Solution verification, goal-oriented adaptive methods for stochastic advection–diffusion problems

A goal-oriented analysis of linear, stochastic advection–diffusion models is presented which provides both a method for solution verification as well as a basis for improving results through adaptation of both the mesh and the way random variables are approximated. A class of model problems with random coefficients and source terms is cast in a variational setting. Specific quantities of interest are specified which are also random variables. A stochastic adjoint problem associated with the quantities of interest is formulated and a posteriori error estimates are derived. These are used to guide an adaptive algorithm which adjusts the sparse probabilistic grid so as to control the approximation error. Numerical examples are given to demonstrate the methodology for a specific model problem.