ABSTRACT
This chapter considers boundary value problems described by nonself-adjoint differential operators. These could be single or multi-variable boundary value problems in single or multi-dimensional space. The steady-state one-dimensional convection-diffusion equation is representative of the energy equation encountered in more complex two- and three-dimensional flows. This chapter considers some special cases and compute solutions for them. Consider a two-element uniform mesh in which each element is a two-node linear element. These results can be compared with those reported by Hughes and Mallet using various upwinding formulations based on the Galerkin method with weak form. Through simple model problems we are able to demonstrate the most significant aspects of VIC and VC integral forms. In all applications with non-self-adjoint differential operators, the findings reported for the 1D and 2D convection-diffusion equations hold without exception. VC integral forms yield unconditionally stable finite element computational processes, hence are completely free of upwinding processes.
