The authors establish error estimates for recently developed finite-element methods for incompressible viscous flow in domains with no-slip boundary conditions.The methods arise by discretization of a well-posed exten...The authors establish error estimates for recently developed finite-element methods for incompressible viscous flow in domains with no-slip boundary conditions.The methods arise by discretization of a well-posed extended Navier-Stokes dynamics for which pressure is determined from current velocity and force fields.The methods use C1 elements for velocity and C0 elements for pressure.A stability estimate is proved for a related finite-element projection method close to classical time-splitting methods of Orszag,Israeli,DeVille and Karniadakis.展开更多
Despite important advances in the mathematical analysis of the Euler equations for water waves,especially over the last two decades,it is not yet known whether local singularities can develop from smooth data in well-...Despite important advances in the mathematical analysis of the Euler equations for water waves,especially over the last two decades,it is not yet known whether local singularities can develop from smooth data in well-posed initial value problems.For ideal free-surface flow with zero surface tension and gravity,the authors review existing works that describe"splash singularities",singular hyperbolic solutions related to jet formation and"flip-through",and a recent construction of a singular free surface by Zubarev and Karabut that however involves unbounded negative pressure.The authors illustrate some of these phenomena with numerical computations of 2D flow based upon a conformal mapping formulation.Numerical tests with a different kind of initial data suggest the possibility that corner singularities may form in an unstable way from specially prepared initial data.展开更多
基金Project supported by the National Science Foundation (Nos.DMS 06-04420 (RLP),DMS 08-11177(JGL))the Center for Nonlinear Analysis (CNA) under National Science Foundation Grant (Nos.0405343,0635983)
文摘The authors establish error estimates for recently developed finite-element methods for incompressible viscous flow in domains with no-slip boundary conditions.The methods arise by discretization of a well-posed extended Navier-Stokes dynamics for which pressure is determined from current velocity and force fields.The methods use C1 elements for velocity and C0 elements for pressure.A stability estimate is proved for a related finite-element projection method close to classical time-splitting methods of Orszag,Israeli,DeVille and Karniadakis.
基金supported by the National Science Foundation under NSF Research Network Grant RNMS11-07444(KI-Net)the NSF Grants DMS-1514826,DMS-1812573,DMS-1515400,DMS-1812609the Simons Foundation under Grant 395796
文摘Despite important advances in the mathematical analysis of the Euler equations for water waves,especially over the last two decades,it is not yet known whether local singularities can develop from smooth data in well-posed initial value problems.For ideal free-surface flow with zero surface tension and gravity,the authors review existing works that describe"splash singularities",singular hyperbolic solutions related to jet formation and"flip-through",and a recent construction of a singular free surface by Zubarev and Karabut that however involves unbounded negative pressure.The authors illustrate some of these phenomena with numerical computations of 2D flow based upon a conformal mapping formulation.Numerical tests with a different kind of initial data suggest the possibility that corner singularities may form in an unstable way from specially prepared initial data.
