Volume 16, Issue 4

This paper formulates and analyzes a weak solution to the coupling of time-dependent Navier–Stokes flow with Darcy flow under certain boundary conditions, one of them being the Beaver–Joseph–Saffman law on the interface. Existence and a priori estimates for the weak solution are shown under additional regularity assumptions. We introduce a fully discrete scheme with the unknowns being the Navier–Stokes velocity, pressure and the Darcy pressure. The scheme we propose is based on a finite element method in space and a Crank–Nicolson discretization in time where we obtain the solution at the first time step using a first order backward Euler method. Convergence of the scheme is obtained and optimal error estimates with respect to the mesh size are derived.

In this paper we present a balancing domain decomposition method of Neumann–Neumann type for the numerical approximation of an age-structured population diffusing in a multilayer environment. We give a quasi optimal convergence result for the algorithm via Fourier analysis in a two-layers case, and we address the variational generalization to an arbitrary number of layers as well as its finite dimensional formulation. We illustrate our approach with some numerical results.

We develop an adaptive finite element concept for elliptic optimal control problems with constraints on the state. In order to generate goal-oriented meshes, we extend the Dual Weighted Residual (DWR) concept proposed by Becker and Rannacher for PDE-constrained optimization to the state constrained case. Using the optimality system of the underlying PDE-constrained optimal control problem we obtain a representation for the error in the objectives. Based on this representation we define local error indicators, whose performance we investigate by means of a numerical example.