THE K-VERSION OF FINITE ELEMENT METHOD FOR NONLINEAR OPERATORS IN BVP
Abstract
In the companion papers [1,2], authors introduced the concepts of k-version of finite element method and k, hk, pk, hkp-processes of the finite element method for boundary value problems described by self-adjoint and non-self adjoint operators using Ĥk,p(Ω) spaces with specific details including numerical studies for weak forms and least square processes. It was demonstrated that a variationally consistent (VC) weak form is possible when the differential operator is self-adjoint, however, in case of non-self-adjoint operators the weak forms are variationally inconsistent (VIC) which lead to degenerate computational processes that can produce spurious oscillations in the computed solutions. In this paper we demonstrate that when the boundary value problems are described by non-linear differential operators, Galerkin processes and weak forms can never be variationally consistent and hence result in degenerate computational processes and suffer from same problems as in the case of non-self-adjoint operators plus more due to the presence of non-linearity. In the proposed mathematical and computational frame-work, upwinding methods are neither required nor used. The k-version of FEM over Ĥk,p spaces for the Galerkin method with weak forms, though meritorious in comparison to Ĥ1,p(Ω) spaces, but it is plagued due to problems arising from variational inconsistency.
We demonstrate that the order of the space k in Ĥk,p(Ω) Hilbert spaces is an independent parameter in all computational processes in addition to the characteristic length h of the discretizations and the degree p of the local approximations. This gives rise to k-version of finite element method and thus, associated k, hk, pk, and hpk processes. The global differentiability of a finite element solution is only dependent on k. The h, p and hp-adaptive processes can not yield global differentiability of order higher than the order of the space containing the local approximations. It is shown that variational consistency of the integral forms and higher order global differentiability of a computed solution by increasing k in Ĥk,p(Ω) spaces are two most important features of mathematical and computational frame work if one wishes the computational process to (1) be non-degenerate and (2) yield solution with the same characteristics in terms of global differentiability as the theoretical solution.
In this paper, we illustrate the important variational aspects of the least squares finite element processes for non-linear partial differential equations of stationary processes. Variationally consistent least squares finite element method (LSFEM) using p-version basis functions in Ĥk,p(Ω) spaces provides a remarkably general framework for numerical simulation of any BVP described by non-linear differential operators in which any desired order of global smoothness or global differentiability is achievable. The proposed mathematical framework indeed allows one to numerically simulate characteristics of the theoretical solutions of any non-linear BVP regardless of the nature of the differential operator. Stationary one-dimensional Burgers equation is used as a model problem to present supporting numerical studies. Many issues and concepts related to upwinding methods and the behaviors of commonly encountered problems in Galerkin method for non-linear differential operators in BVP are considered, discussed and explained.
