International Journal of Engineering Inventions e-ISSN: 2278-7461, p-ISSN: 2319-6491 Volume 4, Issue 6 (November 2014) PP: 39-43
On the Analysis of the Finite Element Solutions of Boundary Value Problems Using Gale kin Method Emenogu N. George1, Oko Nlia2, Okereke R. Ngozi3 1
Department of Mathematics, Michael Okpara University of Agriculture, Umudike, Abia State, Nigeria 2 Department of Mathematics/Statistics, Akanu Ibiam Federal Polytechnic Uwana Afrkpo 3 Department of Mathematics, Michael Okpara University of Agriculture, Umudike, Abia State, Nigeria ABSTRACT: The finite difference method can be considered as a direct discretization of differential equations but in finite element methods, we generate difference equations by using approximate methods with piecewise polynomial solution. In this paper, we use the Galerkin method to obtain the approximate solution of a boundary value problem. The convergence analysis of these solution are also considered. I. Introduction Several researchers including Nasser .M. Abbasi 1 , S.N. Atluri 2 have shown in general how the finite element method could be used to solve both ordinary and partial differential equations. In this paper, we focused particularly on the Galerkin’s method. The Galerkin method is one of the integral parts of the weighted Residual methods. The weighted residual methods are approximate methods which provide analytical procedure for obtaining solution in the form of functions which are close in some sense to the exact solution of the boundary value problem or initial value problem. Suppose they seek an approximate solution u(~đ?‘Ľ ) to a differential equation đ?‘Ľ đ??ż đ?‘˘(~đ?‘Ľ ) = đ?‘“(~đ?‘Ľ ) ~ ∈ â„Ś (1.1) đ?‘Ľ Where đ??ż is a differential operator, đ?‘˘(~ ) is the unknown solution of interest, đ?‘“(~đ?‘Ľ )is a known forcing function, â„Ś is the domain over which the differential equation applies, ∂ â„Ś is the boundary of the domain, ~đ?‘Ľ đ?‘–đ?‘ a vector containing the independent variables and the proper number and type of boundary conditions are specified along ∂ â„Ś. The aforementioned procedure involves the following steps i. Choose a trial space SN of possible approximate solutions. SN should be a finite N- dimensional function space with a set of basis functions Ă˜1 (~đ?‘Ľ ), Ă˜2 (~đ?‘Ľ ), ‌ , Ă˜N (~đ?‘Ľ ) ii. Pick a trail function V(~đ?‘Ľ ), ∈ đ?‘†đ?‘ , so that đ?‘›
�(~� )
âˆ?đ?‘˜ ∅đ?‘˜ (~đ?‘Ľ )
=
(1.2)
đ?‘˜=1
Where âˆ? đ?‘˜ are scalars iii. Minimize the residual đ?‘… đ?‘Ľ = đ??ż đ?‘ˆ đ?‘Ľ − đ?‘“(đ?‘Ľ) (1.3) Which is a measure of the amount by which đ?‘‰(đ?‘Ľ) fails to satisfy the original equation. By doing so, a set of algebraic equations in terms of âˆ?đ?‘– are obtained which are solved to determine the âˆ?đ?‘ . The method of weighted residuals (MWR) seeks to minimize đ?‘…(đ?‘Ľ) by forcing it to zero in a weighted average sense over the entire domain. This requires choosing a set of linearly independent weight functions đ?‘¤1 (~đ?‘Ľ ), đ?‘¤2 (~đ?‘Ľ ), ‌ , đ?‘¤đ?‘š (~đ?‘Ľ ) such that â„Ś
đ?‘… đ?‘Ľ đ?‘¤đ?‘– (~đ?‘Ľ )đ?‘‘ ~đ?‘Ľ = đ?‘œ
(1.4)
T herefore, to apply MWR, two choices must be made, (a) Choice of trial space SN with concomitant definition of basis functions đ?œ™đ?‘–, đ?œ™2‌ , đ?œ™đ?‘ đ?‘š (b) Choice of weight functions đ?‘¤đ?‘– (~đ?‘Ľ ) đ?‘–=1 (c) Different choices of weight functions lead to different MWR approximations. Once a weight function đ?‘¤đ?‘– (~đ?‘Ľ ) is specified, integrations in equation (1.4) can be performed, in conjunction with equation (1.2) and (1.3) to produce a system of algebraic equations in âˆ?đ?‘– . Since our focus is the Galerkin’s method, the weight functions are the basis functions for the trial space đ?‘¤đ?‘– đ?‘Ľ = đ?œ™đ?‘– (đ?‘Ľ) www.ijeijournal.com
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