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.

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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

Numerical dispersion and grid orientation problems with adverse mobility ratios are two of the major difficulties in the numerical simulation of enhanced recovery processes. An efficient method for modeling convection-dominated flows which greatly reduces numerical dispersion and grid orientation problems is presented and applied to miscible displacement in a porous medium. The base method utilizes characteristic flow directions to model convection and finite elements to treat the diffusion and dispersion. The characteristic approach also minimizes certain overshoot difficulties which accompany many finite element methods for problems with sharp fluid interfaces. The truncation error caused by the characteristic time-stepping technique is small, so large stable time-steps can be taken as in fully-implicit methods without the corresponding loss in accuracy.

Simulation of miscible displacement using mixed methods and a modified method of characteristics

Semi-discrete and a family of discrete time locally conservative Discontinuous Galerkin procedures are formulated for approximations to nonlinear parabolic equations. For the continuous time approximations a priori L ∞ (L 2) and L 2(H l) estimates are derived and similarly, l ∞ (L 2) and l 2 (H 1) for the discrete time schemes. Spatial rates in H l and time truncation errors in L 2 are optimal.

A Discontinuous Galerkin Method Applied to Nonlinear Parabolic Equations

In this paper, we examine heuristically the reasons for locking in poroelasticity. As a first step, we first reexamine the problem of locking in linear elasticity. From this, we discover how the problem arises in the poroelasticity setting and how the problem might be overcome.

Overcoming the problem of locking in linear elasticity and poroelasticity: an heuristic approach

We discuss several fully implicit techniques for solving the nonlinear algebraic system arising in an expanded mixed finite element or cell-centered finite difference discretization of two- and three-phase porous media flow. Every outer nonlinear Newton iteration requires solution of a nonsymmetric Jacobian linear system. Two major types of preconditioners, supercoarsening multigrid (SCMG) and two-stage, are developed for the GMRES iteration applied to the solution of the Jacobian system. The SCMG reduces the three-dimensional system to two dimensions using a vertical aggregation followed by a two-dimensional multigrid. The two-stage preconditioners are based on decoupling the system into a pressure and concentration equations. Several pressure preconditioners of different types are described. Extensive numerical results are presented using the integrated parallel reservoir simulator (IPARS) and indicate that these methods have low arithmetical complexity per iteration and good convergence rates.

Iterative Solution Methods for Modeling Multiphase Flow in Porous Media Fully Implicitly

http://web.kaust.edu.sa/faculty/ShuyuSun/pub/angiogenesisBMB2005.pdf

Mary F. Wheeler

Papers

Simulation of miscible displacement using mixed methods and a modified method of characteristics

A Discontinuous Galerkin Method Applied to Nonlinear Parabolic Equations

Overcoming the problem of locking in linear elasticity and poroelasticity: an heuristic approach

Iterative Solution Methods for Modeling Multiphase Flow in Porous Media Fully Implicitly

A deterministic model of growth factor-induced angiogenesis