Corrector estimates for the homogenization of a locally-periodic medium with areas of low and high diffusivity

We prove an upper bound for the convergence rate of the homogenization limit $\epsilon\to 0$ for a linear transmission problem for a advection-diffusion(-reaction) system posed in areas with low and high diffusivity, where $\epsilon$ is a suitable scale parameter. On this way, we justify the formal homogenization asymptotics obtained by us earlier by proving an upper bound for the convergence rate (a corrector estimate). The main ingredients of the proof of the corrector estimate include integral estimates for rapidly oscillating functions with prescribed average, properties of the macroscopic reconstruction operators, energy bounds and extra two-scale regularity estimates. The whole procedure essentially relies on a good understanding of the analysis of the limit two-scale problem.Comment: 19 pages, 1 figur

Improving Pseudo-Time Stepping Convergence for CFD Simulations With Neural Networks

Computational fluid dynamics (CFD) simulations of viscous fluids described by the Navier-Stokes equations are considered. Depending on the Reynolds number of the flow, the Navier-Stokes equations may exhibit a highly nonlinear behavior. The system of nonlinear equations resulting from the discretization of the Navier-Stokes equations can be solved using nonlinear iteration methods, such as Newton's method. However, fast quadratic convergence is typically only obtained in a local neighborhood of the solution, and for many configurations, the classical Newton iteration does not converge at all. In such cases, so-called globalization techniques may help to improve convergence. In this paper, pseudo-transient continuation is employed in order to improve nonlinear convergence. The classical algorithm is enhanced by a neural network model that is trained to predict a local pseudo-time step. Generalization of the novel approach is facilitated by predicting the local pseudo-time step separately on each element using only local information on a patch of adjacent elements as input. Numerical results for standard benchmark problems, including flow through a backward facing step geometry and Couette flow, show the performance of the machine learning-enhanced globalization approach; as the software for the simulations, the CFD module of COMSOL Multiphysics is employed

Computing a partial real ordered generalized Schur form using the Jacobi-Davidson method

Abstract. In this paper, an adapted variant of the Jacobi-Davidson method is presented that is specifically designed for real unsymmetric matrix pencils. Whenever a pencil has a complex conjugated pair of eigenvalues, the method computes the two dimensional real invariant subspace spanned by the two corresponding complex conjugated eigenvectors. This is beneficial for memory costs and in many cases it also accelerates the convergence of the JD method. In numerical experiments, the RJDQZ variant is compared with the original JDQZ method. 1. Introduction. Rea

Crystal precipitation and dissolution in a porous medium : effective equations and numerical experiments

We investigate a two-dimensional microscale model for crystal dissolution and precipitation in a porous medium. The model contains a free boundary and allows for changes in the pore volume. Using a level set formulation of the free boundary, we apply a formal homogenization procedure to obtain upscaled equations. For general microscale geometries, the homogenized model that we obtain falls in the class of distributed microstructure models. For circular initial inclusions the distributed microstructure model reduces to a system of partial differential equations coupled with an ordinary differential equation. In order to investigate how well the upscaled equations describe the behavior of the microscale model, we perform numerical computations for a test problem. The numerical simulations show that for the test problem the solution of the homogenized equations agrees very well with the averaged solution of the microscale model

Crystal precipitation and dissolution in a porous medium: Effective equations and numerical experiments

We investigate a two-dimensional microscale model for crystal dissolution and precipitation in a porous medium. The model contains a free boundary and allows for changes in the pore volume. Using a level set formulation of the free boundary, we apply a formal homogenization procedure to obtain upscaled equations. For general microscale geometries, the homogenized model that we obtain falls in the class of distributed microstructure models. For circular initial inclusions the distributed microstructure model reduces to a system of partial differential equations coupled with an ordinary differential equation. In order to investigate how well the upscaled equations describe the behavior of the microscale model, we perform numerical computations for a test problem. The numerical simulations show that for the test problem the solution of the homogenized equations agrees very well with the averaged solution of the microscale model

Corrector estimates for the homogenization of a locally periodic medium with areas of low and high diffusivity

We prove an upper bound for the convergence rate of the homogenization limit ε → 0 for a linear transmission problem for a advection–diffusion(–reaction) system posed in areas with low and high diffusivity, where ε is a suitable scale parameter. In this way we rigorously justify the formal homogenization asymptotics obtained in [37] (van Noorden, T. and Muntean, A. (2011) Homogenization of a locally-periodic medium with areas of low and high diffusivity. Eur. J. Appl. Math. 22, 493–516). We do this by providing a corrector estimate. The main ingredients for the proof of the correctors include integral estimates for rapidly oscillating functions with prescribed average, properties of the macroscopic reconstruction operators, energy bounds, and extra two-scale regularity estimates. The whole procedure essentially relies on a good understanding of the analysis of the limit two-scale problem

Phase field approximation of a kinetic moving-boundary problem modelling dissolution and precipitation

We present a phase field model which approximates a one-phase Stefan-like problem with a kinetic condition at the moving boundary, and which models a dissolution and precipitation reaction. The concentration of dissolved particles is variable on one side of the free boundary and jumps across the free boundary to a fixed value given by the constant concentration of the particles in the precipitate. Using a formal asymptotic analysis we show that the phase field model approximates the appropriate sharp interface limit. The existence and uniqueness of solutions to the phase field model is studied. By numerical experiments the approximating behaviour of the phase field model is investigated

Phase field approximation of a kinetic moving-boundary problem modelling dissolution and precipitation

We present a phase field model which approximates a one-phase Stefan-like problem with a kinetic condition at the moving boundary, and which models a dissolution and precipitation reaction. The concentration of dissolved ions is variable on one side of the free boundary and jumps across the free boundary to a fixed value given by the constant density of the precipitate. Using a formal asymptotic analysis we show that the phase field model approximates the appropriate sharp interface limit. The existence and uniqueness of solutions to the phase-field model is studied. By numerical experiments the approximating behaviour of the phase-field model is investigated

Computing a partial generalized real Schur form using the Jacobi-Davidson method

In this paper, a new variant of the Jacobi-Davidson (JD) method is presented that is specifically designed for real unsymmetric matrix pencils. Whenever a pencil has a complex conjugate pair of eigenvalues, the method computes the two-dimensional real invariant subspace spanned by the two corresponding complex conjugated eigenvectors. This is beneficial for memory costs and in many cases it also accelerates the convergence of the JD method. Both real and complex formulations of the correction equation are considered. In numerical experiments, the RJDQZ variant is compared with the original JDQZ method