Numerical solution of partial differential equations with random coefficients: a stochastic finite element approach

Mathematical models of engineering systems and physical processes typically take the form of a partial differential equation (PDE). Variability or uncertainty on coefficients of a PDE can be expressed by introducing random variables, random fields or random processes into the PDE . Recently developed stochastic finite element methods enable the construction of high-order accurate solutions of a stochastic PDE , while reducing the high computational cost of more standard uncertainty quantification methods, such as the Monte Carlo simulation method. In this talk, we outline the methodology of the two main variants of the stochastic finite element method, i.e., the stochastic collocation and the stochastic Galerkin method. Both methods transform a stochastic PDE into a system of deterministic PDEs. In the latter case, the number of deterministic PDEs is generally smaller than for the stochastic collocation method, but the system is more complicated to solve. We will point out how these methods can be applied to efficiently solve a small flow problem through random porous media. Multigrid techniques lead to a fast solution of the resulting systems. These simulations can be extended to solve groundwater flow problems or to perform oil reservoir simulations.status: publishe