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

Let $x_m=mh$ be a uniform periodic mesh, with indices interpreted modulo the number of nodes. Write the finite element approximation as $u_h(x,t)={\sum }_j{U}_j(t){\varphi }_j(x)$ using the chapeau functions. The Galerkin method requiresThe mass matrix has $M_{mm}=2h/3$ and $M_{m,m\pm 1}=h/6$. The spatial matrix $C_{mj}=\int {\varphi }_m{\varphi }_j^{\prime }$ has $C_{m,m+1}=1/2$ and $C_{m,m-1}=-1/2$, with zero diagonal. Therefore the semidiscrete equations areor equivalently $M\dot{U}=CU$. The coefficients come from integrating the products of the two overlapping piecewise-linear hat functions on each interval; periodicity supplies the wraparound entries.

The mass matrix is a positive-definite matrix, and integration by parts with periodic boundary conditions gives $C^T=-C$. HenceThis is energy conservation for semidiscrete Galerkin advection: $U^{T}MU=\Vert u_h{\Vert }_{L^2(0,1)}^2$. Thus the semidiscretization is 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 $h$. ConsequentlyThe 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.