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269C, Spring, Vese 

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{\bf Computational Project 2}

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Tentative deadline: Monday, June 3rd.  

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\noindent {\bf (I)} Consider the polynomial function on the unit square defined by 
$$u(x,y)=2^{4p}x^p(1-x)^py^p(1-y)^p,\ \ \ (x,y)\in [0,1]\times[0,1],$$ 
where $p$ is a positive integer. 

(a) Check that $u=0$ on $\partial\Omega$. 

(b) Compute $-\triangle u(x,y)+ u(x,y)$ and denote the result by $f(x,y)$. 

(c) Consider the equation on the unit square $\Omega=(0,1)\times(0,1)$: 
$$-\triangle u +u = f \mbox{ in } \Omega, \ \ \ u=0 \mbox{ on }\partial\Omega,$$ 
and use $P_1$ elements to approximate its solution. You can compare the numerical solution with the exact analytic solution (you can choose $p\leq 10$ or around 10; the accuracy of the approximation will depend on the choice of $p$). 

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\noindent{\bf (II)} Use $P_1$ elements to approximate the solution of 
$$-\triangle u +u=\sin{(2\pi(x+y))},\ (x,y)\in \Omega=\mbox{unit square}$$
with the following boundary conditions: 

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Case (a) $u=0$ for $(x,y)\in \partial \Omega$

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Case (b) $u=0$ for $(x,y)\in \partial \Omega$, $x=0,1$ 

\ \ \ \ \ \ \ \ \ \ \ \ $u_y=0$, for $(x,y)\in \partial \Omega$, $y=0,1$. 

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\noindent{\bf Remarks}

You can base the triangulations on a 10x10 grid. 

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$\bullet$ What you should turn in: the weak formulations, the linear systems, details 
about the discretizations, plots of the results, your computer program. 

$\bullet$ Section 12.2 of the textbook discusses numerical integration or quadrature
formulas, helpful to discretize the load vector, if needed.   

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