(ch_stag)=

# DMSTAG: Staggered, Structured Grid

For structured (aka "regular") grids with staggered data (degrees of freedom potentially living on elements, faces, edges,
and/or vertices), the `DMSTAG` object is available. This can
be useful for problems in many domains, including fluid flow, MHD, and seismology.

It is possible, though extremely cumbersome, to implement a staggered-grid code using multiple `DMDA` objects, or a single
multi-component `DMDA` object where some degrees of freedom are unused.
`DMSTAG` was developed for two main purposes:

1. To help manage some of the burden of choosing and adhering to the complex indexing conventions needed for staggered grids (in parallel)
2. To provide a uniform abstraction for which scalable solvers and preconditioners may be developed (in particular, using `PCFIELDSPLIT` and `PCMG`).

`DMSTAG` is design to behave much
like {ref}`DMDA <sec_struct>`, with a couple of important distinctions, and borrows some terminology
from {doc}`DMPLEX <dmplex>`.

## Terminology

Like a `DMPLEX` object, a `DMSTAG` represents a [cell complex](https://en.wikipedia.org/wiki/CW_complex),
distributed in parallel over the ranks of an `MPI_Comm`. It is, however,
a very regular complex, consisting of a structured grid of $d$-dimensional cells, with $d \in \{1,2,3\}$,
which are referred to as *elements*, $d-1$ dimensional cells defining boundaries between these elements,
and the boundaries of the domain, and in 2 or more dimensions, boundaries of *these* cells,
all the way down to 0 dimensional cells referred to as *vertices*. In 2 dimensions, the 1-dimensional
element boundaries are referred to as *edges* or *faces*. In 3 dimensions, the 2-dimensional element boundaries
are referred to as *faces* and the 1-dimensional boundaries between faces are referred to as *edges*
The set of cells of a given dimension is referred to as a *stratum* (which one can think of as a level in DAG representation of the mesh); a `DMSTAG` object of dimension $d$
represents a complete cell complex with $d+1$ *strata* (levels).

In the description of any:`ch_unstructured` the cells at each level are referred to as *points*. Thus we adopt that terminology uniformly in PETSc and
so furthermore in this document,
point will refer to a cell.

Each stratum has a constant number of unknowns (which may be zero) associated with each point (cell) on that level.
The distinct unknowns associated with each point are referred to as *components*. For a `DMPLEX` there may be a different
number of unknowns for each point on the same level. 

The structured grid, is like with `DMDA`, decomposed via a Cartesian product of decompositions in each dimension,
giving a rectangular local subdomain on each rank. This is extended by an element-wise stencil width
of *ghost* elements to create an atlas of overlapping patches.

## Working with vectors and operators (matrices)

`DMSTAG` allows the user to reason almost entirely about a global indexing of elements.
Element indices are simply 1-3 `PetscInt` values, starting at $0$, in the
back, bottom, left corner of the domain. For instance, element $(1,2,3)$, in 3D,
is the element second from the left, third from the bottom, and fourth from the back
(regardless of how many MPI ranks are used).

To refer to points (elements, faces, edges, and vertices), a value of
`DMStagStencilLocation` is used, relative to the element index. The element
itself is referred to with `DMSTAG_ELEMENT`, the top right vertex (in 2D)
or the top right edge (in 3D) with `DMSTAG_UP_RIGHT`, the back bottom left
corner in 3D with `DMSTAG_BACK_DOWN_LEFT`, and so on.

{numref}`figure_dmstag_indexing` gives a few examples in 2D.

:::{figure} /images/manual/dmstag_indexing.svg
:name: figure_dmstag_indexing

Locations in `DMSTAG` are indexed according to global element indices (here, two in 2D) and a location name. Elements have unique names but other locations can be referred to in more than one way. Element colors correspond to a parallel decomposition, but locations on the grid have names which are invariant to this. Note that the face on the top right can be referred to as being to the left of a "dummy" element $(3,3)$ outside the physical domain.
:::

Crucially, this global indexing scheme does not include any "ghost" or "padding" unknowns outside the physical domain.
This is useful for higher-level operations such as computing norms or developing physics-based solvers. However
(unlike `DMDA`), this implies that the global `Vec` do not have a natural block structure, as different
strata have different numbers of points (e.g. in 1D there is an "extra" vertex on the right). This regular block
structure is, however, very useful for the *local* representation of the data, so in that case *dummy* DOF
are included, drawn as grey in {numref}`figure_dmstag_local_global`.

:::{figure} /images/manual/dmstag_local_global.svg
:name: figure_dmstag_local_global

Local and global representations for a 2D `DMSTAG` object, 3 by 4 elements, with one degree of freedom on each of the three strata: element (squares), faces (triangles), and vertices (circles). The cell complex is parallelized across 4 MPI ranks. In the global representation, the colors correspond to which rank holds the native representation of the unknown. The 4 local representations are shown, with an (elementwise) stencil "box" stencil width of 1. Unknowns are colored by their native rank. Dummy unknowns, which correspond to no global degree of freedom, are colored grey. Note that the local representations have a natural block size of 4, and the global representation has no natural block size.
:::

For working with `Vec` data, this approach is used to allow direct access to a multi-dimensional, regular-blocked
array. To avoid the user having to know about the {ref}`internal numbering conventions used <sec_dmstag_numbering>`,
helper functions are used to produce the proper final integer index for a given location and component, referred to as a "slot".
Similarly to `DMDAVecGetArrayDOF()`, this uses a $d+1$ dimensional array in $d$ dimensions.
The following snippet give an example of this usage.

```{literalinclude} /../src/dm/impls/stag/tests/ex51.c
:end-at: PetscCall(DMStagVecRestoreArray
:start-at: /* Set
```

`DMSTAG` provides a stencil-based method for getting and setting entries of `Mat` and `Vec` objects.
The follow excerpt from DMSTAG Tutorial ex1 demonstrates
the idea. For more, see the manual page for `DMStagMatSetValuesStencil()`.

```{literalinclude} /../src/dm/impls/stag/tutorials/ex1.c
:end-at: PetscCall(DMStagMatSetValuesStencil
:start-at: /* Velocity is either a BC
```

The array-based approach for `Vec` is likely to be more efficient than the stencil-based method just introduced above.

## Coordinates

`DMSTAG`, unlike `DMDA`, supports two approaches to defining coordinates. This is captured by which type of `DM`
is used to represent the coordinates. No default is imposed, so the user must directly or indirectly call
`DMStagSetCoordinateDMType()`.

If a second `DMSTAG` object is used to represent coordinates in "explicit" form, behavior is much like with `DMDA` - the coordinate `DM`
has $d$ DOF on each stratum corresponding to coordinates associated with each point.

If `DMPRODUCT` is used instead, coordinates are represented by a `DMPRODUCT` object referring to a
Cartesian product of 1D `DMSTAG` objects, each of which features explicit coordinates as just mentioned.

Navigating these nested `DM` in `DMPRODUCT` can be tedious, but note the existence of helper functions like
`DMStagSetUniformCoordinatesProduct()` and `DMStagGetProductCoordinateArrays()`.

(sec_dmstag_numbering)=

## Numberings and internal data layout

While `DMSTAG` aims to hide the details of its internal data layout, for debugging, optimization, and
customization purposes, it can be important to know how `DMSTAG` internally numbers unknowns.

Internally, each point is canonically associated with an element (top-level point (cell)). For purposes of local,
regular-blocked storage, an element is grouped with lower-dimensional points left of, below ("down"), and behind ("back") it.
This means that "canonical" values of `DMStagStencilLocation` are `DMSTAG_ELEMENT`, plus all entries consisting only of "LEFT", "DOWN", and "BACK". In general, these are the most efficient values to use, unless convenience dictates otherwise, as they are the ones used internally.

When creating the decomposition of the domain to local ranks, and extending these local domains to handle overlapping halo regions and boundary ghost unknowns, this same per-element association is used. This has the advantage of maintaining a regular blocking, but may not be optimal in some situations in terms of data movement.

Numberings are, like `DMDA`, based on a local "x-fastest, z-slowest" or "PETSc" ordering of elements (see {ref}`sec_ao`), with ordering of locations canonically associated with each element decided by considering unknowns on each point
to be located at the center of their point, and using a nested ordering of the same style. Thus, in 3-D, the ordering of the 8 canonical `DMStagStencilLocation` values associated
with an element is

```
DMSTAG_BACK_DOWN_LEFT
DMSTAG_BACK_DOWN
DMSTAG_BACK_LEFT
DMSTAG_BACK
DMSTAG_DOWN_LEFT
DMSTAG_DOWN
DMSTAG_LEFT
DMSTAG_ELEMENT
```

Multiple DOF associated with a given point are stored sequentially (as with `DMDA`).

For local `Vec`s, this gives a regular-blocked numbering, with the same number of unknowns associated with each element, including some "dummy" unknowns which to not correspond to any (local or global) unknown in the global representation. See {numref}`figure_dmstag_numbering_local` for an example.

In the global representation, only physical unknowns are numbered (using the same "Z" ordering for unknowns which are present), giving irregular numbers of unknowns, depending on whether a domain boundary is present. See {numref}`figure_dmstag_numbering_global` for an example.

:::{figure} /images/manual/dmstag_numbering_global.svg
:name: figure_dmstag_numbering_global

Global numbering scheme for a 2D `DMSTAG` object with one DOF per stratum. Note that the numbering depends on the parallel decomposition (over 4 ranks, here).
:::

:::{figure} /images/manual/dmstag_numbering_local.svg
:name: figure_dmstag_numbering_local

Local numbering scheme on rank 1 (Cf. {numref}`figure_dmstag_local_global`) for a 2D `DMSTAG` object with one DOF per stratum. Note that dummy locations (grey) are used to give a regular block size (here, 4).
:::

It should be noted that this is an *interlaced* (AoS) representation. If a segregated (SoA) representation is required,
one should use `DMCOMPOSITE` collecting several `DMSTAG` objects, perhaps using `DMStagCreateCompatibleDMStag()` to
quickly create additional `DMSTAG` objects from an initial one.
