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

Journal article

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.

Authors: Müller, S; Schweiger, F; Süli, E

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Society for Industrial and Applied Mathematics
• Journal: SIAM Journal on Numerical Analysis
• Volume: 58
• Issue: 1
• Pages: 298–329
• Publication date: 2020-01-14
• Acceptance date: 2019-10-21
• DOI: 10.1137/19M1254313
• EISSN: 1095-7170
• ISSN: 0036-1429
• Keywords: FFR; 65N06 (Primary), 31A30, 31B30 (Secondary); math.NA