A Posteriori Error Bounds for Meshless Methods

We show how to provide safe a–posteriori error bounds for numerical solutions of well-posed operator equations using kernel–based meshless trial spaces. The presentation is kept as simple as possible in order to address a larger community working on applications in Science and Engineering. 1 Operator Equations Most contributions within application–oriented conferences only present numerical results without rigorous arguments concerning error bounds. Since a–priori error bounds are hard to find and to apply, we focus here on a–posteriori error bounds which are obtainable after a solution candidate is found. In order to cover a fairly general range of partial differential equation (PDE) problems arising in Science and Technology, we do not want to confine ourselves here to elliptic problems. The crucial property replacing ellipticity is well-posedness of the problem, or continuous dependence of the solution on the data. In the context of a linear boundary–value problem Lu = fΩ in Ω ⊂ IR d Bu = fΓ in Γ ⊂ ∂Ω (1) on a domain Ω with a linear differential operator L and some linear boundary operator B, continuous dependence means existence of constants CΩ, CΓ such that ‖u‖U ≤ CΩ‖Lu‖F + CΓ‖Bu‖G for all u ∈ U (2) where we use suitable norms in the spaces U, F, G between which the differential operator L and the boundary operator B are defined: L : U → F, B : U → G. (3) This easily generalizes to multiple differential or boundary operators. Note that the choice of spaces U, F, G usually is a mathematically hazardous problem in itself, even for fixed standard operators like L = −∆. A whole scale of trace spaces connected to trace theorems is possible, depending on the smoothness of the expected solution. For instance, a Poisson problem with Dirichlet data on a Institut für Numerische und Angewandte Mathematik, Universität Göttingen, Lotzestraße 16–18, D–37083 Göttingen, Germany http://www.num.math.uni-goettingen.de/schaback/research/group.html