MHB Finite element method for the construction of the approximation of the solution

AI Thread Summary
The discussion focuses on applying the finite element method to solve a specified two-point boundary value problem involving a differential equation. The proposed approach involves using a finite element space of continuous and partially linear functions, leading to a weak formulation of the problem. The user seeks validation of their method, particularly whether the chosen test function \( g \) meets the boundary conditions \( g(0)=0 \) and \( g'(1)=mg(1) \). There is a request for clarification on whether this method is correct or if alternative approaches exist. The conversation emphasizes the importance of ensuring that the test functions adhere to the problem's constraints.
mathmari
Gold Member
MHB
Messages
4,984
Reaction score
7
Hey! :o

Given the following two-point problem:
$$-y''(x)+(by)'(x)=f(x), \forall x \in [0,1]$$
$$y(0)=0, y'(1)=my(1)$$
where $ b \in C^1([0,1];R), f \in C([0,1];R)$ and $ m \in R$ a constant.
Give a finite element method for the construction of the approximation of the solution $y$ of the problem above, where the finite element space ($S$) consists of continuous and partially linear functions.

My idea is the following:
$ u \in S:$
$$ -\int_0^1{u''g}dx+ \int_0^1{bu'g}dx= \int_0^1{fg}dx$$
$$ -u'g|_0^1+ \int_0^1{u'g'}dx+ \int_0^1{bu'g}dx= \int_0^1{fg}dx$$
$$-mu(1)+g(1)+ \int_0^1{u'g'}dx+ \int_0^1{bu'g}dx= \int_0^1{fg}dx$$
$ \forall g \in S$

Could you tell me if this is correct?
 
Last edited by a moderator:
Mathematics news on Phys.org
To find the method we take a function $g$ of $S$, right? Does this function satisfy the conditions of the problem? I mean $g(0)=0, g'(1)=mg(1)$... Or is there an other way to find the method?
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
Back
Top