We consider geometric biomembranes governed by an L 2-gradient flow for bending energy subject to area and volume constraints (Helfrich model). We give a concise derivation of a novel vector formulation, based on shape differential calculus, and corresponding discretization via parametric FEM using quadratic isoparametric elements and a semi-implicit Euler… (More)
We analyze an adaptive discontinuous finite element method (ADFEM) for symmetric second order linear elliptic operators. The method is formulated on nonconforming meshes made of simplices or quadrilaterals, with any polynomial degree and in any dimension ≥ 2. We prove that the ADFEM is a contraction for the sum of the energy error and the scaled error… (More)
We present a mixed finite element method for a model of the flow in a Hele-Shaw cell of 2-D fluid droplets surrounded by air driven by surface tension and actuated by an electric field. The application of interest regards a micro-fluidic device called ElectroWetting on Dielectric (EWOD). Our analysis first focuses on the time-discrete (continuous in space)… (More)
In this paper we consider a finite element discretization of the Oldroyd-B model of viscoelastic flows. The method uses standard continuous polynomial finite element spaces for velocities, pressures and stresses. Inf-sup stability and stability for convection-dominated flows are obtained by adding a term penalizing the jump of the solution gradient over… (More)
A new paradigm of adaptivity is to execute refinement, coarsening, and smoothing of meshes on manifolds with incomplete information about their geometry and yet preserve position and curvature accuracy. We refer to this collectively as geometrically consistent mesh modification. We discuss the concept of discrete geometric consistency, show the failure of… (More)
We propose and analyze an approximation technique for the Maxwell eigenvalue problem using H 1-conforming finite elements. The key idea consists of considering a mixed method controlling the divergence of the electric field in a fractional Sobolev space H −α with α ∈ (1 2 , 1). The method is shown to be convergent and spectrally correct.
We present and study a novel numerical algorithm to approximate the action of T β := L −β where L is a symmetric and positive definite unbounded operator on a Hilbert space H 0. The numerical method is based on a representation formula for T −β in terms of Bochner integrals involving (I + t 2 L) −1 for t ∈ (0, ∞). To develop an approximation to T β , we… (More)