explain fundamental concepts in mathematical modelling with partial differential equation, and fundamental properties for elliptic, parabolic and hyperbolic equations;
formulate and with a computer solve second order elliptic boundary value problems in one and two spatial dimension using the finite element method.
derive error bounds for elliptic equations in one and two spatial dimensions, and be able to use these error bounds to construct adaptive algorithms for local mesh refinement.
solve parabolic and hyperbolic partial differential equations using the finite element method in space and finite differences in time, and to compare different time stepping algorithms and choose appropriate algorithms for the problem at hand.
use finite element software to solve more complicated problems, such as coupled systems of equations.
evaluate different techniques for solving problems and be able to motivate when to use existing software and when to write new code.
The content in the course is built up around a design problem, to solve a coupled physics model. This includes modifing the geometry and ensuring that the accuracy is at least as good as prescribed accuracy. Classes of problems covered in the course are elliptic boundary value problems, hyperbolic and parabolic time dependent problems. The Finite element tools used are surface representation in CAD systems, discrete finite element spaces in 1D and 2D, piecewise polynomial approximation (interpolation and projection, quadratures), mesh generation (triangulation in 1D and 2D, local mesh refinement, Delaunay and Voronoi). Variational formulation, Galerkin FEM (Finite Element Method) in 1D and 2D including time-dependent problems, standard stability estimates, a priori and a posteriori error estimates in 1D for elliptic problems, a priori for 2D for elliptic problems).
Lectures, computer labs and assignments.
Written exam (3hp) and assignments (2hp).
week 30, 2015
Larsson, M. G.;
A first course in finite elements: lecture notes