Conforming Virtual Element Method for Biharmonic Poisson-Nernst-Planck Navier-Stokes systems

Mathematics > Numerical Analysis [Submitted on 16 Jun 2026] Title:Conforming Virtual Element Method for Biharmonic Poisson-Nernst-Planck Navier-Stokes systems View PDF HTML (experimental)Abstract:We develop and analyze a conforming Virtual Element Method (VEM) for the fourth-order Poisson-Nernst-Planck-Navier-Stokes (PNP-NS) system. The proposed scheme is based on compatible discretizations of each component: an \(H^2\)-conforming VEM for a fourth-order electrostatic potential equation, an \(H^1\)-conforming VEM for the Nernst--Planck equations, and a \textit{divergence-free} and \textit{pressure-robust} VEM for the Navier--Stokes equations. Time integration is performed using a backward Euler scheme to ensure stability. We establish the well-posedness of the continuous problem up to three-dimensions and also establish existence and uniqueness of the fully discrete solution via a fixed-point argument. Further, we derive a priori error estimates showing that the electrostatic potential, concentration and velocity converge optimally in Bochner norms \(L^\infty\bigl(0,T; H^2(\Omega)\bigr)\), \(L^2\bigl(0,T; H^1(\Omega)\bigr)\), and \(L^2\bigl(0,T; \bm{H}^1(\Omega)\bigr)\), respectively. The analysis requires a sophisticated argument to avoid any restrictive assumptions on the coercivity and continuity constants and to handle the trilinear form involving three different variables. The pressure-robust design permits the use of lowest-order pressure approximations without compromising convergence rates of the other variables, reducing computational cost. Numerical experiments confirm the theoretical convergence rates for various polynomial orders and demonstrate the scheme's robustness, including in low-viscosity regimes. Current browse context: math.NA References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.