Two-dimensional finite element method solution of a class of integro-differential equations: Application to non-Fickian transport in disordered media

A finite element method is developed to solve a class of integro-differential equations and demonstrated for the important specific problem of non-Fickian contaminant transport in disordered porous media. This transient transport equation, derived from a continuous time random walk approach, includes a memory function. An integral element is the incorporation of the well-known sum-of-exponential approximation of the kernel function, which allows a simple recurrence relation rather than storage of the entire history. A two-dimensional linear element is implemented, including a streamline upwind Petrov-Galerkin weighting scheme. The developed solver is compared with an analytical solution in the Laplace domain, transformed numerically to the time domain, followed by a concise convergence assessment. The analysis shows the power and potential of the method developed here. Copyright (C) 2017 John Wiley & Sons, Ltd.