Monotone Meshfree Methods for Linear Elliptic Equations in Non-divergence Form via Nonlocal Relaxation

We design a monotone meshfree finite difference method for linear elliptic equations in the non-divergence form on point clouds via nonlocal relaxation method. The key idea is novel combination of integral PDE problem with robust discretization clouds. Minimal positive stencils are obtained through local $$l_1$$ -type optimization procedure that automatically guarantees stability and, therefore, convergence equations. A major theoretical contribution existence consistent and given cloud geometry. provide sufficient conditions by finding neighbors within an ellipse (2d) or ellipsoid (3d) surrounding each interior point, generalizing study Poisson’s equation Seibold (Comput Methods Appl Mech Eng 198(3–4):592–601, 2008). It well-known wide general needed constructing schemes Our result represents significant improvement stencil width estimate positive-type methods near-degenerate regime (when ellipticity constant becomes small), compared to previously known works this area. Numerical algorithms practical guidance provided eye case small constant. At end, we present numerical results performance our both 2d 3d, examining range constants including regime.