DAMUT lunchbag colloquium

Date: 06 May 2026 

Location: RA 2334 

Speaker:  Gregor Gantner (MAGNUS-MACS)

Title: "Adaptive Finite Element Method"

Abstract: 
The finite element method (FEM) is a technique for the numerical solution of partial differential equations (PDEs) and thus for the simulation of many processes in nature and engineering.
In this method, the domain under consideration is subdivided into a finite number of elements, e.g., triangles in 2D and tetrahedra in 3D. 
The unknown exact solution is then approximated by piecewise polynomials on the resulting mesh.
While uniform mesh refinement always guarantees convergence of the approximation to the exact solution, in practice, the solution of the PDE varies much more strongly at some locations than at others and may even be singular. 
Therefore, it is usually advisable not to choose all mesh elements to be of the same size.
In this talk, the so‑called adaptive FEM is presented, whose goal is to automatically adapt the mesh to the behavior of the solution.
To this end, the local error between the solution and its approximation is estimated on all elements, and the mesh is iteratively refined where the estimated error is large.
Compared to uniform meshes (with elements of equal size), this can save an enormous amount of computational effort. 
In fact, under certain assumptions one can even prove mathematically that adaptive FEM converges optimally to the solution, that is, it generates the best possible meshes for approximation.