Reference Element Types and Connectivities
The reference element types supported by MFEM are documented in the file geom.hpp. This file contains the reference vertices and their order as well as edge and face definitions when appropriate.
0D Element
Zero dimensional elements are used as boundary elements or interior "faces" of one dimensional domains.
Point Element
A Point element is simply a single vertex at the origin of reference space.
1D Element
One dimensional elements can be used within the domain of 1D meshes. They also arise as boundary elements or interior "faces" of 2D domains.
Segment Element
A Segment element is a line segment from the origin of a 1D reference space,
$x=0$, to the point $x=1$.
2D Elements
Two dimensional elements can be used within the domain of 2D meshes. They also arise as boundary elements or interior "faces" of 3D domains.
Triangle Element
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 |
|---|---|
| Vertices | Edges |
A Triangle element consists of three vertices in a 2D reference space:
| v0 | v1 | v2 |
| (0,0) | (1,0) | (0,1) |
The Edge-to-Vertex connectivity is:
| e0 | e1 | e2 |
| {v0, v1} | {v1, v2} | {v2, v0} |
Square Element
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 |
|---|---|
| Vertices | Edges |
A Square element, a.k.a Quadrilateral, consists of four vertices in a 2D reference space:
| v0 | v1 | v2 | v3 |
| (0,0) | (1,0) | (1,1) | (0,1) |
The Edge-to-Vertex connectivity is:
| e0 | e1 | e2 | e3 |
| {v0, v1} | {v1, v2} | {v2, v3} | {v3, v0} |
3D Elements
Three dimensional elements can be used within the domain of 3D meshes.
Tetrahedron Element
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 |
 |
|---|---|---|
| Vertices | Edges | Faces |
A Tetrahedron element consists of four vertices in a 3D reference space:
| v0 | v1 | v2 | v3 |
| (0,0,0) | (1,0,0) | (0,1,0) | (0,0,1) |
The Edge-to-Vertex connectivity is:
| e0 | e1 | e2 | e3 | e4 | e5 |
| {v0, v1} | {v0, v2} | {v0, v3} | {v1, v2} | {v1, v3} | {v2, v3} |
The Face-to-Vertex connectivity is:
| f0 | f1 | f2 | f3 |
| {v1, v2, v3} | {v0, v3, v2} | {v0, v1, v3} | {v0, v2, v1} |
Cube Element
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 |
 |
|---|---|---|
| Vertices | Edges | Faces |
A Cube element, a.k.a. Hexahedron, consists of eight vertices in a 3D
reference space:
| v0 | v1 | v2 | v3 | v4 | v5 | v6 | v7 |
| (0,0,0) | (1,0,0) | (1,1,0) | (0,1,0) | (0,0,1) | (1,0,1) | (1,1,1) | (0,1,1) |
The Edge-to-Vertex connectivity is:
| e0 | e1 | e2 | e3 | e4 | e5 | e6 | e7 | e8 | e9 | e10 | e11 |
| {v0, v1} | {v1, v2} | {v3, v2} | {v0, v3} | {v4, v5} | {v5, v6} | {v7, v6} | {v4, v7} | {v0, v4} | {v1, v5} | {v2, v6} | {v3, v7} |
The Face-to-Vertex connectivity is:
| f0 | f1 | f2 | f3 | f4 | f5 |
| {v3, v2, v1, v0} | {v0, v1, v5, v4} | {v1, v2, v6, v5} | {v2, v3, v7, v6} | {v3, v0, v4, v7} | {v4, v5, v6, v7} |
Prism Element
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 |
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|---|---|---|
| Vertices | Edges | Faces |
A Prism element, a.k.a. Wedge, consists of six vertices in a 3D
reference space:
| v0 | v1 | v2 | v3 | v4 | v5 |
| (0,0,0) | (1,0,0) | (0,1,0) | (0,0,1) | (1,0,1) | (0,1,1) |
The Edge-to-Vertex connectivity is:
| e0 | e1 | e2 | e3 | e4 | e5 | e6 | e7 | e8 |
| {v0, v1} | {v1, v2} | {v2, v0} | {v3, v4} | {v4, v5} | {v5, v3} | {v0, v3} | {v1, v4} | {v2, v5} |
The Face-to-Vertex connectivity is:
| f0 | f1 | f2 | f3 | f4 |
| {v0, v2, v1} | {v3, v4, v5} | {v0, v1, v4, v3} | {v1, v2, v5, v4} | {v2, v0, v3, v5} |
Pyramid Element
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 |
 |
|---|---|---|
| Vertices | Edges | Faces |
A Pyramid element consists of five vertices in a 3D reference space:
| v0 | v1 | v2 | v3 | v4 |
| (0,0,0) | (1,0,0) | (1,1,0) | (0,1,0) | (0,0,1) |
The Edge-to-Vertex connectivity is:
| e0 | e1 | e2 | e3 | e4 | e5 | e6 | e7 |
| {v0, v1} | {v1, v2} | {v3, v2} | {v0, v3} | {v0, v4} | {v1, v4} | {v2, v4} | {v3, v4} |
The Face-to-Vertex connectivity is:
| f0 | f1 | f2 | f3 | f4 |
| {v3, v2, v1, v0} | {v0, v1, v4} | {v1, v2, v4} | {v2, v3, v4} | {v3, v0, v4} |