# A quasi-optimal adaptive spline-based finite element method for the bi-Laplace operator using Nitsche's method

## TL;DR

This paper develops and proves the convergence of an adaptive spline-based finite element method for solving the bi-Laplace operator, employing Nitsche's method to weakly impose boundary conditions.

## Contribution

It introduces a quasi-optimal adaptive finite element approach using polynomial B-splines for fourth order elliptic problems with boundary conditions.

## Key findings

- Proves convergence of the proposed method.
- Demonstrates quasi-optimality of the adaptive algorithm.
- Validates effectiveness through theoretical analysis.

## Abstract

We establish the convergence of an adaptive spline-based finite element method of a fourth order elliptic problem with weakly-imposed Dirichlet boundary conditions using polynomial B-splines.