First-Order System in One Dimension

Abstract

In fluid dynamics another important phenomenon is diffusion or conduction which is described by second-order derivatives in the governing equations. The Laplace or Poisson equation can be considered as the standard form of an equation describing isotropic diffusion in all space directions. For the Laplace or Poisson equation, the Rayleigh—Ritz method yields a symmetric and positive-definite system of linear algebraic equations and has an optimal rate of convergence. However, in practice, one is often interested in not only the primal variable (e.g., the temperature in heat conduction, the potential in irrotational incompressible flows, and the electric or magnetic potential in electromagnetics), but also the dual variable (e.g., the flux in heat conduction, the velocity in fluid flows, and the electric or magnetic field intensity in electromagnetics). The solution of dual variables computed by a posteriori numerical differentiation has low accuracy in general and is discontinuous across the element boundary.