arXiv subjects

Xianmin Xu

Publications and source records attributed to Xianmin Xu.

26 records · Page 2Linked to original sources

arXiv2017

Sharp-interface limits of a phase-field model with a generalized Navier slip boundary condition for moving contact lines

The sharp-interface limits of a phase-field model with a generalized Navier slip boundary condition for moving contact line problem are studied by asymptotic analysis and numerical simulations. The effects of the {mobility} number as well as a phenomenological relaxation parameter in the boundary condition are considered. In asymptotic analysis, we focus on the case that the {mobility} number is the same order of the Cahn number and derive the sharp-interface limits for several setups of the boundary relaxation parameter. It is shown that the sharp interface limit of the phase field model is the standard two-phase incompressible Navier-Stokes equations coupled with several different slip boundary conditions. Numerical results are consistent with the analysis results and also illustrate the different convergence rates of the sharp-interface limits for different scalings of the two parameters.

math.AP↗

arXiv2017

A trace finite element method for PDEs on evolving surfaces

In this paper, we propose an approach for solving PDEs on evolving surfaces using a combination of the trace finite element method and a fast marching method. The numerical approach is based on the Eulerian description of the surface problem and employs a time-independent background mesh that is not fitted to the surface. The surface and its evolution may be given implicitly, for example, by the level set method. Extension of the PDE off the surface is not required. The method introduced in this paper naturally allows a surface to undergo topological changes and experience local geometric singularities. In the simplest setting, the numerical method is second order accurate in space and time. Higher order variants are feasible, but not studied in this paper. We show results of several numerical experiments, which demonstrate the convergence properties of the method and its ability to handle the case of the surface with topological changes.

math.NA↗

arXiv2016

An efficient threshold dynamics method for wetting on rough surfaces

The threshold dynamics method developed by Merriman, Bence and Osher (MBO) is an efficient method for simulating the motion by mean curvature flow when the interface is away from the solid boundary. Direct generalization of MBO-type methods to the wetting problem with interfaces intersecting the solid boundary is not easy because solving the heat equation in a general domain with a wetting boundary condition is not as efficient as it is with the original MBO method. The dynamics of the contact point also follows a different law compared with the dynamics of the interface away from the boundary. In this paper, we develop an efficient volume preserving threshold dynamics method for simulating wetting on rough surfaces. This method is based on minimization of the weighted surface area functional over an extended domain that includes the solid phase. The method is simple, stable with $O(N \log N)$ complexity per time step and is not sensitive to the inhomogeneity or roughness of the solid boundary.

math.NA↗

arXiv2016

Modified Wenzel and Cassie equations for wetting on rough surfaces

We study a stationary wetting problem on rough and inhomogeneous solid surfaces. We derive a new formula for the apparent contact angle by asymptotic two-scale homogenization method. The formula reduces to a modified Wenzel equation for geometrically rough surfaces and a modified Cassie equation for chemically inhomogeneous surfaces. Unlike the classical Wenzel and Cassie equations, the modified equations correspond to local minimizers of the total interface energy in the solid-liquid-air system, so that they are consistent with experimental observations. The homogenization results are proved rigorously by a variational method.

math.AP↗

arXiv2015

Volterra-composition operators on the weighted Bergman space with exponential type weights

The properties of Volterra-composition operators on the weighted Bergman space with exponential type weights are investigated in this paper. We state some necessary and sufficient conditions that a Volterra-composition operator from the weighted Bergman space to Bloch type space ( or little Bloch type space ) must satisfy for the Volterra-composition operator to be bounded or compact.

math.FA↗

arXiv2014

An Eulerian space-time finite element method for diffusion problems on evolving surfaces

In this paper, we study numerical methods for the solution of partial differential equations on evolving surfaces. The evolving hypersurface in $\Bbb{R}^d$ defines a $d$-dimensional space-time manifold in the space-time continuum $\Bbb{R}^{d+1}$. We derive and analyze a variational formulation for a class of diffusion problems on the space-time manifold. For this variational formulation new well-posedness and stability results are derived. The analysis is based on an inf-sup condition and involves some natural, but non-standard, (anisotropic) function spaces. Based on this formulation a discrete in time variational formulation is introduced that is very suitable as a starting point for a discontinuous Galerkin (DG) space-time finite element discretization. This DG space-time method is explained and results of numerical experiments are presented that illustrate its properties.

math.NA↗

arXiv2013

A stabilized finite element method for advection-diffusion equations on surfaces

A recently developed Eulerian finite element method is applied to solve advection-diffusion equations posed on hypersurfaces. When transport processes on a surface dominate over diffusion, finite element methods tend to be unstable unless the mesh is sufficiently fine. The paper introduces a stabilized finite element formulation based on the SUPG technique. An error analysis of the method is given. Results of numerical experiments are presented that illustrate the performance of the stabilized method.

math.NA↗

arXiv2013

On surface meshes induced by level set functions

The zero level set of a piecewise-affine function with respect to a consistent tetrahedral subdivision of a domain in $\mathbb{R}^3$ is a piecewise-planar hyper-surface. We prove that if a family of consistent tetrahedral subdivions satisfies the minimum angle condition, then after a simple postprocessing this zero level set becomes a consistent surface triangulation which satisfies the maximum angle condition. We treat an application of this result to the numerical solution of PDEs posed on surfaces, using a $P_1$ finite element space on such a surface triangulation. For this finite element space we derive optimal interpolation error bounds. We prove that the diagonally scaled mass matrix is well-conditioned, uniformly with respect to $h$. Furthermore, the issue of conditioning of the stiffness matrix is addressed.

math.NA↗