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## Speaker

Post-Doc Research Fellow Philipp Öffner, Johannes Gutenberg-University Mainz Institute of Mathematics, Numerics Group

## Title

Function Space Summation by Parts Operators and their Applications in Numerical Schemes

## Abstract

Summation-by-parts (SBP) operators are popular building blocks for sys- tematically developing stable and high-order accurate numerical methods for time-dependent differential equations. The main idea behind existing SBP op- erators is that the solution is assumed to be well approximated by polynomials up to a certain degree, and the SBP operator should therefore be exact for them. However, polynomials might not provide the best approximation for some prob- lems, and other approximation spaces may be more appropriate, cf. Figure 1. In this talk, we present the theory of SBP operators based on general function spaces (FSBP), demonstrate that most classical results for polynomial-based SBP operators carry over to FSBP operators and apply them in various numer- ical schemes. We focus especially on radial basis function (RBF) methods and construct stable RBF methods for time-dependent PDEs.

Figure 1: Trigonometric and exponential functions

Institute of Mathematics, Johannes-Gutenberg University Mainz, Germany. Email: poeffner@uni-mainz.de.

**Acknowledgment**

P.O. gratefully acknowledge support of the Gutenberg Research College, JGU

Mainz and the Oberwohlfach research fellow program.