# First order finite volume spatial discretization of the heat equation on an unstructured triangle mesh

Consider a scalar field $u$ on an unstructured triangle mesh which is constant on each face. Let $A_i$ be the area of triangle $T_i$, $N(i)$ the set of triangles sharing an edge with $T_i$, and $L_{ij}$ the length of the edge between $T_i$ and $T_j$. I want the simplest possible spatial discretization of the heat equation $$u_t = k \Delta u$$ The natural choice is $$A_i \frac{du_i}{dt} = k \sum_{j \in N(i)} L_{ij} \frac{u_j - u_i}{h_{ij}}$$ where $h_{ij}$ measures "how far apart" the triangles are.

The question: What are good choices for $h_{ij}$? In particular, are there choices which behave reasonably in the presence of obtuse/skinny triangles?

A two-point flux like this is not convergent if the mesh is not "orthogonal", in the sense that the edge/face between two cells is orthogonal to the line segment joining the cell centroids. If your mesh is orthogonal, you would use the distance between centroids for $h_{ij}$ above.

If you would like a method to work on more general meshes within the cell-centered finite volume framework, you have to do a bit of extra work. A classical approach is to reconstruct inside cells, as outline in these notes. This is not very robust, especially in the presence of irregular coefficients.

Another alternative is to consider the mixed finite element method using a BDM-1 space (constant pressure, linear velocity on faces/edges) and choose a reduced quadrature consisting of one point per vertex. That special quadrature causes the velocity mass matrix to be block diagonal, with one block per vertex, so the velocity degrees of freedom can be eliminated, returning you to a cell-centered non-mixed formulation. This is illustrated in Figure 2.3 from Wheeler and Yotov (2006).

After velocity is eliminated, the cell-centered operator is SPD and cells are coupled only to other cells they share a vertex with, which is sparser than the gradient reconstruction method mentioned earlier. With this method, pressure is second order superconvergent and cell centers and velocity is first order convergent, even on irregular meshes. If the mesh is within $O(h^2)$ of affine, velocity is also second order superconvergent at face centers. This method naturally handles general tensor-valued coefficients. This method is a multipoint flux approximation.

It is known (see page 10 of Edwards) that 9-point schemes for regular quadrilateral meshes cannot be monotone for general tensor-valued coefficients. One approach to maintain monotonicity is to use a nonlinear discretization.

If you use the discontinuous Galerkin method with piecewise constant shape functions, you end up with a scheme just like yours. It provides a systematic way of constructing the weights as well as higher order schemes if you want.

• True, though there are many choices for the elliptic flux, none of which are deeply satisfying. – Jed Brown Mar 29 '13 at 12:34
• Yes, that is correct. I simply wanted to point out that there is a way at that problem that comes from an entirely different direction than the finite volume methods that were the starting point of the original question. – Wolfgang Bangerth Mar 30 '13 at 1:33