This paper concerns the theoretical and numerical analysis of a free boundary problem for the Laplace equation, with a curvature condition on the free boundary. This boundary is described as the graph of a function, and contact angles are imposed at the moving contact points. The equations are set in the framework of classical Sobolev Banach spaces, and existence and uniqueness of the solution are proved via a fixed-point iteration, exploiting a suitably defined lifting operator from the free boundary. The free-boundary function and the bulk solution are approximated by piecewise linear finite elements, and the well-posedness and convergence of the discrete problem are proved. This proof hinges upon a stability result for the Riesz projection onto the discrete space, which is separately proven and has an interest per se.
Numerical Approximations for a Phase-Field Moving Contact Line Model with Variable Densities and Viscosities
NCMIS & LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science（计算数学与科学工程计算研究所，中国科学院数学与系统科学研究院）；Department of Mathematics, University of South Carolina（南卡罗来纳大学数学系）
A structure--preserving ALE--BGN--MDR method for Navier--Stokes free boundary problems with moving contact lines and gravity
Universität Regensburg（雷根斯堡大学）；The Hong Kong Polytechnic University（香港理工大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
Decoupled, Energy Stable Scheme for Hydrodynamic Allen-Cahn Phase Field Moving Contact Line Model
Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）；School of Science, Beijing University of Posts and Telecommunications（北京邮电大学理学院）；Department of Mathematics, University of South Carolina（南卡罗来纳大学数学系）；School of Mathematical Sciences, Beijing Normal University（北京师范大学数学科学学院）；Laboratory of Mathematics and Complex Systems, Ministry of Education（教育部数学与复杂系统重点实验室）
Efficient Second Order Unconditionally Stable Schemes for a Phase-field Moving Contact Line Model Using Invariant Energy Quadratization Approach
University of South Carolina（南卡罗来纳大学）；University of Electronic Science and Technology of China（电子科技大学）；Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所）；University of Chinese Academy of Sciences（中国科学院大学）
Primal-dual splitting methods for phase-field surfactant model with moving contact lines
School of Mathematical Sciences, University of Electronic Science and Technology of China（电子科技大学数学科学学院）；Department of Mathematics, National Center for Applied Mathematics (Shenzhen), Southern University of Science and Technology（南方科技大学应用数学中心（深圳）数学系）
