In the context of the finite-element method we use a finite-element space $`V`$ with the set of basis functions $`\{\phi_i\}`$. Within AMDiS this concept is realized by the class `ParallelGlobalBasis`, contained in the header file `amdis/functions/ParallelGlobalBasis.hpp`. This extends the interface of the underlying `Dune::Functions::DefaultGlobalBasis<Impl>` with an automatic update mechanism used in several places within AMDiS. We strongly advice to always use an `AMDiS::ParallelGlobalBasis` or a user-defined derived class instead of the DUNE data structure for the update mechanism to work properly.
In the context of the finite-element method we use a finite-element space $`V`$ with the set of basis functions $`\{\phi_i\}`$. Within AMDiS this concept is realized by the class `GlobalBasis`, contained in the header file `amdis/functions/GlobalBasis.hpp`. This extends the interface of the underlying `Dune::Functions::DefaultGlobalBasis<Impl>` with an automatic update mechanism used in several places within AMDiS. We strongly advice to always use an `AMDiS::GlobalBasis` or a user-defined derived class instead of the DUNE data structure for the update mechanism to work properly.
For more information on the class interface visit the API documentation.
## PreBasis classes and basis trees
Many finite-element spaces in applications can be constructed as a product of simple spaces $`V = V_1 \times V_2 \times \dots \times V_k`$. For example the Taylor-Hood-Element can be constructed as $`V_{TH} = V_{v} \times V_{p}`$ with the space of velocity functions $`V_v`$ and pressure functions $`V_p`$. The velocity space can again be decomposed into the vector components $`V_v = V_{v_1} \times \dots \times V_{v_n}`$. If we use second-degree lagrange basis functions for the velocity space and first-order lagrange basis functions for the pressure we get a decomposition $`V_{TH} = V_{v_1} \times V_{v_n} \times V_p = L_2^n \times L_1`$.
The underlying numerics environment of AMDiS, DUNE, attempts to model the situation above. Hence a `ParallelGlobalBasis` in AMDiS can be defined by a tree structure using the Dune-PreBasis class, which itself is composed of nodes in a tree. The leaf nodes in this tree contain the implementation details of the simple basis, while inner nodes indicate composition of either identical children (called power nodes) or arbitrary children (called composite nodes).
The underlying numerics environment of AMDiS, DUNE, attempts to model the situation above. Hence a `GlobalBasis` in AMDiS can be defined by a tree structure using the Dune-PreBasis class, which itself is composed of nodes in a tree. The leaf nodes in this tree contain the implementation details of the simple basis, while inner nodes indicate composition of either identical children (called power nodes) or arbitrary children (called composite nodes).
### Making a PreBasis
When we want to build a `PreBasis` we must first identify the structure of the basis we want to construct with a tree. We can then build that structure in AMDiS by nesting the functions `Dune::Functions::BasisFactory::composite(Nodes...[, MergingStrategy])` for composite nodes, `Dune::Functions::BasisFactory::power(Node[, MergingStrategy])` for power nodes and implementations (e.g. `Dune::Functions::BasisFactory::lagrange<k>())`). The second optional argument `MergingStrategy` provides merging strategies to the inner nodes, specifying how the indices of the simple leaf basis should be merged to obtain the indexing of the global basis. Currently only flat indexing is supported by AMDiS.
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@@ -37,24 +37,24 @@ auto taylorHoodPreBasisFactoryWithMergingStrategy =
```
## Making a global basis
Using a PreBasisFactory we can easily make a global basis by defining the set of grid elements on which the basis functions of the FE-space should live. This can be done by providing a GridView and using the `ParallelGlobalBasis` constructors. An optional name can be provided that can be used to pass initfile parameters to the parallel communication class.
Using a PreBasisFactory we can easily make a global basis by defining the set of grid elements on which the basis functions of the FE-space should live. This can be done by providing a GridView and using the `GlobalBasis` constructors. An optional name can be provided that can be used to pass initfile parameters to the parallel communication class.
```c++
// Name, Grid and Dune::DefaultGlobalBasis arguments
If we use a ProblemStat object most of the work will be done automatically for us. The PreBasis is specified via a Traits template parameter with the most frequently used cases already included in `amdis/ProblemStatTraits.hpp`. Those include
The `ParallelGlobalBasis` provides access to element indices and basis functions in the same way as a Dune-basis does, it is even derived from the Dune::DefaultGlobalBasis.
The `GlobalBasis` provides access to element indices and basis functions in the same way as a Dune-basis does, it is even derived from the Dune::DefaultGlobalBasis.
### Getting the total number of DOFs
If we are simple interested in getting the total number of DOFs the basis contains we can simply call `basis.dimension()`. This can be useful for preparing matrices and vectors for insertion.
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@@ -198,4 +198,4 @@ For this purpose the update method exists, which takes a GridView of the Grid af
Usually we do not need to call this function - out of the box AMDiS with a ProblemStat doing most of the work will automatically call it for us. If user code is working with the underlying Dune-Grid directly there is no way for AMDiS to detect if any changes happen to it, therefore update must be called manually in such a scenario.
### The globalRefineCallback function
Certain grid managers support the use of a callback function when doing global refinement. Using a ParallelGlobalBasis in this context is currently not supported.
Certain grid managers support the use of a callback function when doing global refinement. Using a GlobalBasis in this context is currently not supported.
