 2023 (source code)

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= 2023
{scope}
{title2=C6.4}
{title2=A12223W1}

Solutions to all three questions of the <finite element methods for partial differential equations course of the University of Oxford> Part C examination, sat on 6 June 2023. Source: https://www.whatdotheyknow.com/request/request_for_mathematics_examinat/response/3129786/attach/3/FOI%20202507%20740%20Data.pdf#page=325[original released PDF, bundle pages 325–328; paper pages 1–4]. The original numbering is retained; question statements are not reproduced.

All <function spaces> and <bilinear forms> below are real. Write $(p,q)_0=\int_\Omega pq$ for the <L2 inner product>, and $\|p\|_1^2=\|p\|_0^2+\|\nabla p\|_0^2$ for the first-order <Sobolev norm>; the notation applies componentwise to <vector fields>. We use the <Poincaré inequality> explicitly allowed by the paper. On an arbitrary unbounded <Lipschitz domain> that inequality would need an additional assumption. The convergence assertions use the usual full <conforming finite element spaces>, or <vector subspaces> containing the indicated <interpolants>; merely being a <vector subspace> of the stated <piecewise polynomial functions> is insufficient. Part 1(d) explains the intended meaning of positivity for its <nonsymmetric matrix>.

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