Incremental unknowns in finite differences in three space dimensions
The incremental unknowns have been applied to numerous problems, see the introduction of the present paper. The aim of this paper is to study the Laplace equation on the cube \((0,1)^3\). The second-order incremental unknowns are introduced and studied. By deriving suitable prior estimates it is proved that incremental unknowns are small. Then, the condition number of the matrix corresponding to the five-point discretization of the Laplace equation is analyzed. The authors show that this number is \(O(h^{-1} \ln(h)^4)\) instead of \(O(h^{-2})\) when the usual nodal unknowns are used, with \(h\) being the fine grid mesh size.
scientific article; zbMATH DE number 854781