M8: Multiscale methods based on unstructured data
| Researcher: | Patricio Farrell |
| Team Leader(s): | Dr Kathryn Gillow & Prof. Holger Wendland, Universität Bayreuth (External Supervisor) |
| Collaborators: | N/A |
Background
The need to understand and model multiscale phenomena has gained importance in recent years. It is inherent to turbulent fluid flows as well as geological, geophysical and biological data. Often the data are scattered and have little structure. The aim of this project is to develop and analyse mesh-free multiscale methods that can be used for interpolation problems, as well as for solving differential equations.
Techniques and
Challenges
Radial basis functions (RBFs) are the key tool we employ to design such multiscale algorithms. These functions have excellent approximation properties without needing to generate a grid. In order to keep the cost of solving the arising linear systems low, we employ compactly-supported RBFs. Although there has been numerical evidence that these multiscale algorithms converge, until recently there was no mathematical theory explaining why they did.
Results
We have managed to extend a recent convergence result for plain interpolation to a mesh-free multiscale collocation algorithm for linear second order elliptic partial differential equations (PDEs). By implementing some test problems we were able to verify the convergence speed. Unfortunately there is a mismatch: even though the convergence order for this algorithm is influenced by the PDE operator, the stability of the collocation matrices is determined by the boundary operator.
The Future
For this reason we are currently investigating how to improve the condition numbers of the collocation matrices. There seem to be at least two possibilities: either by employing an effective preconditioner, or by splitting the linear system for both operators into two systems – one for each operator.
References
Wendland H.: Multiscale Analysis of Sobolev Spaces on Bounded Domains, Numer. Math. 116, no. 3, 493–517, 2010
Floater M.S., Iske A.: Multistep Scattered Data Interpolation using Compactly Supported Radial Basis Functions, J. Comput. Appl. Math. 73, no. 1-2, 65–78, 1996