## Incremental unknowns in finite differences in three space dimensions.(English)Zbl 0841.65089

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.
Reviewer: L.G.Vulkov (Russe)

### MSC:

 65N06 Finite difference methods for boundary value problems involving PDEs 65F35 Numerical computation of matrix norms, conditioning, scaling 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation