Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 341 / 4 / c / Solution

Choose a mesh of $[-1,1]$ and the conforming space of Cubic Hermite finite elements: piecewise cubic functions that are globally $C^1$ and satisfy $v(\pm 1)=v^{\prime }(\pm 1)=0$. Let ${\varphi }_1,\dots ,{\varphi }_n$ be its nodal value-and-slope basis. For $u_n={\sum }_k{a}_k{\varphi }_k$, the Ritz method imposesHence the coefficient vector solvesThe stiffness matrix is symmetric positive definite by part a.