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Original solutions to past examinations of the finite element methods for partial differential equations course of the University of Oxford.

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: 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 $\Vert p{\Vert }_1^2=\Vert p{\Vert }_0^2+\Vert \nabla p{\Vert }_0^2$ 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.