'(1)=0\}$. 

(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 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.) 


(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 basis functions. 

(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. 


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