An Edge-Based Unstructured Finite Volume Method For The Solution Of Potential Problems.

Abstract

The FV formulation is very flexible to deal with any kind of control volume and so
any kind of unstructured meshes, including triangular, quadrilateral, mixed or even dual
meshes. The element-based finite volume methods are usually either a node/vertex centered,
where the unknowns are defined at the nodes of the mesh, or element/cell centered where the
unknowns are defined within the element, usually at the element centroid. Both options have
advantages and disadvantages, but in two-dimensional applications all of them have basically
the same computational cost, which is proportional to the number of edges of the mesh. However,
the node-centered formulation has a strong connection with an edge-based finite element
formulation, when linear triangular elements are used, and requires less memory and
computations when extended for three-dimensional tetrahedral meshes. In this article an unstructured
finite volume node centered formulation, implemented using an edge-based data
structure, is adapted and detailed for the solution of two-dimensional potential problems. The
whole formulation is fully described considering triangular meshes, but it can directly be extended
and applied to any conform two-dimensional meshes. A straight extension for threedimension
is also possible but not attempted here. In order to demonstrate the potentiality of
the presented procedure some model problems are investigated.