The k-Version of Finite Element Method for Initial Value Problems: Mathematical and Computational Framework

Abstract

This paper presents a mathematical and computational framework for initial value problems (IVP) in which the numerical approximations can be of higher order global differentiability in space and time and the resulting computational processes are unconditionally stable. This is accomplished using Hk, p scalar product spaces containing basis functions of degree p = (p1,p2), p1 and p2 being the degrees of local approximation in space and time and order k = (k1, k2), k1 and k2 being orders of the scalar product space in space and time and ensuring that the integral forms are space-time integral forms that are space-time variationally consistent (STVC). It is shown that order of the scalar product space k (in space and time) is an intrinsically important independent parameter in all finite element computations for IVP in addition to the discretization length h and the degree p of the local approximations, thus in all finite element computations all quantities are dependent on h, p and k. Hence, we have k-version of finite element method and associated k, hk, pk and hpk processes in addition to h, p and hp processes for IVP. Space-time meshes as well as space-time, time marching approaches are discussed and it is demonstrated that space-time, time marching processes are superior in all aspects compared to space-time meshes. Significant features of the proposed mathematical and computational framework are presented and discussed mathematically in context with Galerkin method, Petrov Galerkin method, Weighted Residual method, Galerkin method with weak form and Least Squares Processes.