Finite element approximations of a phase field model, based on the Cahn-Hilliard equation in the presence of an electric field and kinetics

We consider fully practical finite element approximations of the nonlinear parabolic Cahn-Hilliard system [Mathematical equation appears here. To view, please open pdf attachment] subject to an initial condition u0(.) ∈ [−1, 1] on the conserved order parameter u ∈ [−1, 1], and mixed boundary conditions. Here γ ∈ R>0 is the interfacial parameter, [Symbol appears here. To view, please open pdf attachment]∈ R>0 is a time-scaling parameter, α ∈ R≥0 is the field strength parameter, [Symbol appears here. To view, please open pdf attachment] is the obstacle potential, and c(x, u) and b(x, u) are the diffusion coefficients. Furthermore, w is the chemical potential, φ is the electro-static potential and {v, p} are the velocity and pressure. The system, in the limit γ → 0, models the evolution of an unstable interface between two dielectric media in the presence of an electric field, which is quasi-static, and a Stokes flow for the dielectric media. Our goal is to produce stable fully practical finite element approximations to the phase field model above. Additionally, we would like to reproduce the morphologies observed in studies by Buxton and Clarke in [28], and Kim and Lu in [53, 56]. The presence of the electric field and kinetics should drive the interface growth. Initially restricting ourselves to the case without kinetics, we consider coupled and decoupled finite element approximations of the Cahn-Hilliard system. A coupled system is a non-linear algebraic system where the constituent systems are solved simultaneously at each time-step. A decoupled system splits the constituent systems so that they are solved separately and sequentially. Existence, stability and convergence results are presented for a coupled scheme and numerical results are given in two space dimensions. To develop a computationally efficient approximation we present a decoupled scheme with conditional stability in two space dimensions. Numerical results demonstrate that it is a suitable approximation to the coupled scheme. Introducing kinetics to the system requires the careful consideration of both the boundary conditions and mass conservation of the system. A modified coupled scheme admits existence, stability and convergence results. We investigate the applicability of several fast solution methods for the Stokes system. We also present evidence that the MINI-element for the velocity space is more computationally efficient than the Taylor-Hood element. Using further optimisation techniques, such as solving the Stokes system on a coarser mesh, we are able compute results in three dimensions efficiently. Numerous numerical results are presented in two and three dimensions

Weak Form of Stokes-Dirac Structures and Geometric Discretization of Port-Hamiltonian Systems

We present the mixed Galerkin discretization of distributed parameter port-Hamiltonian systems. On the prototypical example of hyperbolic systems of two conservation laws in arbitrary spatial dimension, we derive the main contributions: (i) A weak formulation of the underlying geometric (Stokes-Dirac) structure with a segmented boundary according to the causality of the boundary ports. (ii) The geometric approximation of the Stokes-Dirac structure by a finite-dimensional Dirac structure is realized using a mixed Galerkin approach and power-preserving linear maps, which define minimal discrete power variables. (iii) With a consistent approximation of the Hamiltonian, we obtain finite-dimensional port-Hamiltonian state space models. By the degrees of freedom in the power-preserving maps, the resulting family of structure-preserving schemes allows for trade-offs between centered approximations and upwinding. We illustrate the method on the example of Whitney finite elements on a 2D simplicial triangulation and compare the eigenvalue approximation in 1D with a related approach.Comment: Copyright 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0

Continuous, Semi-discrete, and Fully Discretized Navier-Stokes Equations

The Navier--Stokes equations are commonly used to model and to simulate flow phenomena. We introduce the basic equations and discuss the standard methods for the spatial and temporal discretization. We analyse the semi-discrete equations -- a semi-explicit nonlinear DAE -- in terms of the strangeness index and quantify the numerical difficulties in the fully discrete schemes, that are induced by the strangeness of the system. By analyzing the Kronecker index of the difference-algebraic equations, that represent commonly and successfully used time stepping schemes for the Navier--Stokes equations, we show that those time-integration schemes factually remove the strangeness. The theoretical considerations are backed and illustrated by numerical examples.Comment: 28 pages, 2 figure, code available under DOI: 10.5281/zenodo.998909, https://doi.org/10.5281/zenodo.99890

On the role of commutator arguments in the development of parameter-robust preconditioners for Stokes control problems

The development of preconditioners for PDE-constrained optimization problems is a field of numerical analysis which has recently generated much interest. One class of problems which has been investigated in particular is that of Stokes control problems, that is the problem of minimizing a functional with the Stokes (or Navier-Stokes) equations as constraints. In this manuscript, we present an approach for preconditioning Stokes control problems using preconditioners for the Poisson control problem and, crucially, the application of a commutator argument. This methodology leads to two block diagonal preconditioners for the problem, one of which was previously derived by W. Zulehner in 2011 (SIAM. J. Matrix Anal. & Appl., v.32) using a nonstandard norm argument for this saddle point problem, and the other of which we believe to be new. We also derive two related block triangular preconditioners using the same methodology, and present numerical results to demonstrate the performance of the four preconditioners in practice

POD model order reduction with space-adapted snapshots for incompressible flows

We consider model order reduction based on proper orthogonal decomposition (POD) for unsteady incompressible Navier-Stokes problems, assuming that the snapshots are given by spatially adapted finite element solutions. We propose two approaches of deriving stable POD-Galerkin reduced-order models for this context. In the first approach, the pressure term and the continuity equation are eliminated by imposing a weak incompressibility constraint with respect to a pressure reference space. In the second approach, we derive an inf-sup stable velocity-pressure reduced-order model by enriching the velocity reduced space with supremizers computed on a velocity reference space. For problems with inhomogeneous Dirichlet conditions, we show how suitable lifting functions can be obtained from standard adaptive finite element computations. We provide a numerical comparison of the considered methods for a regularized lid-driven cavity problem