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
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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?
 
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