Two-grid finite element methods for the steady Navier-Stokes/Darcy model are considered. Stability and optimal error estimates in the H1-norm for velocity and piezometric approximations and the L2-norm for pressure are established under mesh sizes satisfying h = H2. A modified decoupled and linearised two-grid algorithm is developed, together with some associated optimal error estimates. Our method and results extend and improve an earlier investigation, and some numerical computations illustrate the efficiency and effectiveness of the new algorithm.

Models of two phase flows in porous media, used in petroleumengineering, lead to a system of two coupled equations with ellipticand parabolic degenerate terms, and two unknowns,the saturation and the pressure.For the purpose of their approximation, a coupled scheme, consisting ina finite volume method together witha phase-by-phase upstream weighting scheme, is used in the industrial setting.This paper presents a mathematical analysis of this coupled scheme, first showingthat it satisfies some a priori estimates:the saturation is shown to remain in a fixed interval, anda discrete L2(0,T;H1 (Ω)) estimate is proved for both the pressureand a function of the saturation. Thanks to these properties,a subsequence of the sequence of approximate solutions is shown toconverge to a weak solutionof the continuous equationsas the size of the discretization tends to zero.

Mires are terrestrial ecosystems which conserve organic production in the form of peat because their soils are waterlogged. Scotland's damp climate makes it rich in mires, with fens, raised mires and blanket mires occurring widely. Intact examples are mostly treeless due to wind exposure. They have two functional layers of soil: a shallow acrotelm overlying the catotelm or peat deposit proper. The acrotelm is the main region of diagenesis and contains the water table, which lies close to the surface for most of the time. Evapotranspiration depends on water table depth and surface vegetation. In the catotelm, water transmission is described by Darcy's law with sufficient exactness to allow prediction of the water table profile. An analysis of seepage using Dupuit–Forchheimer theory predicts a hemi-elliptical profile whose curvature is governed by the water balance in dry years and which, in its turn, closely controls the overall shape of the mire. Water flow is more rapid in the acrotelm, where it creates a regular series of different soil physical regimes which are linked with distinct combinations of surface topography and vegetation arranged sequentially from the centre of a mire to its edge. Such regular arrangements are most clearly seen in the Flowe Country of E Sutherland and Caithness. Acrotelm structure also permits variation in flow rate with depth, so that the amplitude of water table oscillations is narrowly confined and the mire thus protected against both desiccation and sheet flow.