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

Table of contents
- Solution b

Solution

Let

{ϕ_{j}}_{j = 1}^{N - 1}

be the piecewise-linear hat functions on a mesh, and put

u_{h} = \sum_{j = 1}^{N - 1} U_{j} ϕ_{j} .

The Ritz method requires

\sum_{j = 1}^{N - 1} K_{ij} U_{j} = F_{i}, i = 1, \dots, N - 1,

where, for

f \equiv 1

,

K_{ij} = \int_{0}^{1} [(1 + x^{2}) ϕ_{j}^{'} ϕ_{i}^{'} + x^{2} ϕ_{j} ϕ_{i}] d x, F_{i} = \int_{0}^{1} ϕ_{i} d x .

Local support makes

K

symmetric tridiagonal, and coercivity makes it positive definite.

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