Christian Kahle

Jessica Bosch, Christian Kahle, Martin Stoll

Recently, Garcke et al.[Garcke, Hinze, Kahle, A stable and linear time discretization for a thermodynamically consistent model for two-phase incompressible flow, Applied Numerical Mathematics 99, pp. 151-171, 2016] developed a consistent discretization scheme for a thermodynamically consistent diffuse interface model for incompressible two-phase flows with different densities. At the heart of this method lies the solution of large and sparse linear systems that arise in a semismooth Newton method. We propose the use of preconditioned Krylov subspace solvers using effective Schur complement approximations. Numerical results illustrate the efficiency of our approach. In particular, our preconditioner is shown to be robust with respect to parameter changes.

Christian Kahle, Kei Fong Lam, Jonas Latz et al.

We consider the inverse problem of parameter estimation in a diffuse interface model for tumour growth. The model consists of a fourth-order Cahn-Hilliard system and contains three phenomenological parameters: the tumour proliferation rate, the nutrient consumption rate, and the chemotactic sensitivity. We study the inverse problem within the Bayesian framework and construct the likelihood and noise for two typical observation settings. One setting involves an infinite-dimensional data space where we observe the full tumour. In the second setting we observe only the tumour volume, hence the data space is finite-dimensional. We show the well-posedness of the posterior measure for both settings, building upon and improving the analytical results in [C. Kahle and K.F. Lam, Appl. Math. Optim. (2018)]. A numerical example involving synthetic data is presented in which the posterior measure is numerically approximated by the sequential Monte Carlo approach with tempering.

Dominik Hafemeyer, Christian Kahle, Johannes Pfefferer

This work is concerned with quasi-optimal a-priori finite element error estimates for the obstacle problem in the $L^2$-norm. The discrete approximations are introduced as solutions to a finite element discretization of an accordingly regularized problem. The underlying domain is only assumed to be convex and polygonally or polyhedrally bounded such that an application of point-wise error estimates results in a rate less than two in general. The main ingredient for proving the quasi-optimal estimates is the structural and commonly used assumption that the obstacle is inactive on the boundary of the domain. Then localization techniques are used to estimate the global $L^2$-error by a local error in the inner part of the domain, where higher regularity for the solution can be assumed, and a global error for the Ritz-projection of the solution, which can be estimated by standard techniques. We validate our results by numerical examples.

Christian Kahle

Phase field models are widely used to describe multiphase systems. Here a smooth indicator function, called phase field, is used to describe the spatial distribution of the phases under investigation. Material properties like density or viscosity are introduced as given functions of the phase field. These parameters typically have physical bounds to fulfil, e.g. positivity of the density. To guarantee these properties, uniform bounds on the phase field are of interest. In this work we derive a uniform bound on the solution of the Cahn--Hilliard system, where we use the double-obstacle free energy, that is relaxed by Moreau--Yosida relaxation.

Stefan Dohr, Christian Kahle, Sergejs Rogovs et al.

We study the numerical approximation of linear-quadratic optimal control problems subject to the fractional Laplace equation with its spectral definition. We compute an approximation of the state equation using a discretization of the Balakrishnan formula that is based on a finite element discretization in space and a sinc quadrature approximation of the additionally involved integral. A tailored approach for the numerical solution of the resulting linear systems is proposed. Concerning the discretization of the optimal control problem we consider two schemes. The first one is the variational approach, where the control set is not discretized, and the second one is the fully discrete scheme where the control is discretized by piecewise constant functions. We derive finite element error estimates for both methods and illustrate our results by numerical experiments.

Harald Garcke, Michael Hinze, Christian Kahle

We propose to model physical effects at the sharp density interface between atmosphere and ocean with the help of diffuse interface approaches for multiphase flows with variable densities. We use the variable-density model proposed in \cite{m6:AbelsGarckeGruen_CHNSmodell}. This results in a Cahn-Hilliard/Navier-Stokes type system which we complement with tangential Dirichlet boundary conditions to incorporate the effect of wind in the atmosphere. Wind is responsible for waves at the surface of the ocean, whose dynamics have an important impact on the $CO_2-$exchange between ocean and atmosphere. We tackle this mathematical model numerically with fully adaptive and integrated numerical schemes tailored to the simulation of variable density multiphase flows governed by diffuse interface models. Here, {\it fully adaptive, integrated, efficient, and reliable} means that the mesh resolution is chosen by the numerical algorithm according to a prescribed error tolerance in the {\it a posteriori} error control on the basis of residual-based error indicators, which allow to estimate the true error from below (efficient) and from above (reliable). Our approach is based on the work of \cite{m6:HintermuellerHinzeKahle_adaptiveCHNS,m6:GarckeHinzeKahle_CHNS_AGG_linearStableTimeDisc}, where a fully adaptive efficient and reliable numerical method for the simulation of two-dimensional multiphase flows with variable densities is developed. We incorporate the stimulation of surface waves via appropriate boundary conditions.