Holger Wendland

Tobias Ramming, Holger Wendland

We derive a new discretisation method for first order PDEs of arbitrary spatial dimension, which is based upon a meshfree spatial approximation. This spatial approximation is similar to the SPH (smoothed particle hydrodynamics) technique and is a typical kernel-based method. It differs, however, significantly from the SPH method since it employs an Eulerian and not a Lagrangian approach. We prove stability and convergence for the resulting semi-discrete scheme under certain smoothness assumptions on the defining function of the PDE. The approximation order depends on the underlying kernel and the smoothness of the solution. Hence, we also review an easy way of constructing smooth kernels yielding arbitrary convergence orders. Finally, we give a numerical example by testing our method in the case of a one-dimensional Burgers equation.

Sara Avesani, Rüdiger Kempf, Michael Multerer et al.

We study multiscale scattered data interpolation schemes for globally supported radial basis functions with focus on the Matérn class. The multiscale approximation is constructed through a sequence of residual corrections, where radial basis functions with different lengthscale parameters are combined to capture varying levels of detail. We prove that the condition numbers of the the diagonal blocks of the corresponding multiscale system remain bounded independently of the particular level, allowing us to use an iterative solver with a bounded number of iterations for the numerical solution. Employing an appropriate diagonal scaling, the multiscale system becomes well conditioned. We exploit this fact to derive a general error estimate bounding the consistency error issuing from a numerical approximation of the multiscale system. To apply the multiscale approach to large data sets, we suggest to represent each level of the multiscale system in samplet coordinates. Samplets are localized, discrete signed measures exhibiting vanishing moments and allow for the sparse approximation of generalized Vandermonde matrices issuing from a vast class of radial basis functions. Given a quasi-uniform set of $N$ data sites, and local approximation spaces with exponentially decreasing dimension, the samplet compressed multiscale system can be assembled with cost $\mathcal{O}(N \log^2 N)$. The overall cost of the proposed approach is $\mathcal{O}(N \log^2 N)$. The theoretical findings are accompanied by extensive numerical studies in two and three spatial dimensions.

Peter Giesl, Holger Wendland

In this paper, we discuss the solution of certain matrix-valued partial differential equations. Such PDEs arise, for example, when constructing a Riemannian contraction metric for a dynamical system given by an autonomous ODE. We develop and analyse a new meshfree discretisation scheme using kernel-based approximation spaces. However, since these approximation spaces have now to be matrix-valued, the kernels we need to use are fourth order tensors. We will review and extend recent results on even more general reproducing kernel Hilbert spaces. We will then apply this general theory to solve a matrix-valued PDE and derive error estimates for the approximate solution. The paper ends with a typical example from dynamical systems.