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

This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software. The presentation spans mathematical background, software design and the use of FEniCS in applications. Theoretical aspects are complemented with computer code which is available as free/open source software. The book begins with a special introductory tutorial for beginners. Followingare chapters in Part I addressing fundamental aspects of the approach to automating the creation of finite element solvers. Chapters in Part II address the design and implementation of the FEnicS software. Chapters in Part III present the application of FEniCS to a wide range of applications, including fluid flow, solid mechanics, electromagnetics and geophysics.

Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book

Introduction to linear and nonlinear programming

A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

Finite element exterior calculus is an approach to the design and understand- ing of finite element discretizations for a wide variety of systems of partial differential equations. This approach brings to bear tools from differential geometry, algebraic topology, and homological algebra to develop discretiza- tions which are compatible with the geometric, topological, and algebraic structures which underlie well-posedness of the PDE problem being solved. In the finite element exterior calculus, many finite element spaces are re- vealed as spaces of piecewise polynomial differential forms. These connect to each other in discrete subcomplexes of elliptic differential complexes, and are also related to the continuous elliptic complex through projections which commute with the complex differential. Applications are made to the finite element discretization of a variety of problems, including the Hodge Lapla- cian, Maxwell's equations, the equations of elasticity, and elliptic eigenvalue problems, and also to preconditioners.

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Finite element exterior calculus, homological techniques, and applications

http://amasud.web.engr.illinois.edu/Papers/Stabilized_Methods-F-H-M-FEM-Book-2004.pdf

Stabilized Finite Element Methods

On some techniques for approximating boundary conditions in the finite element method

A new mixed finite element formulation for the equations of linear elasticity is considered. In the formulation the variables approximated are the displacement, the unsymmetric stress tensor and the rotation. The rotation act as a Lagrange multiplier introduced in order to enforce the symmetry of the stress tensor. Based on this formulation a new family of both two-and three-dimensional mixed methods is defined. Optimal error estimates, which are valid uniformly with respect to the Poisson ratio, are derived. Finally, a new postprocessing scheme for improving the displacement is introduced and analyzed.

A family of mixed finite elements for the elasticity problem

The recent technique of stabilizing mixed finite element methods by augmenting the Galerkin formulation with least squares terms calculated separately on each element is considered. The error analysis is performed in a unified manner yielding improved results for some methods introduced earlier. In addition, a new formulation is introduced and analyzed. method is used within two different contexts. The first is in connection with the plain strain or three-dimensional problems near the incompressible limit for which the Poisson ratio equals one half. It is well known that for these problems a standard lower-order displacement finite element method will break down. By now, it is also well known how a working method can be obtained; a new unknown, the "pressure," is introduced and discretized with a finite element space different from that used for the displacements. The finite element method obtained falls into the class of saddlepoint problems for which there is a general theory developed by Brezzi and Babuska (5), (7). The application of this theory shows that the method converges optimally if the discrete spaces for the displacement and pressure satisfy a stability condition. For a particular choice of displacement and pressure approximations it can be a difficult task to verify this "Babuska-Brezzi" or "inf-sup" condition. During the last decade this problem has been intensively studied and the performances of many combinations are now known; cf. (18) and (23), (26) for some recent extensions. In addition there exists a general technique, utilizing "bubble functions," by which a stable method can

Error analysis of some Galerkin least squares methods for the elasticity equations

In this note, we propose and analyse a method for handling interfaces between non-matching grids based on an approach suggested by Nitsche (1971) for the approximation of Dirichlet boundary conditions. The exposition is limited to self-adjoint elliptic problems, using Poisson's equation as a model. A priori and a posteriori error estimates are given. Some numerical results are included.

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Rolf Stenberg

Papers

Stabilized Finite Element Methods

On some techniques for approximating boundary conditions in the finite element method

A family of mixed finite elements for the elasticity problem

Error analysis of some Galerkin least squares methods for the elasticity equations

A finite element method for domain decomposition with non-matching grids