Galerkin/Least-Squares Finite Element Processes for BVP in h, p, k Mathematical Framework

Abstract

This paper presents an investigation of the details of mathematical and computational aspects of the Galerkin/least-squares and the Galerkin/weak form least-squares finite element process within the mathematical and computational framework [1, 2, 3] based on h, p, k as independent computational parameters and requiring that the integral forms be variationally consistent (VC). Higher-order global differentiability of order (k−1) defined by the order k of the approximation space is essential for incorporating correct physics of the processes in the computations and that k is an independent parameter in addition to h and p in all finite element computations. In this paper the attributes of the Galerkin method, the Galerkin method with weak form, and least-squares processes in h, p, k framework with variationally consistent (VC) or variationally inconsistent (VIC) integral forms are utilized to investigate the mathematical features of the Galerkin/least-squares processes (GAL/LSP) and Galerkin/weak form least-squares process (GAL/WF/LSP) to establish (1) when such processes have a sound mathematical basis (2) role of functionals resulting from the Galerkin method, the Galerkin method with weak form and least-squares processes in GAL/LSP and GAL/WF/LSP (3) Importance of minimally conforming spaces and role of higher order spaces in GAL/LSP and GAL/WF/LSP. It is concluded that GAL/LSP only have sound mathematical basis for self adjoint differential operators when viewed within h,p,k framework with the requirement that the integral forms be variationally consistent. For non-self adjoint and non-linear differential operators GAL/LSP and GAL/WF/LSP do not have a sound mathematical basis within the proposed framework. Currently used finite element processes based on GAL/WF/LSP utilizing local approximations of class C 0 violate basic mathematical principles, are in violation with physics and hence do not provide a valid and mathematically sound approach. Investigation of the GAL/LSP and GAL/WF/LSP when the local approximations are in minimally conforming spaces or in spaces of order higher than minimally conforming spaces, shows that the functionals from GAL and GAL/WF processes only have detrimental affect on the LSP process. Numerical studies are presented to demonstrate various aspects and features.