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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-341/4/c/solution