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A new model of geophysical prospecting based on a modification of Maxwell's equations governing low frequency electromagnetic wave propagation in a polarisable medium is considered. The main difficulty of the mathematical and numerical study of the problem is the treatment of the convolution term containing the information on the polarisable properties of the medium. A powerful method to treat the convolution term numerically is proposed and given a firm mathematical basis. Numerical simulations demonstrating the efficiency of the method are presented.

This is the second paper in a series in which we construct and analyze a class of TVB (total variation bounded) discontinuous Galerkin finite element methods for solving conservation laws u t + ∑ i = 1 d ( f i ( u ) ) x i = 0 {u_t} + \sum \nolimits _{i = 1}^d {{{({f_i}(u))}_{{x_i}}} = 0} . In this paper we present a general framework of the methods, up to any order of formal accuracy, using scalar one-dimensional initial value and initial-boundary problems as models. In these cases we prove TVBM (total variation bounded in the means), TVB, and convergence of the schemes. Numerical results using these methods are also given. Extensions to systems and/or higher dimensions will appear in future papers.

The stability of Newton's method applied to implicit numerical schemes for scalar conservation laws requires, in the general case, an upper bound on the size of the timesteps. A simple locally defined projection, a total variation diminishing (TVD)-projection, is introduced, and it is used to obtain always-stable extensions of Newton's method. A numerical study of this method applied to Godunov's implicit scheme is presented.

We study a model of interest in petroleum prospection by electromagnetic methods. We propose a method for taking into account the polarization phenomenon (memory effect) using Pade-like approximations. We also show how to treat the air-substratum interface through an integral boundary condition. We give a robust numerical method and present simulation results.

In this paper, we give a simple introduction to the devising of discontinuous Galerkin (DG) methods for nonlinear hyperbolic conservation laws. These methods have recently made their way into the main stream of computational fluid dynamics and are quickly finding use in a wide variety of applications. The DG methods, which are extensions of finite volume methods, incorporate into a finite element framework the notions of approximate Riemann solvers, numerical fluxes and slope limiters coined during the remarkable development of the high-resolution finite difference and finite volume methods for nonlinear hyperbolic conservation laws. We start by stressing the fact that nonlinear hyperbolic conservation laws are usually obtained from well-posed problems by neglecting terms modeling nondominant features of the model which, nevertheless, are essential in crucial, small parts of the domain; as a consequence, the resulting problem becomes ill-posed. The main difficulty in devising numerical schemes for these conservation laws is thus how to re-introduce the neglected physical information in order to approximate the physically relevant solution, usually called the entropy solution. For the classical case of the entropy solution of the nonlinear hyperbolic scalar conservation law, we show how to carry out this process for two prototypical DG methods. The first DG method is the so-called shock-capturing DG method, which does not use slope limiters and is implicit; the second is the Runge–Kutta DG method, which is an explicit method that does not employ a shock-capturing term but uses a slope limiter instead. We then focus on the Runge–Kutta DG methods and show how to obtain a key stability property which holds independently of the accuracy of the scheme and of the nonlinearity of the conservation law; we also show some computational results.

We provide a framework for the analysis of a large class of discontinuous methods for second-order elliptic problems. It allows for the understanding and comparison of most of the discontinuous Galerkin methods that have been proposed over the past three decades for the numerical treatment of elliptic problems.

We introduce the hybridizable discontinuous Galerkin (HDG) methods in the framework of steady-state diffusion problems and show why they can be implemented more efficiently than any other DG method and why they are also more accurate. We then give an overview of the application of these methods to several problems including wave propagation, linear and nonlinear elasticity, convection-diffusion and the incompressible and compressible Navier-Stokes equations.

In this paper, we present a Hybridizable Discontinuous Galerkin (HDG) method for the solution of the compressible Euler and Navier-Stokes equations. The method is devised by using the discontinuous Galerkin approximation with a special choice of the numerical fluxes and weakly imposing the continuity of the normal component of the numerical fluxes across the element interfaces. This allows the approximate conserved variables defining the discontinuous Galerkin solution to be locally condensed, thereby resulting in a reduced system which involves only the degrees of freedom of the approximate traces of the solution. The HDG method inherits the geometric flexibility and arbitrary high order accuracy of Discontinuous Galerkin methods, but offers a significant reduction in the computational cost as well as improved accuracy and convergence properties. In particular, we show that HDG produces optimal converges rates for both the conserved quantities as well as the viscous stresses and the heat fluxes. We present some numerical results to demonstrate the accuracy and convergence properties of the method.

We provide a common framework for the understanding, comparison, and analysis of several discontinuous Galerkin methods that have been proposed for the numerical treatment of elliptic problems. This class includes the recently introduced methods of Bassi and Rebay (together with the variants proposed by Brezzi, Manzini, Marini, Pietra and Russo), the local discontinuous Galerkin methods of Cockburn and Shu, and the method of Baumann and Oden. It also includes the so-called interior penalty methods developed some time ago by Douglas and Dupont, Wheeler, Baker, and Arnold among others.