## Error estimate for a finite volume scheme in a geometrical multi-scale domain.(English)Zbl 1317.65225

The author considers a finite volume scheme, introduced in a previous paper [G. Panasenko and M.-C. Viallon, Math. Methods Appl. Sci. 36, No. 14, 1892–1917 (2013; Zbl 1273.65165)], to solve an elliptic linear partial differential equation appearing in a rod structure. The rod-structure is two-dimensional (2D) and consists of a central node and several outgoing branches. The branches are assumed to be one-dimensional (1D). So the domain is partially 1D, and partially 2D. For this reason, the considered structure is called a geometrical multi-scale domain. A discrete Poincaré inequality in terms of a specific $$H^1$$ norm defined on this geometrical multi-scale 1D-2D domain is established. The stated Poincaré inequality is valid for functions that satisfy a Dirichlet condition on the boundary of the 1D part of the domain and a Neumann condition on the boundary of the 2D part of the domain. An $$L^2$$ error estimate between the solution of the equation and its numerical finite volume approximation is derived.

### MSC:

 65N15 Error bounds for boundary value problems involving PDEs 65N08 Finite volume methods for boundary value problems involving PDEs 74S10 Finite volume methods applied to problems in solid mechanics 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 74K10 Rods (beams, columns, shafts, arches, rings, etc.)

Zbl 1273.65165

### Software:

F.E.M; Chemotaxis
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### References:

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