Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 40E / b

Let $A_h$ be the symmetric Dirichlet second-difference matrix. Its eigenvalues are nonpositive, so every eigenvalue ofhas modulus at most one. Thus $\Vert R_h{\Vert }_{2,h}\le 1$.

Insert the exact smooth solution into the scheme. The local truncation error per step has discrete norm $O(k^3+k{h}^2)$, from the order-three trapezoidal time residual and the order-two centred spatial residual. If $e^n$ is the grid error, stability and iteration giveThere are at most $1/k$ steps and $e^0=0$, henceThis proves convergence for every $\mu =k/h^2>0$ without invoking the Lax equivalence theorem.