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\noindent{\bf Math 269C Spring. Instructor: L. Vese. HW \#1.} 

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\noindent{\bf [1]} Consider the two-point boundary value problem 
$$-u''(x)=x^2-3x+2\ \ \ \mbox{ for } x \in (0,1),$$ 
$$u(0)=u(1)=0.$$
Find the exact (unique) solution of the problem (by integrating twice).

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%\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 [2]} Let $V=H_{0}^1(0,1)$, $f\in L^2(0,1)$, and the 1D model problem: 
$$\mbox{(D)} \ \ \ \ \ -u''=f\mbox{ in }(0,1),\ \ u(0)=u(1)=0.$$

{\bf (a)} Construct a finite-dimensional subspace $V_h$ of $V$ consisting of continuous 
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 ? (see lecture notes or Section 3.2 from the textbook by Johnson). 

{\bf (b)} Find the 
corresponding basis functions. 

{\bf (c)} 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. 

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\noindent{\bf [3]} Consider the BVP 
$$\frac{d^4u}{dx^4}=f,\ \ 0<x<1,$$ \ \ 
$$u(0)=u'(0)=u(1)=u'(1)=0,$$ 
for $f\in L^2(0,1)$. 

\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 $W=H^2_0(0,1)$ (functions in $H^2(0,1)$ with $v(0)=v'(0)=v(1)=v'(1)=0$). Recall that $(\cdot,\cdot)$ denotes the $L^2(0,1)$-inner product. 

Hint: multiply the $4th-order$ equation by $v\in W$, integrate both sides over $(0,1)$, and then apply integration by parts twice.  

\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. 

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$\bullet$ Note: For problems [2] and [3], you can use some software that can help find the basis functions and evaluate the integrals needed for the matrix $A$ (since the evaluation by hand would be more tedious). Or you can leave out this part of the question, or just give the formulas for the basis functions and integrals without carrying out the calculations for obtaining their explicit values (see also Section 3.2 from the textbook by Johnson).

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