A Nyström-based finite element method on polygonal elements
This work provides a novel approach for finite element methods on polygonal meshes, potentially benefiting computational mechanics and PDE solvers, but the results are preliminary and incremental.
The paper introduces a finite element method on polygonal meshes where basis functions are defined implicitly via local Poisson problems and evaluated using Nyström discretizations, enabling handling of curvilinear polygons and non-polynomial boundary data. Experiments demonstrate approximation quality.
We consider families of finite elements on polygonal meshes, that are defined implicitly on each mesh cell as solutions of local Poisson problems with polynomial data. Functions in the local space on each mesh cell are evaluated via Nyström discretizations of associated integral equations, allowing for curvilinear polygons and non-polynomial boundary data. Several experiments demonstrate the approximation quality of interpolated functions in these spaces.