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

Table of contents
- Solution 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^{'} (\pm 1) = 0

. Let

ϕ_{1}, \dots, ϕ_{n}

be its nodal value-and-slope basis. For

u_{n} = \sum_{k} a_{k} ϕ_{k}

, the Ritz method imposes

a (u_{n}, ϕ_{j}) = \int_{- 1}^{1} f ϕ_{j} d x (1 \leq j \leq n) .

Hence the coefficient vector solves

K a = F, K_{jk} = \int_{- 1}^{1} (p ϕ_{k}^{''} ϕ_{j}^{''} + q ϕ_{k}^{'} ϕ_{j}^{'} + r ϕ_{k} ϕ_{j}) d x, F_{j} = \int_{- 1}^{1} f ϕ_{j} d x .

The stiffness matrix is symmetric positive definite by part a.

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