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Subsurface flows are commonly modeled by advection-diffusion equations.Insufficient measurements or uncertain material procurement may be accountedfor by random coefficients. To represent, for example, transitions inheterogeneous media, the parameters of the equation are spatiallydiscontinuous. Specifically, a scenario with coupled advection- and diffusioncoefficients that are modeled as sums of continuous random fields anddiscontinuous jump components are considered. For the numerical approximationof the solution, an adaptive, pathwise discretization scheme based on a FiniteElement approach is introduced. To stabilize the numerical approximation andaccelerate convergence, the discrete space-time grid is chosen with respect tothe varying discontinuities in each sample of the coefficients, leading to astochastic formulation of the Galerkin projection and the Finite Element basis.

Numerical analysis for time-dependent advection-diffusion problems with random discontinuous coefficients