Minimal positive stencils in meshfree finite difference methods for linear elliptic equations in non-divergence form
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We design a monotone meshfree finite difference method for linear elliptic equations in the non-divergence form on point clouds via a nonlocal relaxation method. Nonlocal approximations of linear elliptic equations are first introduced to which a meshfree finite difference method applies. Minimal positive stencils are obtained through a local l_1-type optimization procedure that automatically guarantees the stability and, therefore, the convergence of the meshfree discretization for linear elliptic equations. The key to the success of the method relies on the existence of positive stencils for a given point cloud geometry. We provide sufficient conditions for the existence of positive stencils by finding neighbors within an ellipse (2d) or ellipsoid (3d) surrounding each interior point, generalizing the study for Poisson's equation by Seibold in 2008. It is well-known that wide stencils are in general needed for constructing consistent and monotone finite difference schemes for linear elliptic equations. Our study improves the known theoretical results on the existence of positive stencils for linear elliptic equations when the ellipticity constant becomes small. Numerical algorithms and practical guidance are provided with an eye on the case of small ellipticity constant. We present numerical results in 2d and 3d at the end.
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