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

Introduction. A Simple Model Problem. Abstract Nonlinear Equations. Finite Element Discretizations of Elliptic PDEs. Practical Implementation. Bibliography. Subject Index.

A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques

Fibroblasts can condense a hydrated collagen lattice to a tissue-like structure 1/28th the area of the starting gel in 24 hr. The rate of the process can be regulated by varying the protein content of the lattice, the cell number, or the con- centration of an inhibitor such as Colcemid. Fibroblasts of high population doubling level propagated in vitro, which have left the cell cycle, can carry out the contraction at least as efficiently as cycling cells. The potential uses of the system as an immu- nologically tolerated "tissue" for wound hea ing and as a model for studying fibroblast function are discussed.

Production of a tissue-like structure by contraction of collagen lattices by human fibroblasts of different proliferative

In this paper, we review the development of the Runge–Kutta discontinuous Galerkin (RKDG) methods for non-linear convection-dominated problems. These robust and accurate methods have made their way into the main stream of computational fluid dynamics and are quickly finding use in a wide variety of applications. They combine a special class of Runge–Kutta time discretizations, that allows the method to be non-linearly stable regardless of its accuracy, with a finite element space discretization by discontinuous approximations, that incorporates the ideas of numerical fluxes and slope limiters coined during the remarkable development of the high-resolution finite difference and finite volume schemes. The resulting RKDG methods are stable, high-order accurate, and highly parallelizable schemes that can easily handle complicated geometries and boundary conditions. We review the theoretical and algorithmic aspects of these methods and show several applications including nonlinear conservation laws, the compressible and incompressible Navier–Stokes equations, and Hamilton–Jacobilike equations.

Review Article Runge-Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems

In this paper, we review the development of the Runge–Kutta discontinuous Galerkin (RKDG) methods for non-linear convection-dominated problems. These robust and accurate methods have made their way into the main stream of computational fluid dynamics and are quickly finding use in a wide variety of applications. They combine a special class of Runge–Kutta time discretizations, that allows the method to be non-linearly stable regardless of its accuracy, with a finite element space discretization by discontinuous approximations, that incorporates the ideas of numerical fluxes and slope limiters coined during the remarkable development of the high-resolution finite difference and finite volume schemes. The resulting RKDG methods are stable, high-order accurate, and highly parallelizable schemes that can easily handle complicated geometries and boundary conditions. We review the theoretical and algorithmic aspects of these methods and show several applications including nonlinear conservation laws, the compressible and incompressible Navier–Stokes equations, and Hamilton–Jacobi-like equations.

Runge-Kutta discontinuous Galerkin methods for convection-dominated problems

A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems

We discuss a moving finite element method for solving vector systems of time dependent partial differential equations in one space dimension. The mesh is moved so as to equidistribute the spatial component of the discretization error in $H^1 $. We present a method of estimating this error by using p-hierarchic finite elements. The error estimate is also used in an adaptive mesh refinement procedure to give an algorithm that combines mesh movement and refinement.We discretize the partial differential equations in space using a Galerkin procedure with piecewise linear elements to approximate the solution and quadratic elements to estimate the error. A system of ordinary differential equations for mesh velocities are used to control element motions. We use existing software for stiff ordinary differential equations for the temporal integration of the solution, the error estimate, and the mesh motion. Computational results using a code based on our method are presented for several examples.

A moving finite element method with error estimation and refinement for one-dimensional time dependent partial differential equations

https://www7320.nrlssc.navy.mil/pubs/2006/massey_2006.pdf

Slimane Adjerid

Papers

A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems

A moving finite element method with error estimation and refinement for one-dimensional time dependent partial differential equations

Superconvergence of discontinuous Galerkin solutions for a nonlinear scalar hyperbolic problem

A posteriori discontinuous finite element error estimation for two-dimensional hyperbolic problems

A p-th degree immersed finite element for boundary value problems with discontinuous coefficients