There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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.

/pdf/unified-analysis-of-discontinuous-galerkin-methods-for-9numgtph62.pdf

Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems

The purpose of this paper is to provide a comprehensive presentation and interpretation of the Ensemble Kalman Filter (EnKF) and its numerical implementation. The EnKF has a large user group, and numerous publications have discussed applications and theoretical aspects of it. This paper reviews the important results from these studies and also presents new ideas and alternative interpretations which further explain the success of the EnKF. In addition to providing the theoretical framework needed for using the EnKF, there is also a focus on the algorithmic formulation and optimal numerical implementation. A program listing is given for some of the key subroutines. The paper also touches upon specific issues such as the use of nonlinear measurements, in situ profiles of temperature and salinity, and data which are available with high frequency in time. An ensemble based optimal interpolation (EnOI) scheme is presented as a cost-effective approach which may serve as an alternative to the EnKF in some applications. A fairly extensive discussion is devoted to the use of time correlated model errors and the estimation of model bias.

The Ensemble Kalman Filter: Theoretical formulation and practical implementation

In this paper, we study the local discontinuous Galerkin (LDG) methods for nonlinear, time-dependent convection-diffusion systems. These methods are an extension of the Runge--Kutta discontinuous Galerkin (RKDG) methods for purely hyperbolic systems to convection-diffusion systems and share with those methods their high parallelizability, high-order formal accuracy, and easy handling of complicated geometries for convection-dominated problems. It is proven that for scalar equations, the LDG methods are L2-stable in the nonlinear case. Moreover, in the linear case, it is shown that if polynomials of degree k are used, the methods are kth order accurate for general triangulations; although this order of convergence is suboptimal, it is sharp for the LDG methods. Preliminary numerical examples displaying the performance of the method are shown.

/pdf/the-local-discontinuous-galerkin-method-for-time-dependent-23o5v9uy8q.pdf

The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems

This is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation. In this paper, a general recovery technique is developed for determining the derivatives (stresses) of the finite element solutions at nodes. The implementation of the recovery technique is simple and cost effective. The technique has been tested for a group of widely used linear, quadratic and cubic elements for both one and two dimensional problems. Numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent. One order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements. In particular, an O(h4) convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time. The performance of the proposed technique is compared with the widely used smoothing procedure of global L2 projection and other methods. It is found that the derivatives recovered at interelement nodes, by using L2 projection, are also superconvergent for linear elements but not for quadratic elements. Numerical experiments on the convergence of the recovered solutions in the energy norm are also presented. Higher rates of convergence are again observed. The results presented in this part of the paper indicate clearly that a new, powerful and economical process is now available which should supersede the currently used post-processing procedures applied in most codes.

The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique

In the recent past the Discontinuous Galerkin Method has attracted the interest of researchers in numerical seismology due in part to the flexibility to implement boundary conditions and media discontinuities, the ability to model wave field discontinuities and to approximate the wave field with arbitrary high order polynomials, and because the method can be energy conservative and it is suitable for parallel implementations. In this paper we investigate the applicability of the Interior Penalty Discontinuous Galerkin Method (IP-DGM) for elastic wave propagation by analyzing its grid dispersion. We consider different types of basis functions that naturally yield a diagonal mass matrix. This is relevant to seismology because a diagonal mass matrix is tantamount to an explicit and efficient time marching scheme. We find that the Legendre basis functions that are traditionally used in IP-DGM introduce numerical dispersion and anisotropy. Furthermore, we find that using Lagrange basis functions along with the Gauss nodes has attractive advantages for numerical wave propagation.

Grid Dispersion of the Discontinuous Galerkin Method For Elastic Wave Propagation

Modeling Energized and Foam Fracturing Using the Phase Field Method

SUMMARY Seismic history matching has been used to reduce uncertainty and increase the accuracy in reservoir characterization. In this paper we propose a joint inversion scheme for quantitative reservoir petrophysics characterization and inside fluid flow imaging, which integrates as many data sources as available, such as timelapse seismic data, production data and sensor information. We demonstrate that this integrated framework leads to accurate seismic history matching and reservoir parameter estimation and that the incorporation of time-lapse seismic data does help speed up the convergence process. In addition, the imaging of the inside fluid flow is also obtained. Considering the unique feature of Bayesian inference in data integration and uncertainty analysis, we also propose a formulation to simultaneously integrate all data sources in a Bayesian framework and solve the problem by stochastically constructing the posterior probability distribution (PPD) using Markov Chain Monte Carlo (MCMC) methods. The coefficients of rock fluid physics models are also incorporated. This means that the coefficients are determined stochastically based on data rather from lab measurements or empirical relationships. Based on MCMC samples, uncertainty can be correctly quantified.

Integrated time-lapse seismic inversion for reservoir petrophysics and fluid-flow imaging

A set of discontinuous Galerkin (DG) finite element methods are proposed to solve 
the two-phase flow equations, such as the air-water system that arises in shallow 
subsurface flow problems. The two time-splitting approaches detailed incorporate 
primal formulations, such as the Oden-Baumann-Babuska DG (OBB-DG), Symmetric 
Interior Penalty Galerkin (SIPG), Non-Symmetric Interior Penalty Galerkin (NIPG), 
and Incomplete Interior Penalty Galerkin (IIPG), as well as a local discontinuous 
Galerkin (LDG) method applied to the saturation equation. The two-phase flow 
equations presented are split into sequential and implicit pressure/explicit 
saturation (IMPES) formulations.

The IMPES formulation introduced in this work uses a primal DG formulation to solve 
the pressure equation implicitly at every time step, and then uses an explicit LDG 
scheme for saturation equation. The LDG scheme employed advances in time via 
explicit Runge-Kutta time stepping, while employing a Kirchoff transformation for 
the local solution of the degenerate diffusion term. 

DG finite element methods are naturally suited to problems subsurface flow and 
transport. They can handle general meshes which may be non-conforming, they can 
treat higher order approximations, and they are locally mass conservative, among 
many other desirable properties. In particular, due to different rock properties 
at the interface between two materials, fluid saturations may be discontinuous. 
Therefore DG methods are a natural fit for this class of problems. Computational 
results showing that the IMPES method proposed will hold capillary barriers while 
using two capillary pressure curves in different materials are presented.

http://proceedings.dtu.dk/fedora/repository/dtu:1562/OBJ/article.pdf

Mary F. Wheeler

Papers

Grid Dispersion of the Discontinuous Galerkin Method For Elastic Wave Propagation

Modeling Energized and Foam Fracturing Using the Phase Field Method

Integrated time-lapse seismic inversion for reservoir petrophysics and fluid-flow imaging

Discontinuous Galerkin Methods Applied to Two-Phase Flow Problems

Parameter Estimation for the Coupled Biot-Lamé-Navier Problem in Subsurface Modeling