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Unique solvability and stability analysis of a generalized particle method for a Poisson equation in discrete Sobolev norms. (English) Zbl 1524.65930

In this paper, the stability of a generalized particle method is analyzed for a Poisson equation with a source term given in divergence form. The Poisson equation appears in formulations of particle methods for the incompressible Navier-Stokes equations, with a source term. The unique solvability of the discretized Poisson equation is obtained by introducing a connectivity condition for particle distributions, i.e. the \(h\)-connectivity condition. Furthermore, the stability of the discretized Poisson equation is established based on the semi-regularity condition of a family of discrete parameters and discrete Sobolev norms with properties such as integration by parts.

MSC:

65N75 Probabilistic methods, particle methods, etc. for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
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