Heiko Berninger

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

We derive and analyze 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 nonstandard quadrature points. Our approach allows for nonlinear outflow or seepage boundary conditions of Signorini type. We show convergence of the saturation and, in the nondegenerate 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

On Nonlinear Dirichlet—Neumann Algorithms for Jumping Nonlinearities

L^2

Unsaturated subsurface flow with surface water and nonlinear in- and outflow conditions

-convergence rates of order

Non-overlapping Domain Decomposition for the Richards Equation via Superposition Operators

\mathcal{O}(h^2)

Substructuring of a Signorini-type problem and Robin’s method for the Richards equation in heterogeneous soil

and

Convergence Behaviour of Dirichlet–Neumann and Robin Methods for a Nonlinear Transmission Problem

H^1

Derivation of the Isotropic Diffusion Source Approximation (IDSA) for Supernova Neutrino Transport by Asymptotic Expansions

-convergence rates of order

A multidomain discretization of the Richards equation in layered soil

\mathcal{O}(h)

The 2-Lagrange multiplier method applied to nonlinear transmission problems for the Richards equation in heterogeneous soil with cross points

as well as robust convergence behavior of the multigrid method with respect to extreme choices of soil parameters.

Heterogeneous Substructuring Methods for Coupled Surface and Subsurface Flow

We consider a quasilinear elliptic transmission problem where the nonlinearity changes discontinuously across two subdomains. By a reformulation of the problem via Kirchhoff transformation we first obtain linear problems on the subdomains together with nonlinear transmission conditions and then a nonlinear Steklov– Poincare interface equation. We introduce a Dirichlet–Neumann iteration for this problem and prove convergence to a unique solution in one space dimension. Finally we present numerical results in two space dimensions suggesting that the algorithm can be applied successfully in more general cases.