# Graded Parametric CutFEM and CutIGA for Elliptic Boundary Value Problems in Domains with Corners

## TL;DR

This paper introduces a graded parametric CutFEM and CutIGA approach for elliptic boundary value problems with corner singularities, achieving optimal stability and convergence without needing to identify the corner angle.

## Contribution

The method uses a mesh grading strategy via a suitable mapping that does not require knowing the corner's opening angle, enhancing robustness and applicability.

## Key findings

- Proves stability and optimal convergence of the method.
- Achieves accurate solutions near corners without angle identification.
- Employs structured meshes in the reference domain.

## Abstract

We develop a parametric cut finite element method for elliptic boundary value problems with corner singularities where we have weighted control of higher order derivatives of the solution to a neighborhood of a point at the boundary. Our approach is based on identification of a suitable mapping that grades the mesh towards the singularity. In particular, this mapping may be chosen without identifying the opening angle at the corner. We employ cut finite elements together with Nitsche boundary conditions and stabilization in the vicinity of the boundary. We prove that the method is stable and convergent of optimal order in the energy norm and $L^2$ norm. This is achieved by mapping to the reference domain where we employ a structured mesh.