The <mass matrix> is a <positive-definite matrix>, and <integration by parts> with periodic boundary conditions gives $C^T=-C$. Hence
$$
\frac{d}{dt}(U^TMU)=2U^TCU=0.
$$
This is <energy conservation for semidiscrete Galerkin advection>: $U^TMU=\|u_h\|_{L^2(0,1)}^2$. Thus the semidiscretization is \b[stable and conserves its finite element <L2 norm>].

For a mesh-independent comparison with the grid norm, the <Fourier symbol> of $M$ is $h(2+\cos\theta)/3$, between $h/3$ and $h$. Consequently
$$
\frac h3\sum_m|U_m|^2\leq U^*MU\leq h\sum_m|U_m|^2.
$$
The conserved energy therefore gives a uniform bound in $\sqrt{h\sum_m|U_m|^2}$ as well. Equivalently, each <Fourier mode> evolves with the purely imaginary exponent $\sigma(\theta)=3i\sin\theta/[h(2+\cos\theta)]$. This concerns the semidiscrete method; a chosen time integrator must separately control those imaginary <eigenvalues>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 341 / 5 / b / Solution

For periodic $u_t=u_x$, the conforming <Galerkin method> has $M\dot U=CU$, with <mass matrix> $M_{ij}=\int\phi_i\phi_j$ and $C_{ij}=\int\phi_i\phi_j'$. <Integration by parts> and periodicity give $C^T=-C$. Therefore
$$
\frac{d}{dt}(U^TMU)=2U^TCU=0.
$$
The conserved quantity is the squared <L2 norm> of the <finite element> function. On a uniform one-dimensional mesh of spacing $h$, the <piecewise-linear hat functions> give $M_{ii}=2h/3$, $M_{i,i\pm1}=h/6$, $C_{i,i+1}=1/2$ and $C_{i,i-1}=-1/2$.

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 Incoming links: Energy conservation for semidiscrete Galerkin advection