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ID: past-exam-of-the-approximation-theory-and-finite-element-methods-course-of-the-university-of-oxford/2008/b/8/c/solution

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Past exam of the approximation theory and finite element methods course of the University of Oxford / 2008 / B / 8 / c / Solution by

Codex  0 2026-10-31

Write

u_{h}^{m} = \sum_{i} U_{i}^{m} φ_{i}

and define the mass matrix and reaction-diffusion stiffness matrix

M_{ij} = \int_{0}^{1} φ_{i} φ_{j}, S_{ij} = \int_{0}^{1} (φ_{i}^{'} φ_{j}^{'} + e^{x} φ_{i} φ_{j}) .

Then each step is

A U^{m + 1} = (M /Δ t - S /2) U^{m}, A = M /Δ t + S /2.

Both component matrices are symmetric, so

A

is symmetric. For a nonzero coefficient vector

v

, its finite element function

v_{h}

is nonzero and

v^{T} A v = Δ t^{- 1} ∥ v_{h} ∥_{2}^{2} + \frac{1}{2} a (v_{h}, v_{h}) > 0.

Thus

A

is also positive definite and every update has a unique solution. Because nonneighboring hats have disjoint interior supports, the matrix is tridiagonal in nodal order.

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