Reference documentation for deal.II version GIT relicensing12g89ba5bf9db 20240224 00:00:03+00:00

Library of integrals over cells and faces. More...
Namespaces  
namespace  Advection 
Local integrators related to advection along a vector field and its DG formulations.  
namespace  Divergence 
Local integrators related to the divergence operator and its trace.  
namespace  Elasticity 
Local integrators related to elasticity problems.  
namespace  GradDiv 
namespace  L2 
Local integrators related to L^{2}inner products.  
namespace  Laplace 
Local integrators related to the Laplacian and its DG formulations.  
namespace  Maxwell 
Local integrators related to curl operators and their traces.  
namespace  Patches 
Integrators writing patches with values in quadrature points.  
Library of integrals over cells and faces.
This namespace contains application specific local integrals for bilinear forms, forms and error estimates. It is a collection of functions organized into namespaces devoted to certain applications. For instance, the namespace Laplace contains functions for computing cell matrices and cell residuals for the Laplacian operator, as well as functions for the weak boundary conditions by Nitsche or the interior penalty discontinuous Galerkin method. The namespace Maxwell does the same for curlcurl type problems.
The namespace L2 contains functions for mass matrices and L^{2}inner products.
In most cases, the action of a function in this namespace can be described by a single integral. We distinguish between integrals over cells Z and over faces F. If an integral is denoted as
\[ \int_Z u \otimes v \,dx, \]
it will yield the following results, depending on the type of operation
We will use regular cursive symbols \(u\) for scalars and bold symbols \(\mathbf u\) for vectors. Test functions are always v and trial functions are always u. Parameters are Greek and the face normal vectors are \(\mathbf n = \mathbf n_1 = \mathbf n_2\).
Functions in this namespace follow a generic signature. In the simplest case, you have two related functions
There is typically a pair of functions for the same operator, the function cell_residual
implementing the mapping of the operator from the finite element space into its dual, and the function cell_matrix
generating the bilinear form corresponding to the Frechet derivative of cell_residual
.
The first argument of these functions is the return type, which is
The next argument is the FEValuesBase object representing the finite element for integration. If the integrated operator maps from one finite element space into the dual of another (for instance an offdiagonal matrix in a block system), then first the FEValuesBase for the trial space and after this the one for the test space are specified.
This list is followed by the set of required data in the order
The local integrators can be used wherever a local integration loop would have been implemented instead. The following example is from the implementation of a Stokes solver, using MeshWorker::Assembler::LocalBlocksToGlobalBlocks. The matrices are
With these matrices, the function called by MeshWorker::loop() could be written like
See step39 for a worked out example of this code.