Anisotropic polygonal and polyhedral discretizations in finite element analysis
This work provides theoretical foundations and practical tools for finite element analysis on general anisotropic meshes, which is important for problems requiring high directional resolution.
The paper develops new interpolation operators for anisotropic polygonal/polyhedral meshes, derives a priori error estimates, and proposes an adaptive mesh refinement method that produces highly anisotropic discretizations in 2D and 3D.
New interpolation and quasi-interpolation operators of Clément- and Scott-Zhang-type are analyzed on anisotropic polygonal and polyhedral meshes. Since no reference element is available, an appropriate linear mapping to a reference configuration plays a crucial role. A priori error estimates are derived respecting the anisotropy of the discretization. Finally, the found estimates are employed to propose an adaptive mesh refinement based on bisection which leads to highly anisotropic and adapted discretizations with general element shapes in two- and three-dimensions.