## Virtual Seminar Series - One World Numerical Analysis Series

This is the webpage for the One World Numerical Analysis Series. The seminars every two weeks, on a Monday (14:00-15:00 British Summer Time).

The first seminar will be held on 20th July followed by a break until September at which point the series will resume.

Organisers are Lehel Banjai (Heriot-Watt Univerity), Emmanuil Georgoulis (University of Leicester/NTU Athens), Charalambos Makridakis (IACM/University of Sussex/Crete) and Daniel Peterseim (University of Augsburg).

**To sign up for this seminar series, please complete this form.**

Zoom links will be sent at 11:00 BST on the day of each seminar

### Future Seminars

21 September

Ilaria Perugia(Vienna)

Space-time discontinuous Galerkin methods for wave propagation

We present discontinuous Galerkin approximations of the wave equation based on the space-time paradigm. The main drawback of space-time methods is their increased complexity, as compared to time stepping approaches. Two ideas to reduce this complexity, asymptotically, to that of one elliptic solve in the space domain will be considered. The first idea is to use Trefftz polynomial spaces, instead of complete polynomial spaces, in combination with a tent-pitching mesh design. The second idea exploits the unconditional well-posedness of suitably designed space-time discontinuous Galerkin methods, and consists in applying the so-called combination formula to a sequence of anisotropic space-time discretisations.This talk is based on the following references:

[1] A. Moiola and I. Perugia, A space-time Trefftz discontinuous Galerkin method for the acoustic wave equation in first-order formulation, Num. Math., 139 (2018), 389-435.

[2] I. Perugia, J. Schöberl, P. Stocker, and Ch. Wintersteiger, Tent pitching and Trefftz-DG method for the acoustic wave equation, Comput. Math. with Appl., 70 (2020), 2987-3000.

[3] P. Bansal, A. Moiola, I. Perugia and Ch. Schwab, Space-time discontinuous Galerkin approximation of acoustic waves with point singularities, arXiv: 2002.11575 [math.NA].

05 October

Christian Lubich(University of Tubingen)Title:

Convergent evolving surface finite element algorithms for geometric evolution equationsAbstract: Geometric flows of closed surfaces are important in a variety of applications, ranging from the diffusion-driven motion of the surface of a crystal to models for biomembranes and tumour growth. Basic geometric flows are mean curvature flow (described by a spatially second-order evolution equation) and Willmore flow and the closely related surface diffusion flow (described by spatially fourth-order evolution equations). Devising provably convergent surface finite element algorithms for such geometric flows of closed two-dimensional surfaces has long remained an open problem, going back to pioneering work by Dziuk in 1988. Recently, Balázs Kovács, Buyang Li and I arrived at a first solution to this problem for the geometric flows mentioned above. The proposed algorithms discretize evolution equations for geometric quantities along the flow, in our cases the normal vector and mean curvature, and use these evolving geometric quantities in the velocity law interpolated to the finite element space. This numerical approach admits a convergence analysis in the case of continuous finite elements of polynomial degree at least two. The error analysis combines stability estimates and consistency estimates to yield optimal-order H^1-norm error bounds for the computed surface position, velocity, normal vector and mean curvature. The stability analysis is based on the matrix-vector formulation of the finite element method and does not use geometric arguments. The geometry only enters into the consistency estimates.

References: our papers

B. Kovács, Buyang Li, C. Lubich, A convergent evolving finite element algorithm for mean curvature flow of closed surfaces, Numer. Math. 143 (2019), 797-853.---, A convergent algorithm for forced mean curvature flow driven by diffusion on the surface, arXiv:1912.05924---, A convergent evolving finite element algorithm for Willmore flow of closed surfaces, arXiv:2007.15257

The reviews

G. Dziuk and C.M. Elliott. Finite element methods for surface PDEs. Acta Numerica 22 (2013), 289-396. J. W. Barrett, H. Garcke, R. Nürnberg, Parametric finite element approximations of curvature driven interface evolutions, arXiv:1903.09462 A. Bonito and R. H. Nochetto (eds.), Handbook of numerical analysis. Vol. XXI: Geometric Partial Differential Equations - Part I. Elsevier/North-Holland, Amsterdam, 2020.

19 October

Virginie Ehrlacher(CERMICS - ENPC)02 November

Houman Owhadi(Caltech)16 November

Desmond Higham(University of Edinburgh)

### Previous Seminars

07 September

Christoph Schwab(SAM, ETH Zurich)

Exponential Convergence of hp-FEM for Spectral Fractional Diffusion in PolygonsSlides for talk are here

20 July

Alfio Quarteroni(Politecnico di Milano/Ecole Polytechnique Fédérale de Lausanne (EPFL))

The mathematical heart: a computational model for the simulation of the heart function

This is a One World Seminar Series.

Zoom is the online platform being used to deliver this seminar series

**This seminar is supported as part of the ICMS Online Mathematical Sciences Seminars.**