Strong and Weak Form of the Governing Differential Equations in Least Squares Finite Element Processes in h,p,k Framework

Abstract

This paper presents an investigation of performance of the least squares finite element process in hpk mathematical framework utilizing: (i) governing differential equations (GDEs) containing highest order derivatives of the dependent variables (strong form of GDEs); (ii) GDEs containing only first order derivatives of the dependent variables and hence constituting a system of first order equations (weak form of GDEs) derived either directly from conservation laws or derived using auxiliary variables or auxiliary equations. It is shown that while the weak form of the GDEs may appear perfectly legitimate, the auxiliary equations in such forms cause irrecoverable inconsistencies in the resulting computational processes due to the fact that local approximations for the dependent variables and the auxiliary variables always remain inconsistent regardless of the choices p-level and the orders of the approximation spaces. The inconsistency of local approximations in the auxiliary equations may introduce spuriousness in the numerically computed solution that may progressively grow if the theoretical solution gradients are severe and isolated and may even cause failure of the computational process. It is shown that strong forms of the GDEs have no such problems and are highly meritorious over the weak forms in all aspects and hence are always the preferred choice but require the use of higher order local approximation spaces. One dimensional steady state convection diffusion equation (BVP) and one dimensional Riemann shock tube (IVP) are used as model problems to illustrate various concepts and to present numerical studies.