Reference documentation for deal.II version GIT 9042b9283b 2023-12-02 14:50:02+00:00
The deal.II Library Documentation

This is the main starting page for the deal.II class and function documentation. Documentation on other aspects, such as the build system, can be found elsewhere. In addition, there are Tutorial programs on the use of the library.

Many of the classes in the deal.II library can be grouped into modules (see the Modules page or the corresponding entry in the menu at the top of this page). These modules form around the building blocks of any finite element program. An outline of how the primary groups of classes in deal.II interact is given by the following clickable graph, with a more detailed description below (gray boxes denote a subset of the optional external libraries, gray ovals a subset of the optional external applications with which deal.II can interact):

These groups are all covered in the tutorial programs, with a first overview of how they fit together given in step-3. The following is a guide to this classification of groups, as well as links to the documentation pertaining to each of them:

1. Triangulation: Triangulations are collections of cells and their lower-dimensional boundary objects. Cells are images of the reference hypercube [0,1]dim under a suitable mapping in the module on Mappings between reference and real cell.

The triangulation stores geometric and topological properties of a mesh: how are the cells connected and where are their vertices. A triangulation doesn't know anything about the finite elements that you may want to used on this mesh, and a triangulation does not even know anything about the shape of its cells: in 2d it only knows that a cell has 4 faces (lines) and 4 vertices (and in 3d that it has 6 faces (quadrilaterals), 12 lines, and 8 vertices), but everything else is defined by a mapping class.

The properties and data of triangulations are almost always queried through loops over all cells, possibly querying all faces of each cell as well. Most of the knowledge about a mesh is therefore hidden behind iterators, i.e. pointer-like structures that one can iterate from one cell to the next, and that one can ask for information about the cell it presently points to.

The classes that describe triangulations and cells are located and documented in the Grids and Triangulations module. Iterators are described in the Iterators on mesh-like containers module.

2. Manifold: Manifolds describe the shape of cells and, more generally, the geometry of the domain on which one wants to solve an equation. They use the language of differential geometry. More information can be found in Manifold description for triangulations.

3. Finite Element: Finite element classes describe the properties of a finite element space as defined on the unit cell. This includes, for example, how many degrees of freedom are located at vertices, on lines, or in the interior of cells. In addition to this, finite element classes of course have to provide values and gradients of individual shape functions at points on the unit cell.

The finite element classes are described in the Finite elements module.

4. Quadrature: As with finite elements, quadrature objects are defined on the unit cell. They only describe the location of quadrature points on the unit cell, and the weights of quadrature points thereon.

The documentation of the classes describing particular quadrature formulas is found in the Quadrature formulas module.

5. DoFHandler: DoFHandler objects are the confluence of triangulations and finite elements: the finite element class describes how many degrees of freedom it needs per vertex, line, or cell, and the DoFHandler class allocates this space so that each vertex, line, or cell of the triangulation has the correct number of them. It also gives them a global numbering.

A different viewpoint is this: While the mesh and finite element describe abstract properties of the finite dimensional space $$V_h$$ in which we seek the discrete solution, the DoFHandler classes enumerate a concrete basis of this space so that we can represent the discrete solution as $$u_h(\mathbf x)= \sum_j U_j \varphi_i(\mathbf x)$$ by an ordered set of coefficients $$U_j$$.

Just as with triangulation objects, most operations on DoFHandlers are done by looping over all cells and doing something on each or a subset of them. The interfaces of the two classes are therefore rather similar: they allow to get iterators to the first and last cell (or face, or line, etc) and offer information through these iterators. The information that can be gotten from these iterators is the geometric and topological information that can already be gotten from the triangulation iterators (they are in fact derived classes) as well as things like the global numbers of the degrees of freedom on the present cell. On can also ask an iterator to extract the values corresponding to the degrees of freedom on the present cell from a data vector that stores values for all degrees of freedom associated with a triangulation.

It is worth noting that, just as triangulations, DoFHandler classes do not know anything about the mapping from the unit cell to its individual cells. It is also ignorant of the shape functions that correspond to the degrees of freedom it manages: all it knows is that there are, for example, 2 degrees of freedom for each vertex and 4 per cell interior. Nothing about their specifics is relevant to the DoFHandler class with the exception of the fact that they exist.

The DoFHandler class and its associates are described in the Degrees of Freedom module. In addition, there are specialized versions that can handle multilevel and hp-discretizations. These are described in the Multilevel support and hp-finite element support modules. Finite element methods frequently imply constraints on degrees of freedom, such as for hanging nodes or nodes at which boundary conditions apply; dealing with such constraints is described in the Constraints on degrees of freedom module.

6. Mapping: The next step in a finite element program is that one would want to compute matrix and right hand side entries or other quantities on each cell of a triangulation, using the shape functions of a finite element and quadrature points defined by a quadrature rule. To this end, it is necessary to map the shape functions, quadrature points, and quadrature weights from the unit cell to each cell of a triangulation. This is not directly done by, but facilitated by the Mapping and derived classes: they describe how to map points from unit to real space and back, as well as provide gradients of this derivative and Jacobian determinants.

These classes are all described in the Mappings between reference and real cell module.

7. FEValues: The next step is to actually take a finite element and evaluate its shape functions and their gradients at the points defined by a quadrature formula when mapped to the real cell. This is the realm of the FEValues class and siblings: in a sense, they offer a point-wise view of the finite element function space.

This seems restrictive: in mathematical analysis, we always write our formulas in terms of integrals over cells, or faces of cells, involving the finite element shape functions. One would therefore think that it is necessary to describe finite element spaces as continuous spaces. However, in practice, this is not necessary: all integrals are in actual computations replaced by approximations using quadrature formula, and what is therefore really only necessary is the ability to evaluate shape functions at a finite number of given locations inside a domain. The FEValues classes offer exactly this information: Given finite element, quadrature, and mapping objects, they compute the restriction of a continuous function space (as opposed to discrete, not as opposed to discontinuous) to a discrete number of points.

There are a number of objects that can do this: FEValues for evaluation on cells, FEFaceValues for evaluation on faces of cells, and FESubfaceValues for evaluation on parts of faces of cells. All these classes are described in the Finite element access/FEValues classes module.

8. Linear Systems: If one knows how to evaluate the values and gradients of shape functions on individual cells using FEValues and friends, and knows how to get the global numbers of the degrees of freedom on a cell using the DoFHandler iterators, then the next step is to use the bilinear form of the problem to assemble the system matrix (and right hand side) of the linear system. We will then determine the solution of our problem from this linear system.

To do this, we need to have classes that store and manage the entries of matrices and vectors. deal.II comes with a whole set of classes for this purpose, as well as with interfaces to other software packages that offer similar functionality. Documentation to this end can be found in the Linear algebra classes module.

9. Linear Solvers: In order to determine the solution of a finite-dimensional, linear system of equations, one needs linear solvers. In finite element applications, they are frequently iterative, but sometimes one may also want to use direct or sparse direct solvers. deal.II has quite a number of these. They are documented in the Linear solver classes module.

10. Output: Finally, once one has obtained a solution of a finite element problem on a given triangulation, one will often want to postprocess it using a visualization program. This library doesn't do the visualization by itself, but rather generates output files in a variety of graphics formats understood by widely available visualization tools.

A description of the classes that do so is given in the Graphical output module.

In addition, deal.II has a number of groups of classes that go beyond the ones listed here. They pertain to more refined concepts of the hierarchy presented above, or to tangential aspects like handling of input and output that are not necessarily specific to finite element programs, but appear there as well. These classes are all listed in the Classes and Namespaces views reachable from the menu bar at the top of this page, and are also grouped into modules of their own (see the Modules link at the top of this page).

We provide the Doxygen tag file for those of you who would like to directly link the documentation of application programs to the deal.II online documentation. The tag file is at deal.tag. For each release of deal.II, it resides in the directory right above the Doxygen reference documentation. In order to use the tag file, you have to download it into a place where Doxygen can find it. After that, find the key TAGFILES in your Doxygen options file and write something like

   TAGFILES = deal.tag=https://www.dealii.org/X.Y.Z/doxygen/deal.II


where X.Y.Z refers to the release you want to link to. Be sure you use the matching tag file. In theory, you can also link against the developing revisions of deal.II, but then you have to fear that your links may become invalid if the deal.II structure changes.