Andrea Bonito

Numerical simulation of 3D viscoelastic flows with free surfaces

A numerical model is presented for the simulation of viscoelastic flows with complex free surfaces in three space dimensions. The mathematical formulation of the model is similar to that of the volume of fluid (VOF) method, but the numerical procedures are different.A splitting method is used for the time discretization. The prediction step consists in solving three advection problems, one for the volume fraction of liquid (which allows the new liquid domain to be obtained), one for the velocity field, one for the extra-stress. The correction step corresponds to solving an Oldroyd-B fluid flow problem without advection in the new liquid domain.Two different grids are used for the space discretization. The three advection problems are solved on a fixed, structured grid made out of small cubic cells, using a forward characteristics method. The Oldroyd-B problem without advection is solved using continuous, piecewise linear stabilized finite elements on a fixed, unstructured mesh of tetrahedrons.Efficient post-processing algorithms enhance the quality of the numerical solution. A hierarchical data structure reduces the memory requirements.Convergence of the numerical method is checked for the pure extensional flow and the filling of a tube. Numerical results are presented for the stretching of a filament. Fingering instabilities are obtained when the aspect ratio is large. Also, results pertaining to jet buckling are reported.

Quasi-Optimal Convergence Rate of an Adaptive Discontinuous Galerkin Method

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

Parametric FEM for geometric biomembranes

\geq2

Approximation of the eigenvalue problem for the time harmonic Maxwell system by continuous Lagrange finite elements

. We prove that the ADFEM is a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops. We design a refinement procedure that maintains the level of nonconformity uniformly bounded and prove that the approximation classes using continuous and discontinuous finite elements are equivalent. The geometric decay and the equivalence of classes are instrumental in deriving the optimal cardinality of the ADFEM. We show that the ADFEM (and the AFEM on nonconforming meshes) yields a decay rate of energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity.

Mathematical and numerical analysis of a simplified time-dependent viscoelastic flow

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 method. We document the performance of the new parametric FEM with a number of simulations leading to dumbbell, red blood cell and toroidal equilibrium shapes while exhibiting large deformations.

Adaptive Finite Element Methods for Elliptic Problems with Discontinuous Coefficients

We propose and analyze an approximation technique for the Maxwell eigenvalue problem using H1-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.

Geometrically Consistent Mesh Modification

A time-dependent model corresponding to an Oldroyd-B viscoelastic fluid is considered, the convective terms being disregarded. Global existence in time is proved in Banach spaces provided the data are small enough, using the implicit function theorem and a maximum regularity property for a three fields Stokes problem. A finite element discretization in space is then proposed. Existence of the numerical solution is proved for small data, so as a priori error estimates, using again an implicit function theorem.

Mixed finite element method for electrowetting on dielectric with contact line pinning

Elliptic PDEs with discontinuous diffusion coefficients occur in application domains such as diffusions through porous media, electromagnetic field propagation on heterogeneous media, and diffusion processes on rough surfaces. The standard approach to numerically treating such problems using finite element methods is to assume that the discontinuities lie on the boundaries of the cells in the initial triangulation. However, this does not match applications where discontinuities occur on curves, surfaces, or manifolds, and could even be unknown beforehand. One of the obstacles to treating such discontinuity problems is that the usual perturbation theory for elliptic PDEs assumes bounds for the distortion of the coefficients in the

A Continuous Interior Penalty Method for Viscoelastic Flows

L_\infty

Time-discrete higher order ALE formulations: a priori error analysis

norm and this in turn requires that the discontinuities are matched exactly when the coefficients are approximated. We present a new approach based on distortion of the coefficients in an