Anisotropic polygonal and polyhedral discretizations in finite element analysis

TL;DR

This paper develops new interpolation operators and error estimates for anisotropic polygonal and polyhedral meshes in finite element analysis, enabling adaptive mesh refinement with complex element shapes in 2D and 3D.

Contribution

It introduces novel anisotropic interpolation operators and error estimates for polygonal/polyhedral meshes, facilitating adaptive refinement without reference elements.

Findings

Derived a priori error estimates accounting for anisotropy.

Proposed an adaptive mesh refinement strategy based on bisection.

Achieved highly anisotropic, shape-robust discretizations in 2D and 3D.

Abstract

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.