Meshfree methods

About: Meshfree methods is a research topic. Over the lifetime, 2216 publications have been published within this topic receiving 69596 citations.

Adaptive Meshfree Method of Backward Characteristics for Nonlinear Transport Equations

Abstract: In previous work, a new adaptive meshfree advection scheme for numerically solving linear transport equations has been proposed. The scheme, being a combination of an adaptive semi-Lagrangian method and local radial basis function interpolation, is essentially a method of backward characteristics. The adaptivity of the meshfree advection scheme relies on customized rules for the refinement and coarsening of scattered nodes. In this paper, the method is extended to nonlinear transport equations. To this end, in order to be able to model shock propagation, an artificial viscosity term is added to the scheme. Moreover, the local interpolation method and the node adaption rules are modified accordingly. The good performance of the resulting method is finally shown in the numerical examples by using two specific nonlinear model problems: Burgers equation and the Buckley-Leverett equation, the latter describing a two-phase fluid flow in a porous medium.

All well-posed problems have uniformly stable and convergent discretizations

Abstract: This paper considers a large class of linear operator equations, including linear boundary value problems for partial differential equations, and treats them as linear recovery problems for functions from their data. Well-posedness of the problem means that this recovery is continuous. Discretization recovers restricted trial functions from restricted test data, and it is well-posed or stable, if this restricted recovery is continuous. After defining a general framework for these notions, this paper proves that all well-posed linear problems have stable and refinable computational discretizations with a stability that is determined by the well-posedness of the problem and independent of the computational discretization, provided that sufficiently many test data are used. The solutions of discretized problems converge when enlarging the trial spaces, and the convergence rate is determined by how well the data of the function solving the analytic problem can be approximated by the data of the trial functions. This allows new and very simple proofs of convergence rates for generalized finite elements, symmetric and unsymmetric Kansa-type collocation, and other meshfree methods like Meshless Local Petrov---Galerkin techniques. It is also shown that for a fixed trial space, weak formulations have a slightly better convergence rate than strong formulations, but at the expense of numerical integration. Since convergence rates are reduced to those coming from Approximation Theory, and since trial spaces are arbitrary, this also covers various spectral and pseudospectral methods. All of this is illustrated by examples.