Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 341 / 5

Write the operator in Sturm-Liouville form:The natural solution space is the Sobolev space $H_0^1(0,1)$. Multiplication by a test function $v\in H_0^1(0,1)$ and integration by parts gives the symmetric bilinear formIt is bounded andby the Poincare inequality. Thus $L$ is positive definite and $a$ is coercive. The Lax-Milgram theorem gives a unique weak solution of the variational problem

Let $\{{\varphi }_j{\}}_{j=1}^{N-1}$ be the piecewise-linear hat functions on a mesh, and putThe Ritz method requireswhere, for $f\equiv 1$,Local support makes $K$ symmetric tridiagonal, and coercivity makes it positive definite.