Meshfree Methods for Partial Differential Equations

Meshfree methods for the solution of partial differential equations gained much attention in recent years, not only in the engineering but also in the mathematics community. One of the reasons for this development is the fact that meshfree discretizations and particle models are often better suited to cope with geometric changes of the domain of interest, e.g. free surfaces and large deformations, than classical discretization techniques such as finite differences, finite elements or finite volumes. Another obvious advantage of meshfree discretizations is their independence of a mesh so that the costs of mesh generation are eliminated. Also, the treatment of time-dependent PDEs from a Lagrangian point of view and the coupling of particle models and continuous models gained enormous interest in recent years from a theoretical as well as from a practial point of view. This volume consists of articles which address the different meshfree methods (SPH, PUM, GFEM, EFGM, RKPM etc.) and their application in applied mathematics, physics and engineering.

Table of Contents:
Meshless and Generalized Finite Element Methods: A Survey of Some Major  Results.- Adaptive Meshfree Method of Backward Characteristics for Nonlinear Transport Equations.- New Methods for Discontinuity and Crack Modeling in EFG.- SPH Simulations of MHD Shocks Using a Piecewise Constant Smoothing Length Profile.- On the Numerical Solution of Linear Advection-Diffusion Equation using Compactly Supported Radial Basis Functions.- New RBF Collocation Methods and Kernel RBF with Apphcations.- Tuned Local Regression Estimators for the Numerical Solution of Differential Equations.- Approximate Moving Least-Squares Approximation with Compactly Supported Radial Weights.- Coupling Finite Elements and Particles for Adaptivity.- A Hamiltonian Particle-Mesh Method for the Rotating Shallow-Water Equations.- Fast Multi-Level Meshless Methods Based on the Implicit Use of Radial Basis Functions.- A Particle-Partition of Unity Method-Part IV: Parallelization.- Some Studies of the Reproducing Kernel Particle Method.- Consistency by Correcting Coefficients in the Finite-Volume-Particle Method.- Do Finite Volume Methods Need a Mesh?.- An Upwind Finite Pointset Method (FPM) for Compressible Euler and Navier-Stokes Equations.- Adaptive Galerkin Particle Method.- An Adaptivity Procedure Based on the Gradient of Strain Energy Density and its Apphcation in Meshless Methods.- New Developments in Smoothed Particle Hydrodynamics.- The Distinct Element Method — Apphcation to Structures in Jointed Rock.- Advance Diffraction Method as a Tool for Solution of Complex Non-Convex Boundary Problems.- On the Stochastic Weighted Particle Method.- The SPH/MLSPH Method for the Simulation of High Velocity Concrete Fragmentation.- Stability of DPD and SPH.- A New Meshless Method — Finite-Cover Based ElementFree Method.- Finite Pointset Method Based on the Projection Method for Simulations of the Incompressible Navier-Stokes Equations.- LPRH — Local Polynomial Regression Hydrodynamics.- On Multigrid Methods for Generalized Finite Element Methods.- The Convergence of the Finite Mass Method for Flows in Given Force and Velocity Fields.- Survey of Multi-Scale Meshfree Particle Methods.- Appendix. Color Plates.