# Compatible Reconstructions of Vectors and their Application to the Navier-Stokes Equations

Friday, May 14, 2004 - 3:00pm - 4:00pm

Keller 3-180

Blair Perot (University of Massachusetts)

Compatible spatial discretizations of partial differential equations invariably deal with vector components as the primary variables. This use of individual vector components (rather than entire vectors) is fundamental and closely tied to algebraic topology and differential forms. However, there are situations where the entire vector is required. The Navier-Stokes equations for fluid dynamics exhibit two such situations. First, the advective term in the equations requires a velocity vector to be defined. Second, the Navier-Stokes equations have certain vector conservation statements (like conservation of momentum, kinetic energy, and circulation) that one would like to construct discrete analogs for.

Since vectors are not primary quantities, their reconstruction is a numerical approximation and is not uniquely specified when compatible spatial discretizations are used. A number of different vector reconstruction proposals for both structured and unstructured meshes in 2D and 3D are described. The relationship between these reconstructions is explored along with the resulting conservation properties (when they are known to exist). We develop a unified framework in which vector reconstruction is associated with discrete averaging operators, and where compatible vector reconstructions (with conservation statements) have averaging operators which commute with the discrete differential operators and are closely related to them.

This work has application even when vectors are not explicitly required (such as in electromagnetics, elasticity, and Stokes flow). Discrete Hodge star operators with attractive properties can be constructed from compatible vector reconstruction operators (due to the fact that they commute with the differential operators) even when an explicit vector is never required to solve the equations.

Since vectors are not primary quantities, their reconstruction is a numerical approximation and is not uniquely specified when compatible spatial discretizations are used. A number of different vector reconstruction proposals for both structured and unstructured meshes in 2D and 3D are described. The relationship between these reconstructions is explored along with the resulting conservation properties (when they are known to exist). We develop a unified framework in which vector reconstruction is associated with discrete averaging operators, and where compatible vector reconstructions (with conservation statements) have averaging operators which commute with the discrete differential operators and are closely related to them.

This work has application even when vectors are not explicitly required (such as in electromagnetics, elasticity, and Stokes flow). Discrete Hodge star operators with attractive properties can be constructed from compatible vector reconstruction operators (due to the fact that they commute with the differential operators) even when an explicit vector is never required to solve the equations.