The <piecewise-linear hat functions> have disjoint supports unless their nodes coincide or are neighbours. Direct integration on the two adjacent intervals gives
$$
\int\varphi_i'^2=\frac2h,\quad
\int\varphi_i^2=\frac{2h}3,\quad
\int\varphi_i'\varphi_{i+1}'=-\frac1h,\quad
\int\varphi_i\varphi_{i+1}=\frac h6.
$$
Since $x=x_i+(x-x_i)$ and the hat is symmetric about $x_i=ih$, its load is $F_i=x_i\int\varphi_i=x_i h=ih^2$. Thus the <reaction-diffusion finite element matrix> gives the explicit equations
$$
\boxed{\left(\frac2h+\frac{2h}3\right)c_i
+\left(-\frac1h+\frac h6\right)(c_{i-1}+c_{i+1})=ih^2,
\quad 1\leq i\leq M,\quad c_0=c_{M+1}=0.}
$$
The coefficient matrix is a <tridiagonal matrix> and a <positive-definite matrix> by the energy identity in part (b), hence \b[nonsingular]. A separate spectral verification uses the sine basis with $\theta_j=j\pi/(M+1)$ and gives
$$
\lambda_j=\frac{2(1-\cos\theta_j)}h
+\frac h3(2+\cos\theta_j)>0.
$$
Both terms are positive for the Dirichlet modes. This also identifies precisely how the diffusion <stiffness matrix> and reaction <mass matrix> combine.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 341 / 5 / c / Solution

{title2=$K=S+M$}

For $-u''+u=f$ with homogeneous <Dirichlet boundary conditions>, the conforming <piecewise-linear hat functions> on a uniform grid give a diffusion <stiffness matrix> $S$ and a <mass matrix> $M$. Their nonzero entries are $S_{ii}=2/h$, $S_{i,i+1}=-1/h$, $M_{ii}=2h/3$, and $M_{i,i+1}=h/6$, with symmetric counterparts. The sum is a <positive-definite matrix> because its coefficient <quadratic form> is $\int((u_h')^2+u_h^2)$. For forcing $f(x)=x$, the load at $x_i$ is $hx_i$.

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