Entry requirements

120 credits of which at least 45 credits in mathematics, Computer Programming I and Scientific computing III or the equivalent.

Learning outcomes

To pass the course the student shall be able to

• 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 spatial dimension using the finite element method.
• formulate and with a computer solve second order elliptic boundary value problems in two spatial dimensions with Dirichlet, Neumann, and Robin boundary conditions, using the finite element method.
• derive a priori and a posteriori 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.

Content

Discrete function spaces in one and two spatial dimensions. Variational formulation of elliptic boundary value problems. Finite element methods in one and two spatial dimensions. Error bounds for the finite element approximation of elliptic problems. Adaptive mesh refinement. Time dependent problems where finite elements are used in space and finite differences in time. Use existing FEM-software, such as Comsol-Multiphysics.

Instruction

Lectures, laboratory work, compulsory assignments. Guest lecture.

Assessment

Written examination (3 credits) and compulsory assignments (2 credits).