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

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 $H^s(\Omega )\cap H_0^2(\Omega )$, for $\frac{1}{2}\max (5,n)<s\le 4$, 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.