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

Real and Complex Analysis

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

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Introduction To Electrodynamics

This article discusses finite element Galerkin schemes for a number of linear model problems in electromagnetism. The finite element schemes are introduced as discrete differential forms, matching the coordinate-independent statement of Maxwell's equations in the calculus of differential forms. The asymptotic convergence of discrete solutions is investigated theoretically. As discrete differential forms represent a genuine generalization of conventional Lagrangian finite elements, the analysis is based upon a judicious adaptation of established techniques in the theory of finite elements. Risks and difficulties haunting finite element schemes that do not fit the framework of discrete differential forms are highlighted.

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Finite elements in computational electromagnetism

In this paper we are concerned with the efficient solution of discrete variational problems related to the bilinear form ({\bf curl}\! $\cdot$, {\bf curl}\! $\cdot)_{{\fontsize{6}{0pt}\selectfont\textbf{\textit{L}}}^2(\Omega)}$ + ($\cdot,\cdot)_{{\fontsize{6}{0pt}\selectfont\textbf{\textit{L}}}^2(\Omega)}$ defined on \textbf{\textit{H}}$_0$({\bf curl}; $\Omega$). This is a core task in the time-domain simulation of electromagnetic fields, if implicit timestepping is employed. We rely on Nedelec's \textbf{\textit{H}}({\bf curl}; $\Omega$)-conforming finite elements (edge elements) to discretize the problem. 
We construct a multigrid method for the fast iterative solution of the resulting linear system of equations. Since proper ellipticity of the bilinear form is confined to the complement of the kernel of the {\bf curl} operator, Helmholtz decompositions are the key to the design of the algorithm: ${\cal N}$({\bf curl}) and its complement ${\cal N}$({\bf curl})$^{\bot}$ require separate treatment. Both can be tackled by nodal multilevel decompositions, where for the former the splitting is set in the space of discrete scalar potentials.
Under certain assumptions on the computational domain and the material functions, a rigorous proof of the asymptotic optimality of the multigrid method can be given, which shows that convergence does not deteriorate on very fine grids. The results of numerical experiments confirm the practical efficiency of the method.

Multigrid Method for Maxwell's Equations

In this paper, we develop and analyze a general approach to preconditioning linear systems of equations arising from conforming finite element discretizations of H(curl, )- and H(div, )-elliptic variational problems. The preconditioners exclusively rely on solvers for discrete Poisson problems. We prove mesh-independent effectivity of the preconditioners by using the abstract theory of auxiliary space preconditioning. The main tools are discrete analogues of so-called regular decomposition results in the function spaces H(curl, ) and H(div, ). Our preconditioner for H(curl, ) is similar to an algorithm proposed in [R. Beck, Algebraic Multigrid by Component Splitting for Edge Elements on Simplicial Triangulations, Tech. rep. SC 99-40, ZIB, Berlin, Germany, 1999].

Nodal Auxiliary Space Preconditioning in H(curl) and H(div) Spaces

We consider H (curl ;Ω)-elliptic problems that have been discretized by means of Nedelec's edge elements on tetrahedral meshes. Such problems occur in the numerical computation of eddy currents. From the defect equation we derive localized expressions that can be used as a posteriori error estimators to control adaptive refinement. Under certain assumptions on material parameters and computational domains, we derive local lower bounds and a global upper bound for the total error measured in the energy norm. The fundamental tool in the numerical analysis is a Helmholtz-type decomposition of the error into an irrotational part and a weakly solenoidal part.

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Residual based a posteriori error estimators for eddy current computation

The mixed variational formulation of many elliptic boundary value problems involves vector valued function spaces, like, in three dimensions, H(curl; Ω) and H(Div; Ω) Thus finite element subspaces of these function spaces are indispensable for effective finite element discretization schemes Given a simplicial triangulation of the computational domain Ω, among others, Raviart, Thomas and Nedelec have found suitable conforming finite elements for H(Div; Ω) and H(curl; Ω) At first glance, it is hard to detect a common guiding principle behind these approaches We take a fresh look at the construction of the finite spaces, viewing them from the angle of differential forms This is motivated by the well-known relationships between differential forms and differential operators: div, curl and grad can all be regarded as special incarnations of the exterior derivative of a differential form Moreover, in the realm of differential forms most concepts are basically dimension-independent Thus, we arrive at a fairly canonical procedure to construct conforming finite element subspaces of function spaces related to differential forms In any dimension we can give a simple characterization of the local polynomial spaces and degrees of freedom underlying the definition of the finite element spaces With unprecedented ease we can recover the familiar H(Div; Q)- and H(curl; Ω)-conforming finite elements, and establish the unisolvence of degrees of freedom In addition, the use of differential forms makes it possible to establish crucial algebraic properties of the canonical interpolation operators and representation theorems in a single sweep for all kinds of spaces

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Ralf Hiptmair

Papers

Finite elements in computational electromagnetism

Multigrid Method for Maxwell's Equations

Nodal Auxiliary Space Preconditioning in H(curl) and H(div) Spaces

Residual based a posteriori error estimators for eddy current computation

Canonical construction of finite elements