Anisotropic polygonal and polyhedral discretizations in finite element analysis

New interpolation and quasi-interpolation operators of Cl'ement- 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.