# Applied PDEs Working Seminar

## Meetings are held on Tuesdays at 16.00-17.00 in A1.01.

Organisers: Andreas Dedner, Charlie Elliott, Björn Stinner

These meetings provide an opportunity for individuals to discuss in an informal manner progress on their current work, and describe interesting new problems related to PDEs and their applications and computation.

We have chosen this time to allow us to make a pub visit afterwards if we feel inclined.

Group emails can be sent via applied_maths_pde_workgroup at listserv dot warwick dot ac dot uk.

#### Schedule for Term 3, 2012/13:

23nd April (week 1) - The Cahn-Hilliard equation: applications, numerical difficulties and discretizations (Simone Stangalino, Politecnico di Milano)

The Cahn-Hilliard (CH) equation is historically used to describe the phase separation and coarsening phenomena in a melted alloy. Recently it has been applied to many fields, like tumor growth and images inpainting. The CH equation brings several numerical difficulties: it is a fourth order parabolic equation with a non-linear term and it evolves with very different time scales. Furthermore, in order to fully capture the interface dynamics, high spatial resolution is required. In this talk we give an overview of the discretization of the classical equation both with conforming and discontinuous finite element methods. We also focus on the problem of imposing different kind of boundary conditions. Special emphasis is given to dynamic boundary conditions with the discontinuous Galerkin finite element.

30th April (week 2) - Optimal control of nonlinear parabolic PDEs and model order reduction (Sophia Kohle, TU Berlin)

In this talk an optimal control problem governed by a nonlinear parabolic equation with constraints to the control is considered. To solve this problem numerically a semi-smooth Newton method is applied. In this optimization method we must solve two PDEs in every iteration, so solving the problem requires substantial numerical effort. For an efficient treatment of the problem, a reduced order approach is presented, the proper orthogonal decomposition. POD uses so-called snapshots to create a reduced basis, which are used as trialfunctions in the Galerkin approach for the variational formulation. How well does this approach work? To answer this question, we introduce an a-posteriori error estimator which estimates the distance between the solution of the reduced problem and the solution of the original problem. In the nonlinear case we need some information on the lowest eigenvalue of the reduced Hessian to compute the estimator. Finally, numerical comparisons are drawn between POD and FEM.

7th May (week 3) - A parabolic PDE on an evolving curve with finite time singularity (Mike Scott, Warwick)

Geometric PDEs, such as the mean curvature flow equation, that undergo singularities have undergone extensive research in the literature. To the best of the author's knowledge, what has not been studied is the effects of a surface singularity to a PDE with geometric coefficients living on the surface. In this talk we look at an example where the heat equation lives on a time dependent hyperbola, that undergoes a finite time "kink" singularity. We investigate the properties of the solution at the singuarlity, continue the solution past the singularity in some weak sense and (if we have time) investigate what happens when one changes the speed at which we approach the singularity.

This is joint work with Professor Martin Hairer and Professor Charles Elliott.

4th June (week 7) - Deriving Phase Field Crystals (Simon Bignold, Warwick)

In this talk we consider the Phase Field Crystal (PFC) model and several ways of simulating this model. PFC is a recent model for crystallisation proposed in [Elder, Grant, PRE,2004], this model consists of a relatively simple functional which when minimised exhibits a phase transition dependent on two parameters and in one case the minimiser can be chosen to be a periodic lattice.

We consider two numerical approaches for the minimisation of our PFC functional which lead to two partial differential equations. The first is a review of the method of [Elsey, Wirth, M2AN, 2013] which gives a numerical approach to simulating the PFC equation. The second is a new approach which numerically solves a modified gradient flow of the PFC functional. Both methods rely on solving the equation in Fourier space and use periodic domains so that the fast Fourier transform can be used.

Finally we compare the two methods and show that they both obtain the expected minimisation lattice. We also give some possible applications for PFC emphaising the applications in the physics of crystals. This talk should be accessible to anyone with a reasonable degree of mathematical literacy.

#### Schedule for Term 2, 2012/13:

15th January (week 2) - Diffusion on Rapidly Varying Surfaces

Andrew Duncan (Warwick) - In this presentation I will describe a multi-scale approach to analysing the macroscopic behaviour of passive tracers diffusing laterally on a surface with high-frequency low-amplitude fluctuations. This is motivated by various problems in cellular biology involving lateral protein transport over fluctuating cellular membranes. In this talk I will focus specifically on lateral diffusion on a time-independent surface with locally-periodic undulations, leaving the more biologically-motivated models for another time.

22nd January (week 3) - On a Discontinuous Galerkin Method for Surface PDEs

Pravin Madhavan (Warwick) - I will talk about how one can extend a discontinuous Galerkin method onto surfaces, namely by deriving a priori error estimates. This is then verified numerically for a number of test problems.

29th January (week 4) - No seminar

5th February (week 5) - Partial differential equations on evolving surfaces

Amal Alphonse (Warwick) - In this talk, which is based on a paper that I read, I'll go through a well-posedness theory for a PDE on an evolving surface. This will require the construction of relevant functions spaces necessary to pose the problem and the transformation of the PDE onto a fixed domain on which we can apply existing abstract parabolic PDE theory.

12th February (week 6) - On the analysis of a reaction diffusion systems defined on a parttioned domain with nonlinear conditions at the interface

Maha Alhajri (Warwick) - This a work is in progress to address a global existence and uniqueness of a strong solution for a semi-linear reaction diffusion systems. The two system is defined on two adjoint portion of a domain $\Omega_{1}\cup\Omega_{2}=\Omega\subset\mathbb{R}^{N}$ with nonlinear conditions on the interface $\Gamma=\partial\Omega_{1}\cup\partial\Omega_{2}$. We use the strong maximum principle, Hope boundary Lemma, the Schauder fixed point theorem and the fundamental solutions of linear parabolic partial differential equations.

19th Febraury (week 7) - Data inversion in coupled subsurface flow and geomechanics models

Marco Iglesias (Warwick) - In this talk I present a deterministic inverse modeling approach to estimate subsurface properties by jointly inverting surface deformation and pressure data from a fully-coupled geomechanics-flow (single-phase) model. I discuss key aspects of the analysis of the inverse problem and its regularization. Numerical results of synthetic experiments are displayed to demonstrate the capabilities of the proposed technique for the estimation of petrophysical and elastic properties of the subsurface.

26th February (week 8) - A generalised Ladyzhenskaya inequality and a coupled parabolic-elliptic problem

Dave McCormick (Warwick) - In this talk I will discuss a simplification of the standard magnetohydrodynamic (MHD) system in which the fluid evolution is replaced by an elliptic equation. Our aim is to show that the resulting system is well-posed, with a view to making rigorous the method of magnetic relaxation proposed by Moffatt (1985). Despite the apparent simplicity of the equations it turns out that this requires results that are at the limit of what is available, involving elliptic regularity in L^1 and a strengthened form of the Ladyzhenskaya inequality derived using the theory of interpolation

5th March (week 9) - Optimal control of variational inequalities with pointwise objective functionals

Caroline Löbhard (HU Berlin) - We consider a class of optimization problems, where a control variable acts as distributed force on a state variable satisfying a variational inequality. This constraint problem can for instance describe a membrane which is pushed towards an obstacle by the control (and other) forces. The objective then contains the distance of the state in certain evaluation points and furthermore, costs of the control. The point evaluations require for additional regularity of the solution of the variational inequality. A lack of continuity and directional differentiability of the control-to-state operator as a mapping into the more regular space causes difficulties in the problem analysis. We establish the existence of a solution of the optimal control problem and use a smoothed penalization technique to derive first order stationarity conditions.

12th March (week 10) - No seminar

#### Schedule for Term 1, 2012/13:

9th October - PDE optimisation

Charles Brett - We look at some approaches to optimal control of variational inequalities, and discuss how they can be extended to deal with point evaluations in the objective functional.

Chandrashekar Venkataraman - We discuss an algorithm for the optimisation of parameters in a model of cell motility where the objective function depends on experimental data.

16th October - PDEs on surfaces

Andrew Lam - We review a paper on the well-posedness of the solution to an Allen-Cahn type equation with dynamic boundary conditions and singular potentials.

Tom Ranner - I will talk about how one can use the surface finite element method to show existence of solutions to a Cahn-Hilliard equation on an evolving surface.

23rd October - A Degenerately Dispersive Nonlocal Nonlinear Wave Equation

Gideon Simpson (Minnesota) - I will present properties of an equation arising in magma dynamics and highlight challenges in the well posedness theory of the reduced model, and its relation to a full model of the Earth's interior. I will also discuss solitary waves and their stability.

30th October - Surfactants

Andrew Lam - I will talk about how to model surfactants in a two-fluid system using a sharp interface description and a diffuse interface description.

6th and 14th November - Mathematical models arising from John Ockendon's lectures

20th November - Cancelled

27th November - Extra Applied Maths seminar

Desmond J Higham (Strathclyde) - Algorithms and Models for Evolving Networks

The digital revolution is generating novel large scale examples of connectivity patterns that change over time. This scenario may be formalized as a graph with a fixed set of nodes whose edges switch on and off. For example, we may have networks of interacting mobile phone users, emailers, Facebookers or Tweeters. To understand and quantify the key properties of such evolving networks, we can extend classical graph theoretical notions like degree, pathlength and centrality. In this talk I will focus on linear algebra-based algorithms and show that appropriate matrix products can capture various aspects of information flow around an evolving network. I will show how these algorithms performed in a recent case study on Twitter data, where independent influence rankings were available from social media experts. I will also show how classical random graph models can be extended to the time-dependent setting. In particular, a model for triadic closure (friends-of-friends tend to become friends) will be seen to produce a bistability effect.

4th December - No seminar