We present a discontinuous Galerkin method on a fully unstructured grid for the modeling of unsteady incompressible fluid flows with free surfaces. The surface is modeled by embedding and represented by a levelset. We discuss the discretization of the flow equations and the level set equation as well a various ways of advancing the equations in time using velocity projection techniques. The eff...

A number of practical engineering problems require the repeated simulation of unsteady fluid flows. These problems include the control, optimization and uncertainty quantification of fluid systems. To make many of these problems tractable, reduced-order modeling has been used to minimize the simulation requirements. For nonlinear, time-dependent problems, such as the Navier-Stokes equations, re...

The paper presents an unconditionally stable numerical scheme to solve a nonlinear integro-differential equation which arises in mathematical modelling of the penetration of a magnetic field into a substance, if the temperature is kept constant throughout the material. Numerical scheme comprises of the Galerkin finite element method [18] for the spatial discretization followed by an implicit fi...

We analyze the spatially semidiscrete piecewise linear finite element method for a nonlocal parabolic equation resulting from thermistor problem. Our approach is based on the properties of the elliptic projection defined by the bilinear form associated with the variational formulation of the finite element method. We assume minimal regularity of the exact solution that yields optimal order erro...

This paper provides equivalent characterization of the discrete maximum principle for Galerkin solutions of general linear elliptic problems. The characterization is formulated in terms of the discrete Green’s function and the elliptic projection of the boundary data. This general concept is applied to the analysis of the discrete maximum principle for the higher-order finite elements in one-di...

A method for model reduction of dynamical systems with the second order structure is proposed in this paper. The proposed technique preserves the second order structure of the system, and also preserves the stability of the original systems. The method uses the controllability and observability gramians within the time interval to build the appropriate Petrov-Galerkin projection for dynamical s...

Population dynamics in spatially extended systems can be modeled by Coupled Map Lattices (CML). We employ such equations to study the behavior of populations confined to a finite patch surrounded by a completely hostile environment. By means of the Galerkin projection and the normal solution ansatz, we are able to find analytical expressions for the critical patch size and show the existence of...

In this paper we analyse the performance of a low dimensional model for the nonlinear thermo-mechanical waves. The model has been obtained by using proper orthogonal decomposition methods combined with a Galerkin projection. First, we analyse the original PDE model in order to obtain the system states at many time instances. Then, by using an empirical orthogonal basis extracted from our numeri...

The goal of this work is to provide accurate dynamical models of oscillations in the flow past a rectangular cavity, for the purpose of bifurcation analysis and control. We have performed an extensive set of direct numerical simulations which provide the data used to derive and evaluate the models. Based on the method of Proper Orthogonal Decomposition (POD) and Galerkin projection, we obtain l...

We derive optimal a priori error estimates for discontinuous Galerkin (dG) time discrete schemes of any order applied to an advection-diffusion model defined on moving domains and written in the Arbitrary Lagrangian Eulerian (ALE) framework. Our estimates hold without any restrictions on the time steps for dG with exact integration or Reynolds’ quadrature. They involve a mild restriction on the...