IMI Interdisciplinary Mathematics InstituteCollege of Arts and Sciences

Mimetic and convergent discretization of vector fields on unstructured meshes

  • March 30, 2015
  • 1 p.m.
  • LeConte 312

Abstract

Finite difference and finite volume methods are superb at retaining key characteristics of the continuous system, such as positive definiteness, and conservation and transport of certain quantities. This is why FD/FV is popular in certainly areas, particularly in CFD, where specific behaviors of the underlying systems are of great interests for practical purposes. The analysis of FD/FV has lagged behind that of the Finite element methods (FEM), for which two reasons can be recounted. First, unlike the FEM, the FD/FV do not usually fit in the variational formulations. Second, even if they can be re-formulated into the variational formulations, the FD/FV often use piecewise constant functions as the trial and/or test functions, which are not even continuous. In this work, we aim to develop a new theoretical framework for analyzing FD/FV schemes for a wide range of fluid problems. There are two essential ingredients to this framework. The first is the external approximation of function spaces, which seems particular adept in dealing with discontinuous functions. The second is the tracking of divergence and vorticity, instead of individual derivatives. This approach gets rid of the need for a Cartesian coordinate system, and makes this framework applicable to unstructured meshes. Once the framework has been presented, we will apply it to the classical incompressible Stokes problem, and prove that the discrete solutions converges to the solution of the continuous system, without assuming that one actually exists.

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