European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
08:30   MS24: Computational surface PDEs
Chair: Vanessa Styles
08:30
25 mins
A coupled bulk-surface model for cell polarisation
Anotida Madzvamuse, Davide Cusseddu
Abstract: In this talk we will present a model describing the GTPase cycle between its active membrane-bound and inactive cytosolic form. Rho GTPases are key players in cell polarisation, which is required in several cellular activities, such as migration. The intricate reactions network of such proteins can lead to very complex mathematical models that are hard to analyse. Here, we present a simple basic interaction model of the same Rho GTPase protein in three-dimensional geometries, taking into account the different spatial compartmentalisation through the maturing theory of coupled bulk-surface semilinear parabolic equations. In this work the bulk-surface model is a substantial extension of the wave pinning model first proposed by Mori et al (2008, Biophys J.). To understand the theoretical behavior of the model, we carry out detailed asymptotic and local perturbation analysis, which helps to find parameter regions in which polarisation occurs. The geometry effects are naturally taken into account and with the emergent property that polarisation regions become bigger when a cell increases its surface. This last result is particularly meaningful since surface increase typical occurs during cell migration. To provide validation and confirmation of the theoretical results, we proposed and implemented a corresponding bulk-surface finite element method which we use to solve the system of coupled bulk-surface reaction-diffusion equations. Simulation results are shown over simple and more complex three-dimensional geometries and the pattern generation mechanism is in line with theoretical predictions.
08:55
25 mins
PDEs on hypersurafaces with random velocity
Ana Djurdjevac, Lewis Church, Charlie Elliott, Ralf Kornhuber, Thomas Ranner
Abstract: It is well-known that in a variety of applications, especially in the biological modeling, PDEs that appear can be better formulated on evolving curved domains. Most of these equations contain various parameters and often there is a degree of uncertainty regarding the given data. We investigate the uncertainty which comes from geometry. More precisely, we study the PDEs that evolve with a given random velocity. Utilizing the domain mapping method, we transfer the problem into a PDE with random coefficients on a fixed domain. For numerical analysis, we consider surface FEM coupled with Monte Carlo sampling. Our theoretical convergence rates are confirmed by numerical experiments. This work is supported by DFG through project AA1-3 of MATH + .
09:20
25 mins
A unified theory for continuous in time evolving finite element space approximations to partial differential equations in evolving domains
Tom Ranner
Abstract: We develop a unified theory for continuous in time finite element discretisations of partial differential equations posed in evolving domains including the consideration of equations posed on evolving surfaces and bulk domains as well coupled surface bulk systems. We use an abstract variational setting with time dependent function spaces and abstract time dependent finite element spaces. Optimal a priori bounds are shown under usual assumptions on perturbations of bilinear forms and approximation properties of the abstract finite element spaces. The abstract theory is applied to evolving finite elements in both flat and curved spaces. The theory allows us to give precise definitions which relate the abstract theory to concrete constructions and show which assumptions lead to stability and error estimates. Our approach allows an isoparametric approximation of parabolic equations in general domains. Numerical experiments are described which confirm the rates of convergence.
09:45
25 mins
Multiscale bulk-surface problems and mathematical models for signalling processes in biological tissues
Mariya Ptashnyk
Abstract: In order to better understand development, growth and remodelling of biological tissues and organs a better understanding of interactions between cells in a tissue is required. Essential parts of communications between cells, as well as cell responses to external and internal stimuli, are governed by intercellular signalling processes. In this talk we consider derivation and analysis of mathematical models for cellular signalling processes on the level of a single cell. A coupled system of nonlinear bulk-surface partial differential equations is used to model the dynamics of signalling molecules in the inter- and intra-cellular spaces and of cell membrane receptors. Using multiscale analysis techniques we derive macroscopic two-scale model for signalling processes defined on the tissue level. Two-scale numerical method is developed and implemented for simulations of the macroscopic bulk-surface problem. The nonlinear coupling between microscopic and macroscopic scales induces formation of patterns in the dynamics of solutions of the macroscopic model for cellular signalling processes, which may correspond to heterogeneity in cellular response mechanisms.