Meshfree and particle methods are numerical approaches for partial differential equations that have not yet found their way into most standard textbooks. The reasons for this are diverse. For one, these methods do not match the simplicity and generality of grid-based finite difference approaches. Second, they lack the theoretical underpinnings of mesh-based finite element methods. And finally, there is actually a wide variety of different meshfree and particle methods, and unifying principles are few. All these shortcomings being mentioned, it must be pointed out that meshfree and particle methods, when applied in the right way to the right problem, can achieve an accuracy, flexibility, or generality that is unmatched by classical mesh-based methods.

In this talk, I will present two examples of attractive approaches. The first example is a generalized finite difference method for the Poisson equation on irregular domains (and possibly high dimensions) that is based on optimally sparse stencils. The optimal sparsity is achieved by defining approximations to the Laplace operator via $\ell^1$ optimization. The second example is a characteristic particle method for scalar nonlinear conservation laws on networks. The most intriguing feature of this numerical approach is that it solves the underlying equation exactly on each network edge. We demonstrate the performance of the method for the simulation of vehicular traffic flow on highway networks.