Speaker : Lehel Banjai, Maxwell Institute for Mathematical Sciences, School of Mathematical & Computer Sciences, Heriot-Watt University, UK 

Time : 15:00 - 16:00 CET (Rome/Paris)

Hosted at: SISSA, International School of Advanced Studies, Trieste, Italy

Zoom :  A link will appear here

Organizers : Pavan Pranjivan Mehta* (pavan.mehta@sissa.it) and Arran Fernandez** (arran.fernandez@emu.edu.tr)

* SISSA, International School of Advanced Studies, Italy

** Eastern Mediterranean University, Northern Cyprus

Keywords: spectral fractional Laplacian, fractional Helmholtz, hp-FEM, contour integral methods, sinc quadrature.

Abstract: In this talk we consider the Helmholtz equation where the usual Laplacian is replaced by a non-local variant: the spectral fractional Laplacian. The thus obtained fractional Helmholtz equation has applications to geophysical electromagnetics [1], but in this talk we focus on the development and analysis of efficient numerical methods. We describe and analyse two possible approaches. Firstly, a direct contour integral method discretised by a sinc quadrature combined with an hp-finite element method [2]. We will discuss the merits of choosing the contour before discretisation of the Laplacian or after. Secondly a tensor product hp-finite element method applied to an equivalent local problem in a spatially extended domain [3,4]. In both cases we show that the cost is equivalent to solving a single standard Helmholtz equation and a fractional Poisson problem. Relationship to existing works is described [1,5]. The talk ends with numerical experiments showing exponential convergence of both approaches

Biography: Lehel Banjai is a professor at the Mathematics Department of Heriot-Watt University which is part of the Maxwell Institute for Mathematical Sciences in Edinburgh, UK. Before coming to Edinburgh, he was at the Max Planck Institute for Mathematics in the Sciences, Leipzig and at the University of Zurich. He obtained his D.Phil. in the Numerical Analysis Group at the University of Oxford.  

He is best known for his work on numerical methods for time-domain boundary integral equations and on convolution quadrature (see the recent research monograph with Francisco Javier Sayas). In recent years he has been active in researching numerical methods for fractional differential equations -- time and space, a priori and a posteriori analysis. His other interests include numerical wave propagation, data sparse methods such as H-matrices and FMM, conformal mapping etc. 

He has jointly founded and co-organised the One World Numerical Analysis Seminar and is currently acting as an Associate Editor for SINUM. 

Bibliography

[1] C. J. Weiss, B. G. van Bloemen Waanders, and H. Antil. Fractional operators applied to geophysical electromagnetics, Geophys. J. Int., 2020. 

[2] L. Banjai, J.M. Melenk, and Ch. Schwab. hp-FEM for reaction-diffusion equations. II: Robust exponential convergence for multiple length scales in corner domains, IMA JNA, 2023. 

[3] L. Banjai, Jens M. Melenk, Ricardo H. Nochetto, Enrique Otárola, Abner J. Salgado, and Christoph Schwab. Tensor FEM for spectral fractional diffusion. Found. Comput. Math., 19(4):901–962, 2019. 

[4] L. Banjai, J.M. Melenk, and Ch. Schwab. Exponential Convergence of hp-FEM for Spectral Fractional Diffusion in Polygons, Numerische Mathematik, 2023. 

[5] Christian Glusa, Harbir Antil, Marta D'Elia, Bart van Bloemen Waanders, and Chester J. Weiss. A fast solver for the fractional Helmholtz equation, SISC, 2021.