# Is there a minimum angle requirement for cells in the finite volume method?

In his talk "What is a good linear finite element?", Shewchuk states that small dihedral angles in linear tetrahedra elements cause ill-conditioning of the stiffness matrix.

Do small dihedral angles in cells also cause ill-conditioning of the FVM system of equations?

References

1. Shewchuk, J. "What is a good linear finite element? interpolation, conditioning, anisotropy, and quality measures (preprint)." University of California at Berkeley 73 (2002).

I don't remember exactly the formulation in finite-element but I believe you need to compute the determinant of the jacobian when you define the transformation between some element and the reference element. When your element is very "flat", the determinant tends to zero and that is a problem in your computation.

In finite volume, we define the variables as cell-centered. For instance we define $$\overline{u}$$ as $$\overline{u} = \frac{1}{V}\int_V u~\mathrm{d}V$$ and we perform the numerical scheme on this variable. If your cell is very "flat", the volume tends to zero so this quantity is not defined. It does not answer your question regarding the minimum angle requirement but in general you should avoid such elements.

• I think also the distribution of volumes might play a role for the convergence properties. – stephanmg Jul 22 at 17:56

This is the sort of question that is more difficult to answer in the finite volume context than in the finite element context. So I would not be surprised if there are no papers that prove that bad triangles (or other kinds of cells) lead to bad solutions, though I would expect that there are plenty of papers that show it experimentally.

That said, many finite volume schemes can be written as variations or re-interpretations of finite element schemes. Consequently, whatever we know about methods in the finite element context must then also be true for the corresponding finite volume methods. I have no doubt that problems with poor cell shapes is one of these items that can be proven in the finite element context and that subsequently carry over to the finite volume context.

• Doesn't the shape function (the mapping from the unit cell to the physical cell) in FEM cause a problem for poorly shaped elements ( for example, a quad with a nearly 180 degree angle)? The determinant of the Jacobian will be nearly zero at the vertex with the nearly 180 deg angle. FVM doesn't require shape functions (which also allows use of arbitrary polyhedral cells), so it seems to me that this same quad might not cause a problem for FVM. – J. Heller Aug 14 '16 at 21:24
• What I mean is this: there are often a number of different ways how you can derive a given method (method = an algorithm to come up with a matrix and right hand side). For the FEM, we derive things using shape functions. For the FVM, we use integrals. But if two ways of deriving a scheme end up with the same matrix and rhs, then this yields two different ways to analyze this method. For example, if you can derive a particular method both via the FEM and FVM viewpoints, then it will have properties that involve shape functions, even if these don't appear in the typical FVM derivation. – Wolfgang Bangerth Aug 14 '16 at 22:12

I point out some references

http://dx.doi.org/10.1137/0713021 Here they show that angles should be bounded below 180 deg.

http://link.springer.com/article/10.1007/BF03322598 This gives a more recent survey.

http://www.bcamath.org/documentos_public/archivos/actividades_cientificas/TalkBCAMWSonCM20131018Korotov.pdf Nice brief survey, show that maximum angle condition is also not necessary.

Most of this discussion refers to simplicial elements and I am not aware of what happens on quad/hex elements.

For finite volumes, there is so much variety of schemes (e.g., cell-centered vs cell-vertex) that I don't know any general results. But still small angles should not be a problem and in fact, you benefit from using highly anisotropic triangles when computing discontinuous solutions, or those with steep gradients as in boundary layers. For high Reynolds number flows, you have to use highly stretched triangles in the boundary layers which have small angles, nearly right angled, but they do not have large angles.

• The focus of my question is on the condition number of the stiffness matrix. Small dihedral angles in tetrahedra do cause ill-conditioning of the stiffness matrix (see Shewchuk's "What is a good linear finite element"). I want to know if small (or large) dihedral angles in cells also cause ill-conditioning in the FVM system of equations. – J. Heller Aug 14 '16 at 6:57