Random PDEs on (randomly) moving hypersurfaces
A talk in the Bielefeld Stochastic Afternoon series by
Ana Djurdjevac from TU Berlin
| 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. In particular, we investigate the uncertainty which comes from unknown parameters or geometry.
As a starting point, we study the advection-diffusion equation with random coefficients that is posed on an evolving hypersurface. We consider both cases, uniformly bounded and log-normal distributions of the coefficient. We introduce the time-dependent solution space and prove the well-posedness. Furthermore, we will introduce and analyse the evolving surface finite element discretization of the equation, introduced by Dziuk and Elliott. In the uniformly bounded case, we show unique solvability of the resulting semi-discrete problem and prove optimal error bounds for the semi-discrete solution and Monte Carlo of its expectation. Our theoretical convergence rates are confirmed by numerical experiments.
In the second part we study 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 and analyse this equation. Moreover, we discuss the representation of random fields on hypersurfaces. This work is supported by DFG through project AA1-3 of MATH +. |