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We give a short and self-contained proof for rates of convergence of the Allen--Cahn equation towards mean curvature flow, assuming that a classical (smooth) solution to the latter exists and starting from well-prepared initial data. Our approach is based ...

In this paper, we consider the numerical approximation for a phase field model of the coupled two-phase free flow and two-phase porous media flow. This model consists of Cahn--Hilliard--Navier--Stokes equations in the free flow region and Cahn--Hilliard--...

Free boundary problems based on mass conservation and surface tension with application in osmotic swelling are the topic of this contribution. Phase-field approximations of such models are introduced, in order to numerically investigate properties of the ...

We formulate a general shape and topology optimization problem in structural optimization by using a phase field approach. This problem is considered in view of well-posedness, and we derive optimality conditions with minimal regularity assumptions. We ...

The simulation of fluid mixing under the Eulerian framework often suffers from numerical dissipation issues. In this paper, we present a mass-preserving convection scheme that offers direct control on the shape of the interface. The key component of ...

Central in a variational implicit-solvent description of biomolecular solvation is an effective free-energy functional of the solute atomic positions and the solute-solvent interface (i.e., the dielectric boundary). The free-energy functional couples ...

A wavelet analogue of the Ginzburg--Landau energy was recently designed and integrated in variational methods for image processing. In this paper we prove global well-posedness of the gradient descent equation (in the weak sense) and convergence to an ...

We examine the effect of adaptively generated refined meshes on the P¹ - P¹ finite element method with semi-explicit time stepping of part I, which applies to a phase relaxation model with small parameter $\ep>0$. A typical mesh is highly graded in the ...

We study a simple model of phase relaxation which consists of a parabolic PDE for temperature $\theta$ and an ODE with a small parameter $\ep$ and double obstacles for phase variable $\chi$. The model replaces sharp interfaces by diffuse ones and gives ...