Convergence Rates For Non-degenerate Elliptic PDEs On Junctions With Kirchhoff Conditions

Abstract

We present monotone finite-difference schemes for second-order nonlinear elliptic equations on a junction, with Dirichlet conditions at the boundary vertices and a Kirchhoff condition at the interior vertex, see Barles et al. (2025) for a more general problem. In contrast with fully coupled discretizations on the whole junction studied by Morfe in 2020, we propose a decoupling strategy: the network problem is reduced to a family of Dirichlet problems posed on the individual branches and parametrized by the unknown junction value. Each branch problem is solved by a monotone scheme inspired by Crandall and Lions (1984), while the junction value is recovered from a scalar nonlinear flux-balance equation by a Newton-type method. This approach is simple to implement, preserves sparsity, and is well suited to extensions to more general networks. On each branch, the numerical analysis yields first-order convergence for the solution and order 1/2 for the discrete derivative. At the junction level, the reconstruction recovers the classical 1/2 convergence rate obtained for coupled schemes such as the one by Morfe (2020). We illustrate the method for Hamiltonians of absolute-value type, using Lax–Friedrichs and upwind numerical approximations. The resulting nonlinear algebraic systems are solved by a semi-smooth Newton method, in particular Howard’s algorithm, together with recent techniques for nonlinear absolute value equations proposed in Daniilidis et al. (2026).

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