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

Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for this type of system. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.

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Numerical solution of saddle point problems

Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics

Tables of integrals

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

The intended readership includes graduate students and researchers in engineering, numerical analysis, applied mathematics and interdisciplinary scientific computing. The publisher describes the book as follows:

* An excellent introduction to finite elements, iterative linear solvers and scientific computing

* Contains theoretical problems and practical exercises

* All methods and examples use freely available software

* Focuses on theory and computation, not theory for computation

* Describes approximation methods and numerical linear algebra

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Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics

Mixed finite element approximation of the classical Stokes problem describing slow viscous incompressible flow gives rise to symmetric indefinite systems for the discrete velocity and pressure variables. Iterative solution of such indefinite systems is feasible and is an attractive approach for large problems. Part I of this work described a conjugate gradient-like method (the method of preconditioned conjugate residuals) which is applicable to symmetric indefinite problems [A. J. Wathen and D. J. Silvester, SIAM J. Numer. Anal., 30 (1993), pp. 630–649]. Using simple arguments, estimates for the eigenvalue distribution of the discrete Stokes operator on which the convergence rate of the iteration depends are easily derived. Part I discussed the important case of diagonal preconditioning (scaling). This paper considers the more general class of block preconditioners, where the partitioning into blocks corresponds to the natural partitioning into the velocity and pressure variables. It is shown that, provid...

Fast iterative solution of stabilised Stokes systems part II: using general block preconditioners

IFISS is a graphical Matlab package for the interactive numerical study of incompressible flow problems. It includes algorithms for discretization by mixed finite element methods and a posteriori error estimation of the computed solutions. The package can also be used as a computational laboratory for experimenting with state-of-the-art preconditioned iterative solvers for the discrete linear equation systems that arise in incompressible flow modelling. A unique feature of the package is its comprehensive nature; for each problem addressed, it enables the study of both discretization and iterative solution algorithms as well as the interaction between the two and the resulting effect on overall efficiency.

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Algorithm 866: IFISS, a Matlab toolbox for modelling incompressible flow

Mixed finite element approximation of the classical Stokes problem describing slow viscous incompressible flow gives rise to symmetric indefinite systems for the discrete velocity and pressure variables. Iterative solution of such indefinite systems is feasible and is an attractive approach for large problems. The use of stabilisation methods for convenient (but unstable) mixed elements introduces stabilisation parameters. We show how these can be chosen to obtain rapid iterative convergence. We propose a conjugate gradient-like method (the method of preconditioned conjugate residuals) which is applicable to symmetric indefinite problems, describe the effects of stabilisation on the algebraic structure of the discrete Stokes operator and derive estimates of the eigenvalue spectrum of this operator on which the convergence rate of the iteration depends. Here we discuss the simple case of diagonal preconditioning. Our results apply to both locally and globally stabilised mixed elements as well as to elements which are inherently stable. We demonstrate that convergence rates comparable to that achieved using the diagonally scaled conjugate gradient method applied to the discrete Laplacian are approachable for the Stokes problem.

Fast iterative solution of stabilised Stokes systems, part I: using simple diagonal preconditioners

Discretization and linearization of the steady–state Navier-Stokes equations gives rise to a nonsymmetric indefinite linear system of equations. In this paper, we introduce preconditioning techniques for such systems with the property that the eigenvalues of the preconditioned matrices are bounded independently of the mesh size used in the discretization. We confirm and supplement these analytic results with a series of numerical experiments indicating that Krylov subspace iterative methods for nonsymmetric systems display rates of convergence that are independent of the mesh parameter. In addition, we show that preconditioning costs can be kept small by using iterative methods for some intermediate steps performed by the preconditioner.

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David J. Silvester

Papers

Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics

Fast iterative solution of stabilised Stokes systems part II: using general block preconditioners

Algorithm 866: IFISS, a Matlab toolbox for modelling incompressible flow

Fast iterative solution of stabilised Stokes systems, part I: using simple diagonal preconditioners

Fast nonsymmetric iterations and preconditioning for Navier-Stokes equations