Multi-algorithmic Methods for Coupled Hyperbolic-parabolic Problems

This work is motivated by the study of flow and transport phenomena in highly heterogeneous porous media, an important application in the petroleum and environmental industries. An appealing technique for handling such phenomena is the use of a multi-algorithmic strategy based on the decomposition of the spatial domain into multiple non-overlapping subdomains according to the geological, physical and chemical properties of the medium. This promotes the use of a different scheme within each subdomain in order to reduce computational expenses while preserving accuracy. The resulting numerical models are consistent with the underlying equations on the subdomains and physically meaningful conditions are imposed on interfaces between the subdomains. Examples of such domain decomposition approaches include the mortar finite element method employing Lagrange multipliers to weakly impose flux-matching across interfaces [1, 2] and related multi-block multi-physics techniques [3, 4]. We consider the specific case of advective-diffusive transport of a chemical species within strongly contrasting geological layers where the resulting diffusion coefficient varies spatially. As a model problem, we investigate advection-diffusion equations where diffusion is locally degenerate within the computational domain, leading to a coupled hyperbolic-parabolic problem. This situation lends itself to the use of domain-decomposition type coupled continuous Galerkin (CG) and discontinuous Galerkin (DG) methods where the strengths of each method are exploited within