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 $v(\pm1)=v'(\pm1)=0$. Let $\phi_1,\ldots,\phi_n$ be its nodal value-and-slope basis. For $u_n=\sum_ka_k\phi_k$, the <Ritz method> imposes
$$
a(u_n,\phi_j)=\int_{-1}^1f\phi_jdx
\qquad(1\leq j\leq n).
$$
Hence the coefficient vector solves
$$
Ka=F,
\qquad
K_{jk}=\int_{-1}^1
\left(p\phi_k''\phi_j''+q\phi_k'\phi_j'+r\phi_k\phi_j\right)dx,
\qquad
F_j=\int_{-1}^1f\phi_jdx.
$$
The stiffness matrix is symmetric positive definite by part a.

Solution

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

{c}
{wiki=Hermite_interpolation}

A cubic Hermite finite element is piecewise cubic and joins with continuous first derivative. It is therefore conforming for fourth-order variational problems whose energy space lies in $H^2$.


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