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

A Numerical Study of Benefits of Adding Polymer to WAG Processes for a Pilot Case

Locally Conservative Algorithms for Flow

An approximate Jacobian nonlinear solver for multiphase flow and transport

A space–time domain decomposition approach using enhanced velocity mixed finite element method

Phase-field methods have the potential to simulate large scale evolution of networks of fractures in porous media without the need to explicitly track interfaces. Practical field simulations require however that robust and efficient decoupling techniques can be applied for solving these complex systems. In this work, we focus on the mechanics-step that involves the coupling of elasticity and the phase-field variable. We develop a multirate scheme in which a coarser time grid is employed for the mechanics equation (i.e., the displacements) and a finer time grid is taken for the phase-field problem. The performance of this algorithm is demonstrated for two test cases.

Mary F. Wheeler

Papers

A Numerical Study of Benefits of Adding Polymer to WAG Processes for a Pilot Case

Locally Conservative Algorithms for Flow

An approximate Jacobian nonlinear solver for multiphase flow and transport

A space–time domain decomposition approach using enhanced velocity mixed finite element method

A phase-field multirate scheme with stabilized iterative coupling for pressure driven fracture propagation in porous media