# Optimal order finite difference approximation of generalized solutions to the biharmonic equation in a cube

## TL;DR

This paper establishes an optimal order error bound for finite difference methods approximating the biharmonic equation's solutions in multiple dimensions, extending the known convergence results to the maximal Sobolev space range.

## Contribution

It provides the first optimal order error estimate in the discrete $H^2$ norm for generalized solutions with minimal regularity in up to seven dimensions.

## Key findings

- Error bounds are optimal in the discrete $H^2$ norm.
- Results extend convergence theory to the maximal Sobolev space range.
- Applicable to solutions with Sobolev regularity $s$ in the specified range.

## Abstract

We prove an optimal order error bound in the discrete $H^2(\Omega)$ norm for finite difference approximations of the first boundary-value problem for the biharmonic equation in $n$ space dimensions, with $n \in \{2,\dots,7\}$, whose generalized solution belongs to the Sobolev space $H^s(\Omega) \cap H^2_0(\Omega)$, for $\frac{1}{2} \max(5,n) < s \leq 4$, where $\Omega = (0,1)^n$. The result extends the range of the Sobolev index $s$ in the best convergence results currently available in the literature to the maximal range admitted by the Sobolev embedding of $H^s(\Omega)$ into $C(\overline\Omega)$ in $n$ space dimensions.