Fast and Robust Numerical Solution of the Richards Equation in Homogeneous Soil

Abstract

Abstract. We derive and analyse a solver-friendly finite element discretization of a time discrete Richards equation based on Kirchhoff transformation. It can be interpreted as a classical finite element discretization in physical variables with non-standard quadrature points. Our approach allows for nonlinear outflow or seepage boundary conditions of Signorini type. We show convergence of the saturation and, in the non-degenerate case, of the discrete physical pressure. The associated discrete algebraic problems can be formulated as discrete convex minimization problems and, therefore, can be solved efficiently by monotone multigrid methods. In numerical examples for two and three space dimensions we observe L 2 -convergence rates of order O(h 2 ) and H 1 -convergence rates of order O(h) as well as robust convergence behaviour of the multigrid method with respect to extreme choices of soil parameters.

Numerical Solution of Richards' Equation: A Review of Advances and Challenges

“…For example, rather than attempt to extend or modify existing preconditioners like geometric and algebraic multigrid to the linearized systems arising from standard discretizations of Richards' equation, the work of Berninger et al (2011) takes an alternate view and modifies the underlying discretization to realize the performance of multigrid for classical linear self-adjoint problems. That is, introducing a Kirchoff (or matric flux potential) transform for homogeneous media (Williams et al, 2000) ( ) ( ) u K p dp…”

Adaptive Modelling of Coupled Hydrological Processes with Application in Water Management

“…Then, since M : R → R is an increasing function, the weak form of Problem (6) allows an equivalent formulation as a uniquely solvable convex minimization problem on a convex subset of H 1 (Ω p ). Using linear finite elements, we construct a convergent discretization of (6), which is meaningful also in the physical variable p. It can be solved efficiently and robustly by monotone multigrid methods [3]. In layered heterogeneous soils, different functions θ i (·) and kr i (·), i = 1, .…”

A polynomial chaos approach to stochastic variational inequalities

“…Note that the growth condition in (2.4) is satisfied by piecewise quadratic functions occurring, e.g., in the enthalpy formulation of the Stefan problem [14], or by the generalized saturation resulting from Kirchhoff tranformation of the Richards equation describing saturated/unsaturated groundwater flow [5].…”

“…The solution u(x, ω) is illustrated in Figure 4. Related problems typically arise from time discretization of Richards equation modeling saturated/unsaturated groundwater flow, cf., e.g., [5] A parameteric deterministic formulation (3.10) is obtained by replacing the random variables ξ = (ξ 1 , ξ 2 ) by coordinates y = (y 1 , y 2 ) ∈ I = R 2 . The probability density P ξ = pdf 2 (y 1 )pdf 2 (y 2 ) on I is obtained from the standard normal distribution pdf i (y i ) = (2π) −1/2 exp(−y 2 i /2).…”