TITLE:
Error Estimations, Error Computations, and Convergence Rates in FEM for BVPs
AUTHORS:
Karan S. Surana, A. D. Joy, J. N. Reddy
KEYWORDS:
Finite Element, Error Estimation, Convergence Rate, A Priori, A Posteriori, BVP, Variationally Consistent Integral Form, Variationally Inconsistent Integral Form, Differential Operator Classification, Self-Adjoint, Non-Self-Adjoint, Nonlinear
JOURNAL NAME:
Applied Mathematics,
Vol.7 No.12,
July
29,
2016
ABSTRACT: This paper presents derivation of a priori
error estimates and convergence rates of finite element processes for boundary
value problems (BVPs) described by self adjoint, non-self adjoint, and
nonlinear differential operators. A posteriori error estimates are discussed in
context with local approximations in higher order scalar product spaces. A
posteriori error computational framework (without the knowledge of theoretical
solution) is presented for all BVPs regardless of the method of approximation
employed in constructing the integral form. This enables computations of local
errors as well as the global errors in the computed finite element solutions.
The two most significant and essential aspects of the research presented in
this paper that enable all of the features described above are: 1) ensuring
variational consistency of the integral form(s) resulting from the methods of
approximation for self adjoint, non-self adjoint, and nonlinear differential
operators and 2) choosing local approximations for the elements of a
discretization in a subspace of a higher order scalar product space that is
minimally conforming, hence ensuring desired global differentiability of the
approximations over the discretizations. It is shown that when the theoretical
solution of a BVP is analytic, the a priori error estimate (in the asymptotic
range, discussed in a later section of the paper) is independent of the method
of approximation or the nature of the differential operator provided the
resulting integral form is variationally consistent. Thus, the finite element
processes utilizing integral forms based on different methods of approximation
but resulting in VC integral forms result in the same a priori error estimate
and convergence rate. It is shown that a variationally consistent (VC) integral
form has best approximation property in some norm, conversely an integral form
with best approximation property in some norm is variationally consistent. That
is best approximation property of the integral form and the VC of the integral
form is equivalent, one cannot exist without the other, hence can be used
interchangeably. Dimensional model problems consisting of diffusion equation,
convection-diffusion equation, and Burgers equation described by self adjoint,
non-self adjoint, and nonlinear differential operators are considered to
present extensive numerical studies using Galerkin method with weak form
(GM/WF) and least squares process (LSP) to determine computed convergence rates
of various error norms and present comparisons with the theoretical convergence
rates.