25 research outputs found
On the rate of convergence of alternating minimization for non-smooth non-strongly convex optimization in Banach spaces
In this paper, the convergence of the fundamental alternating minimization is established for non-smooth non-strongly convex optimization problems in Banach spaces, and novel rates of convergence are provided. As objective function a composition of a smooth, and a block-separable, non-smooth part is considered, covering a large range of applications. For the former, three different relaxations of strong convexity are considered: (i) quasi-strong convexity; (ii) quadratic functional growth; and (iii) plain convexity. With new and improved rates benefiting from both separate steps of the scheme, linear convergence is proved for (i) and (ii), whereas sublinear convergence is showed for (iii).publishedVersio
Well-posedness analysis of the Cahn-Hilliard-Biot model
We investigate the well-posedness of the recently proposed Cahn-Hilliard-Biot
model. The model is a three-way coupled PDE of elliptic-parabolic nature, with
several nonlinearities and the fourth order term known to the Cahn-Hilliard
system. We show existence of weak solutions to the variational form of the
equations and uniqueness under certain conditions of the material parameters
and secondary consolidation, adding regularizing effects. Existence is shown by
discretizing in space and applying ODE-theory (the Peano-Cauchy theorem) to
prove existence of the discrete system, followed by compactness arguments to
retain solutions of the continuous system. In addition, the continuous
dependence of solutions on the data is established, in particular implying
uniqueness. Both results build strongly on the inherent gradient flow structure
of the model
A Cahn-Hilliard-Biot system and its generalized gradient flow structure
In this work, we propose a new model for flow through deformable porous media, where the solid material has two phases with distinct material properties. The two phases of the porous material evolve according to a generalized Ginzburg–Landau energy functional, with additional impact from both elastic and fluid effects, and the coupling between flow and deformation is governed by Biot’s theory. This results in a three-way coupled system which can be seen as an extension of the Cahn–Larché equations with the inclusion of a fluid flowing through the medium. The model covers essential coupling terms for several relevant applications, including solid tumor growth, biogrout, and wood growth simulation. Moreover, we show that this coupled set of equations follow a generalized gradient flow framework. This opens a toolbox of analysis and solvers which can be used for further study of the model. Additionally, we provide a numerical example showing the impact of the flow on the solid phase evolution in comparison to the Cahn–Larché system.publishedVersio
Efficient Solvers for Nonstandard Models for Flow and Transport in Unsaturated Porous Media
We study several iterative methods for fully coupled flow and reactive transport in porous media. The resulting mathematical model is a coupled, nonlinear evolution system. The flow model component builds on the Richards equation, modified to incorporate nonstandard effects like dynamic capillarity and hysteresis, and a reactive transport equation for the solute. The two model components are strongly coupled. On one hand, the flow affects the concentration of the solute; on the other hand, the surface tension is a function of the solute, which impacts the capillary pressure and, consequently, the flow. After applying an Euler implicit scheme, we consider a set of iterative linearization schemes to solve the resulting nonlinear equations, including both monolithic and two splitting strategies. The latter include a canonical nonlinear splitting and an alternate linearized splitting, which appears to be overall faster in terms of numbers of iterations, based on our numerical studies. The (time discrete) system being nonlinear, we investigate different linearization methods. We consider the linearly convergent L-scheme, which converges unconditionally, and the Newton method, converging quadratically but subject to restrictions on the initial guess. Whenever hysteresis effects are included, the Newton method fails to converge. The L-scheme converges; nevertheless, it may require many iterations. This aspect is improved by using the Anderson acceleration. A thorough comparison of the different solving strategies is presented in five numerical examples, implemented in MRST, a toolbox based on MATLAB.publishedVersio