Past exam of the approximation theory and finite element methods course of the University of Oxford / 2008 / A / 2 / d

Exactness on constants and linear functions gives

a_{1} + a_{2} = \frac{1}{3}, a_{1} x_{1} + a_{2} x_{2} = \frac{1}{4} .

With the nodes just computed, the solution is

a_{1} = \frac{8 - 10}{48}, a_{2} = \frac{8 + 10}{48} .

To prove cubic exactness rather than merely moment matching, divide a polynomial

u

of degree at most three by the monic

ϕ_{2}

u = q_{1} ϕ_{2} + r_{1}, de g q_{1}, de g r_{1} \leq 1.

The weighted integral of

q_{1} ϕ_{2}

vanishes by orthogonality, and its two nodal values vanish because the nodes are roots of

ϕ_{2}

. The remaining

r_{1}

is integrated exactly by the two equations above. This proves Gaussian quadrature exactness through degree three.

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