an encyclopedia of finite element definitions

# Kong–Mulder–Veldhuizen

 Abbreviated names KMV Orders $$0\leqslant k$$ Reference elements triangle, tetrahedron Categories Scalar-valued elements

## Implementations

 Symfem string "KMV"↓ Show Symfem examples ↓ UFL string "KMV"↓ Show UFL examples ↓

## Examples

triangle
order 1
triangle
order 3
tetrahedron
order 1
• $$R$$ is the reference triangle. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$1$$, $$x$$, $$y$$
• $$\mathcal{L}=\{l_0,...,l_{2}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto \frac{1}{6} v(0,0)$$

$$\displaystyle \phi_{0} = - 6 x - 6 y + 6$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{1}:v\mapsto \frac{1}{6} v(1,0)$$

$$\displaystyle \phi_{1} = 6 x$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{2}:v\mapsto \frac{1}{6} v(0,1)$$

$$\displaystyle \phi_{2} = 6 y$$

This DOF is associated with vertex 2 of the reference element.
• $$R$$ is the reference triangle. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$1$$, $$x$$, $$x^{2}$$, $$y$$, $$x y$$, $$y^{2}$$, $$x y \left(- x - y + 1\right)$$
• $$\mathcal{L}=\{l_0,...,l_{6}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto \frac{1}{40} v(0,0)$$

$$\displaystyle \phi_{0} = - 120 x^{2} y + 80 x^{2} - 120 x y^{2} + 280 x y - 120 x + 80 y^{2} - 120 y + 40$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{1}:v\mapsto \frac{1}{40} v(1,0)$$

$$\displaystyle \phi_{1} = 40 x \left(- 3 x y + 2 x - 3 y^{2} + 3 y - 1\right)$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{2}:v\mapsto \frac{1}{40} v(0,1)$$

$$\displaystyle \phi_{2} = 40 y \left(- 3 x^{2} - 3 x y + 3 x + 2 y - 1\right)$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{3}:v\mapsto \frac{1}{15} v(\tfrac{1}{2},\tfrac{1}{2})$$

$$\displaystyle \phi_{3} = 60 x y \left(3 x + 3 y - 2\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{4}:v\mapsto \frac{1}{15} v(0,\tfrac{1}{2})$$

$$\displaystyle \phi_{4} = 60 y \left(3 x^{2} + 3 x y - 4 x - y + 1\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{5}:v\mapsto \frac{1}{15} v(\tfrac{1}{2},0)$$

$$\displaystyle \phi_{5} = 60 x \left(3 x y - x + 3 y^{2} - 4 y + 1\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{6}:v\mapsto \frac{9}{40} v(\tfrac{1}{3},\tfrac{1}{3})$$

$$\displaystyle \phi_{6} = 120 x y \left(- x - y + 1\right)$$

This DOF is associated with face 0 of the reference element.
• $$R$$ is the reference tetrahedron. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$1$$, $$x$$, $$y$$, $$z$$
• $$\mathcal{L}=\{l_0,...,l_{3}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto \frac{1}{24} v(0,0,0)$$

$$\displaystyle \phi_{0} = - 24 x - 24 y - 24 z + 24$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{1}:v\mapsto \frac{1}{24} v(1,0,0)$$

$$\displaystyle \phi_{1} = 24 x$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{2}:v\mapsto \frac{1}{24} v(0,1,0)$$

$$\displaystyle \phi_{2} = 24 y$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{3}:v\mapsto \frac{1}{24} v(0,0,1)$$

$$\displaystyle \phi_{3} = 24 z$$

This DOF is associated with vertex 3 of the reference element.

## References

• Chin-Joe-Kong, M. J. S., Mulder, W. A., and Van Veldhuizen, M. Higher-order triangular and tetrahedral finite elements with mass lumping for solving the wave equation, Journal of Engineering Mathematics 35, 405–426, 1999. [DOI: 10.1023/A:1004420829610] [BibTeX]