Heterogeneous Substructuring Methods for Coupled Surface and Subsurface Flow

The exchange of groundand surface water plays a crucial role in a variety of practically relevant processes ranging from flood protection measures to preservation of ecosystem health in natural and human-impacted water resources systems. Commonly accepted models are based on the shallow water equations for overland flow and the Richards equation for saturated–unsaturated subsurface flow with suitable coupling conditions. Continuity of mass flow across the interface is natural, because it directly follows from mass conservation. Continuity of pressure is typically imposed for simplicity. Mathematically, this makes sense for sufficiently smooth height of surface water as occurring, e.g., in filtration processes [8, 13]. Here we impose Robin-type coupling conditions modelling a thin, nearly impermeable layer at the bottom of the river bed that may cause pressure discontinuities; an effect which is known in hydrology as clogging (see [15] or [7, p. 1376]). From a mathematical perspective, clogging can be regarded as a kind of regularization, because, in contrast to Dirichlet conditions, Robin conditions can be straightforwardly formulated in a weak sense. Existence and uniqueness results for the Richards equation and the shallow water equations are rare and hard to obtain, and nothing seems to be known about solvability of coupled problems. Extending the general framework of heterogeneous Steklov–Poincaré formulations and iterative substructuring [9, 12] to time-dependent problems, we introduce a Robin–Neumann iteration for the continuous coupled problem and motivate its feasibility by well-known existence results for the linear case. As surface and subsurface