Numerical solutions of SPDEs with boundary noise

Abstract

Galerkin finite element method is a technique for approximating solutions to stochastic partial differential equations (SPDEs) that has been extensively studied in the literature. In this thesis, we extend the scheme to solve the case where noise enters through the boundary of the domain. We prove that the optimal convergence rate is achieved for semi-linear parabolic SPDEs with random Neumann boundary conditions. Considering the advection-diffusion equation with boundary noise, we show that solutions are useful for simulating solute dynamics in arteries where the vessel walls are treated as flexible. We also investigate SPDEs with Dirichlet boundary noise. The solutions do not exist in a Sobolev space but in a weighted Sobolev space. For the one-dimensional heat equation with white noise, we show that a numerical scheme that combines Galerkin finite element method in space and discontinuous Galerkin stepping in time converges at an optimal rate.