Nektar++: MultiRegions sublibrary

In the MultiRegions library, all classes and routines are related to the process of assembling a global spectral/hp expansion out of local elemental contributions are bundled together. The most important entities of this library are the base class ExpList and its daughter classes. These classes all are the abstraction of a multi-elemental spectral/hp element expansion. Three different types of multi-elemental expansions can be distinguished:

A collection of local expansions.
This collection is just a list of local expansions, without any coupling between the expansions on the different elements, and can be formulated as:

$u^{\delta}(\boldsymbol{x})=\sum_{e=1}^{{N_{\mathrm{el}}}} \sum_{n=0}^{N^{e}_m-1}\hat{u}_n^e\phi_n^e(\boldsymbol{x})$

where

${N_{\mathrm{el}}}$ is the number of elements,
$N^{e}_m$ is the number of local expansion modes within the element ,
$\phi_n^e(\boldsymbol{x})$ is the $n^{th}$ local expansion mode within the element ,
$\hat{u}_n^e$ is the $n^{th}$ local expansion coefficient
within the element .

These types of expansion are represented by the classes ExpList0D, ExpList1D, ExpList2D and ExpList3D, depending on the dimension of the problem (ExpList0D is used just to deal with boundary conditions for 1D expansions).

A multi-elemental discontinuous global expansion.
The expansions are represented by the classes DisContField1D, DisContField2D and DisContField3D. Objects of these classes should be used when solving partial differential equations using a discontinuous Galerkin approach. These classes enforce a coupling between elements and augment the domain with boundary conditions.

All local elemental expansions are now connected to form a global spectral/hp representation. This type of global expansion can be defined as:

$u^{\delta}(\boldsymbol{x})=\sum_{n=0}^{N_{\mathrm{dof}}-1}\hat{u}_n \Phi_n(\boldsymbol{x})=\sum_{e=1}^{{N_{\mathrm{el}}}} \sum_{n=0}^{N^{e}_m-1}\hat{u}_n^e\phi_n^e(\boldsymbol{x})$

where

$N_{\mathrm{dof}}$ refers to the number of global modes,
$\Phi_n(\boldsymbol{x})$ is the $n^{th}$ global expansion mode,
$\hat{u}_n$ is the $n^{th}$ global expansion coefficient.

Typically, a mapping array to relate the global degrees of freedom $\hat{u}_n$ and local degrees of freedom $\hat{u}_n^e$ is required to assemble the global expansion out of the local contributions.

In order to solve (second-order) partial differential equations, information about the boundary conditions should be incorporated in the expansion. In case of a standard Galerkin implementation, the Dirichlet boundary conditions can be enforced by lifting a known solution satisfying these conditions, leaving a homogeneous Dirichlet problem to be solved. If we denote the unknown solution by $u^{\mathcal{H}}(\boldsymbol{x})$ and the known Dirichlet boundary conditions by $u^{\mathcal{D}}(\boldsymbol{x})$ then we can decompose the solution $u^{\delta}(\boldsymbol{x})$ into the form
$u^{\delta}(\boldsymbol{x}_i)=u^{\mathcal{D}}(\boldsymbol{x}_i)+ u^{\mathcal{H}}(\boldsymbol{x}_i)=\sum_{n=0}^{N^{\mathcal{D}}-1} \hat{u}_n^{\mathcal{D}}\Phi_n(\boldsymbol{x}_i)+ \sum_{n={N^{\mathcal{D}}}}^{N_{\mathrm{dof}}-1} \hat{u}_n^{\mathcal{H}}\Phi_n(\boldsymbol{x}_i).$
Implementation-wise, the known solution can be lifted by ordering the known degrees of freedom $\hat{u}_n^{\mathcal{H}}$ first in the global solution array $\boldsymbol{\hat{u}}$ .

A multi-elemental continuous global expansion.
The discontinuous case is supplimented with a global continuity condition. In this case a $C^0$ continuity condition is imposed across the element interfaces and the expansion is therefore globally continuous.

This type of global continuous expansion which incorporates the boundary conditions are represented by the classes ContField1D, ContField2D and ContField3D. Objects of these classes should be used when solving partial differential equations using a standard Galerkin approach.

Additional classes.
Furthermore, we have two more sets of classes:

The class LocalToGlobalBaseMap and its daughter classes: LocalToGlobalC0ContMap and LocalToGlobalDGMap.
These classes are an abstraction of the mapping from local to global degrees of freedom and contain one or both of the following mapping arrays:
- map[ ][ ]
  This array contains the index of the global degree of freedom corresponding to the $n^{th}$ local expansion mode within the $e^{th}$ element.
- bmap[ ][ ]
  This array contains the index of the global degree of freedom corresponding to the $n^{th}$ local boundary mode within the $e^{th}$ element. Next to the mapping array, these classes also contain routines to assemble the global system from the local contributions, and other routines to transform between local and global level.
The classes GlobalLinSys and GlobalLinSysKey.
The class GlobalLinSys is an abstraction of the global system matrix resulting from the global assembly procedure. Depending of the choice to statically condense the global matrix or not, the relevant blocks are stored as a member of this class. Given a proper right hand side vector, this class also contains a routine to solve the resulting matrix system.
The class GlobalLinSysKey represents a key which uniquely defines a global matrix. This key can be used to construct or retrieve the global matrix associated to a certain key.

More information about the implementation of connectivity between elements in Nektar++ can be found here.

Quasi-3D approach.
The Quasi-3D approach is an extension of the 1D and the 2D spectral/hp element method. This technique permits to study 3D problems combining the spectral/hp element method with a spectral method. In the Quasi-3D approach with 1 homogenous direction, the third dimension (z-axis) is expandend with an harmonic expansion (a Fourier series). In each quadrature point of the Fourier discretisation we can find a 2D plane discretised with a 2D spectral/hp elements expasions. In the case with 2 homogeneous directions a plane is discretised with a 2D Fourier expansion (y-z palne). In each one of the quadrature point of this harmonic expansion there is a 1D spectral/hp element discretisation. The homogenous classes derive directly form ExpList, and they are ExpListHomogeneous1D and ExpListHomogeneous2D. This classes are used to represent the collections of 2D (or 1D) spectral/hp element problems which are located in the Fourier expansions quatradure points to create a 3D problem. As describer above, we can find the find the continuos or discontinuos case, depending on the spectral/hp element approach. ExpList2DHomogeneous1D and ExpList1DHomogeneous2D are used to manage boundary conditions. A description of the Quasi-3D approach usage can be found in Nektar++ XML File Format.