SUMMARYA novel space-time meshfree collocation method (STMCM) for solving systems of non-linear ordinary and partial differential equations by a consistent discretization in both space and time is proposed as an alternative to established mesh-based methods. The STMCM belongs to the class of truly meshfree methods, i.e. the methods that do not have any underlying mesh, but work on a set of nodes only without any a priori node-to-node connectivity. Instead, the neighbouring information is established on-the-fly. The STMCM is constructed using the Interpolating Moving Least-squares technique, which allows a simplified implementation of boundary conditions due to fulfillment of the Kronecker delta property by the kernel functions, which is not the case for the major part of other meshfree methods.The method is validated by several examples ranging from interpolation problems to the solution of PDEs, whereas the STMCM solutions are compared with either analytical or reference ones.
SUMMARYThis paper presents a meshfree interpolating moving least squares (IMLS) method based on singular weights for the solution of partial di erential equations. Due to the speciÿc singular choice of weight functions, which is needed to guarantee the interpolation, there arises a problem of ÿnding the inverse of the occurring singular matrix. The inverse is carried out using a regularization of weight functions. It turns out that a stable inverse is obtained by considering the vanishing regularization parameter.Moreover, the use of this perturbation technique allows the correct evaluation of all necessary derivatives in interpolation points at a reasonable cost.Unlike standard kernel functions used in EFGM, RKPM, etc., the singular kernel functions lead to really interpolating functions which satisfy the Kronecker-delta property. They can be used for enforcement of Dirichlet boundary conditions when solving boundary value problems.Solution to a model BVP as well as an experimental convergence study of the method to analytical solutions, conÿrming the mathematical derivations, are given.
Abstract. In a foregoing paper [Sonar, ESAIM: M2AN 39 (2005) 883-908] we analyzed the Interpolating Moving Least Squares (IMLS) method due to Lancaster andŠalkauskas with respect to its approximation powers and derived finite difference expressions for the derivatives. In this sequel we follow a completely different approach to the IMLS method given by Kunle [Dissertation (2001)]. As a typical problem with IMLS method we address the question of getting admissible results at the boundary by introducing "ghost points". Most interesting in IMLS methods are the finite difference operators which can be computed from different choices of basis functions and weight functions. We present a way of deriving these discrete operators in the spatially one-dimensional case. Multidimensional operators can be constructed by simply extending our approach to higher dimensions. Numerical results ranging from 1-d interpolation to the solution of PDEs are given.Mathematics Subject Classification. 65M06, 65M60, 65F05.
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