This paper is concerned with the finite element method for nonlinear Hamiltonian systems from three aspects: conservation of energy, symplicity, and the global error. To study the symplecticity of the finite element methods, we use an analytical method rather than the commonly used algebraic method. We prove optimal order of convergence at the nodes tn for mid-long time and demonstrate the symp...

In this work we present a general approach for developing high-order compact differencing schemes by utilizing the governing differential equation to help approximate truncation error terms. As an illustrative application we consider the stream-function vorticity form of the Navier Stokes equations, and provide driven cavity results. Some extensions to treat non-constant metric coefficients res...

In this paper we propose a stabilized conforming finite volume element method for the Stokes equations. On stating the convergence of the method, optimal a priori error estimates in different norms are obtained by establishing the adequate connection between the finite volume and stabilized finite element formulations. A superconvergence result is also derived by using a postprocessing projecti...

Two of the authors earlier suggested a method of calculating special grid steps for three point finite-difference schemes which yielded exponential superconvergence of the Neumann-to-Dirichlet map. We apply this approach to solve the two-dimensional timedomain wave problem and the 2.5-D elasticity system in cylindrical coordinates. Our numerical experiments exhibit exponential convergence at pr...

In this paper a simple turning point (y = yc, A = Ac) of the parameterdependent Hammerstein equation y(t) = f(t) + X k(t,s)g(s,y(s))ds, te[a,b], is approximated numerically in the following way. A simple turning point (z = zc, X = Xe) of an equivalent equation for z(t) := Xg(t, y(t)) is computed first. This is done by solving a discretized version of a certain system of equations which has (2e,...

In this paper we review the existing and develop new local discontinuous Galerkin methods for solving time dependent partial differential equations with higher order derivatives in one and multiple space dimensions. We review local discontinuous Galerkin methods for convection diffusion equations involving second derivatives and for KdV type equations involving third derivatives. We then develo...

Three formulations of the Stokes-Cahn–Hilliard (SCH) system are investigated for the simulation of rigid particles in two-phase flows. Using the SCH framework, we assume that the interface between the two fluids is diffuse, whereas the interface between the fluids and the particle is assumed to be sharp. To describe the sharp boundary of the particle, a moving, boundary-fitted mesh is used whic...

Super convergence of the gradient of approximate solutions to second order elhptic équations is analysed and justified for a large ciass of curved isoparametric quadrilatéral éléments Résumé — On analyse et on justifie la super convergence du gradient des solutions approchées obtenues lors de la résolution d'équations elliptiques du second ordre à Vaide d'éléments isoparamétriques courbes de ty...

In this paper, we show a phenomenon, which we named “super-convergence”, where residual networks can be trained using an order of magnitude fewer iterations than is used with standard training methods. One of the key elements of superconvergence is training with cyclical learning rates and a large maximum learning rate. Furthermore, we present evidence that training with large learning rates im...