For elliptical and parabolic differential equations, and briefly for hyperbolic problems, the course addresses how to rewrite the problem in a form suitable for treatment with the finite element method, select appropriate mesh, element, variational formulation and how to implement the finite element method using both self-written code and existing routines. The theoretical part of the course deals with deriving error estimates and stability results given scalar linear partial differential equations.
The course deals with, for example: the weak formulation, mesh generation, function spaces, different element types, the Lax-Milgram theorem, interpolation, a priori error estimates, a posteriori error estimates, adaptivity, stability, accuracy, computational cost and discretization.
An overall goal of the course is to provide the student with both theoretical and practical skills to make reliable and efficient computations using the finite element method for models, described as Partial Differential Equations (PDEs), treated in the course as described in the course content.
After completing the course, the student shall be able to
- given a PDE rewrite it in a form suitable for treatment with the finite element method.
- describe key concepts and basic ideas of the finite element method and be able to use these concepts and ideas to describe advantages and limitations of the finite element methods included in the course.
- describe, apply, and implement the finite element methods included in the course.
- derive error estimates for the finite element solution, stability of the finite element method and well-posedness of the given PDE using theorems and analytical procedures included in the course.