SSFEM : An Introduction to the Spectral Stochastic Finite Element Method

Abstract:

Fluid flow through porous media, and other physical phenomena, are described mathematically using partial differential equations (PDEs). Simulations are typically performed by discretising the PDEs using finite element methods, which approximate the unknown quantities (eg fluid velocity) using piecewise defined polynomials on a mesh of the computational domain. For simplicity, inputs such as porosity coefficients and other material properties, boundary conditions and/or source terms are often assumed to be known exactly. In reality, this is hardly ever the case. For example, the porosity coefficients of rocks in a given computational domain are never explicitly known at every point in space. Although solving the so-called deterministic equations usually provides valuable information, it is more appropriate to model an input that is based on a sample of field measurements using random variables with given statistical properties. For example, they might be generated from Gaussian random fields with stated mean and variance. In this way, the solution of the problem itself becomes a random variable and we require an approximation to its expected value in a single experiment rather than performing multiple realisations with different inputs.

This is the idea behind so-called stochastic finite element methods. The unknown quantities are essentially discretised in physical space and probability space, leading to linear systems with particular structures that differ significantly from those arising in deterministic models. The system size in particular (very large!) is often considered as problematic. This talk will be a user-friendly one and will focus on computational issues. For some very simple problems, I'll give details about the system assembly, numerical linear algebra, data storage and preconditioning and show that the solution of the large, structured algebraic systems of equations is not problematic.