- Lecturer: Patrick Farrell
General Prerequisites:
The course assumes knowledge of vector calculus (divergence theorem) and some elementary knowledge of partial differential equations. Part A Numerical Analysis is useful but is not necessary; students without a background in e.g. interpolation and quadrature should expect to undertake some preliminary background reading. Material on Hilbert and Sobolev spaces will be introduced in the course as needed.
Course Term: Hilary
Course Lecture Information: 16 lectures
Course Weight: 1
Course Level: M
Assessment Type: Written Examination
Course Overview:
The finite element method gives a systematic way to approximate the numerical solution of boundary value problems involving differential equations. It underpins the design of aircraft and spacecraft, the design of bridges and skyscrapers, the development of advanced materials, the prediction of weather and climate, and much more besides. This course gives a mathematical introduction to the finite element method and its error analysis.
Learning Outcomes:
Course Synopsis:
Introduction to Lebesgue and Sobolev spaces. The Lax-Milgram theorem and Céa's Lemma. Fréchet differentiation and Euler-Lagrange equations. Galerkin approximation and function spaces constructed via finite elements. The Poisson, linear elasticity, and biharmonic equations. Local and global assembly. Interpolation error estimates. The Newton-Kantorovich iteration for nonlinear PDEs. Babuška's theorem. Brezzi's theorem. Mixed finite element methods for the Stokes and mixed Poisson equations.