vertexcentered

Vertex-centered refers to a grid-based discretization in which the unknown field values are associated with the vertices (grid points) of the mesh rather than with the centers of cells or elements. This approach is used across several numerical methods, including finite elements, finite differences, and finite volumes, particularly on structured or unstructured meshes. In a vertex-centered formulation, the solution unknowns are defined at the mesh nodes, and the governing equations are discretized to produce a sparse system that couples values at neighboring vertices. In contrast, cell-centered formulations place unknowns at the centers of cells.

On structured grids, a vertex-centered discretization often yields stencils that connect a vertex to its neighboring vertices (for example, a four-point stencil in two dimensions for diffusion problems). On unstructured meshes, vertex-centered methods correspond to nodal discretizations where the degrees of freedom are located at mesh nodes and assembled via element contributions, as in many finite element schemes.

Advantages of vertex-centered methods include natural handling of Dirichlet boundary conditions at boundary vertices, alignment with nodal finite element spaces (Lagrange bases), and straightforward implementation on unstructured meshes. They are also well-suited for problems where field values are naturally defined at nodes and for certain types of linear or nonlinear solvers.

Disadvantages can arise in specific multiphysics or conservation contexts, where fluxes are more naturally associated with cell faces or edges, potentially leading to less local flux control or conditioning challenges. In some applications, cell-centered or staggered arrangements may better preserve conservation properties or avoid numerical decoupling.

Commonly, vertex-centered approaches appear in structural mechanics, heat conduction, electromagnetics, and groundwater flow, and form a core part of nodal finite element discretizations. They are often described as node-centered or nodal discretizations in related literature.