Optimal order finite difference approximation of generalized solutions to the biharmonic equation in a cube
For researchers in numerical analysis, this provides the best possible convergence rate for a known numerical method on a specific problem, extending existing results to the maximal theoretical range.
The paper proves an optimal order error bound for finite difference approximations of the biharmonic equation in a cube, extending the Sobolev index range to the maximal allowed by Sobolev embedding for dimensions 2 to 7.
We prove an optimal order error bound in the discrete $H^2(Ω)$ 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 $H^s(Ω) \cap H^2_0(Ω)$, for $\frac{1}{2} \max(5,n) < s \leq 4$, where $Ω= (0,1)^n$. 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 $H^s(Ω)$ into $C(\overlineΩ)$ in $n$ space dimensions.