FMNN45F

Temporary

Numerical Methods for Partial Differential Equations

7.5

Course given jointly for second and third cycle

FMNN45, NUMN33

7154 (Centre of Mathematical Sciences / Numerical Analysis)

FMNN45, NUMN33

2025-01-07

Maria Sandsten

English

Every spring semester

The course is a continuation of FMNN10 Numerical methods for differential equations and the purpose of the course is to deepen the doctoral student's knowledge of partial differential equations, with an emphasis on numerical approximation of solutions, and to provide practical training in solving relevant computational problems in a python environment with tools in DUNE (Distributed and Unified Numerics Environment), a modular numerical toolbox for partial differential equations.

Theory for ellipic partial differential equations (PDEs): Well-posed problems. Weak theory of solvability. Existence and uniqueness. Stability with respect to data, and approximation of solutions. Regularity of solutions.

Solution with the Finite Element Method in DUNE-FEM: Introduction to DUNE-FEM. Discretization of boundary value problems for PDEs in DUNE-FEM using the Finite Element Method. Construction of finite elements, e.g. discretization grids, reference elements, degree of freedom (DOF) mappings.

Using the Unified Form Language (UFL) in DUNE-FEM for description of weak forms of PDEs,

Parallelization of Finite Element methods using domain decomposition,

Adaptive Finite Elements.

be able to understand how concepts from functional analysis are used to develop and analyse numerical algorithms for elliptic partial differential equations,
be able to explain how partial differential equations may be discretized using the Finite Element Method,
be able to state and derive simple error estimates for elliptic problems,
be able to independently identify an appropriate numerical method and select suitable parameters with regard to accuracy and efficiency requirements,
be able to give examples of important fields of applications in which the algorithms occurring in the course are important.
be able to explain concepts presented in the course with adequate terminology, in writing and orally.

be able to rewrite problem on finite element form and apply suitable solution strategies using DUNE-FEM,
be able to independently proceed from observation and interpretation of results to conclusion, and present the conclusions on a scientific basis in a free report format,
be able to independently present results and conclusions of numerical experiments performed with DUNE-FEM, in written and oral form, with references and other documentation of work carried out in support of their conclusions.

be able to independently evaluate obtained numerical results with DUNE-FEM in relation to the available theory.

Lectures

Written report

Miscellaneous

A number of compulsory projects carried out in
small groups are included in the course. Attendance at all oral group presentations of the project results is mandatory.

Failed, pass

FMAB30 Calculus in Several Variables & FMNN10 Numerical Methods for Differential Equations.

FMAN55 Applied Mathematics.

Larsson, S. & Thomee, V.: Partial Differential Equations with Numerical Methods. Springer Science & Business Media, 2008. ISBN 9783540887058.
Renardy, M. & Rogers, Robert C.: An Introduction to Partial Differential Equations. ISBN 9783540979524.
Brenner, Susanne C. & Scott, L. Ridgway: The Mathematical Theory of Finite Element Methods. ISBN 9783540941934.
Ciarlet, Philippe G.: The Finite Element Method for Elliptic Problems. North-Holland, 1978. ISBN 9780444850287.
Thomee, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer, 2006. ISBN 9783540331216.

FMNN45F

2025-01-07

Maria Sandsten