A formal finite element approach for open boundaries in transport and diffusion ground‐water problems

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Abstract
In the application of the finite element method to diffusion and convection‐dispersion equations over a ground‐water domain, the Galerkin technique was used to incorporate Neumann (or second‐type) and Cauchy (or third‐type) boundary conditions. While mass movement through open boundaries is a priori unknown, these boundaries are usually treated as a zero Neumann condition at some far distance from the domain of interest. Nevertheless, cheaper and better solutions can be obtained if these unknown conditions are adequately incorporated in the weak formulation and in the transient solution schemes (open boundary condition). Theoretical and numerical proofs are given of the equivalences between this approach and a ‘well‐posed’ problem in a semi‐infinite domain with a zero Neumann condition at a boundary placed at infinity. Transport and diffusion equations were applied in one dimension to show the numerical performances and limitations of this procedure for some linear and non‐linear problems. No a priori limitations are foreseen in order to find similar solutions in two or three dimensions. Thus the spatial discretization in the proximity of open boundaries could be drastically reduced to the domain of interest.