Optimal-Order Finite Difference Approximation of Generalized Solutions to the Biharmonic Equation in a Cube
Stefan Müller, Florian Schweiger, ENDRE E. SÜLI · SIAM Journal on Numerical Analysis · 2020
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.