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If I have the following problem $$-\mu u'' + u' = 1$$

with boundary conditions $u(0) = u'(1) = 1$ in the interval $\Omega = (0,1)$. The exact solution is $$u(x) = x + 1$$

Will the FEM approximation with piecewise linear functions ($\mathbb{P}^1$) have a very small (or no) error? If so, how to think about it to have an image in my mind?

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1 Answer 1

up vote 4 down vote accepted

By Céa's lemma, the finite element discretization error is bounded by the best approximation error: $$\|u-u_h\|_{H^1(0,1)} \leq C \min_{v_h\in \mathbb{P}^1} \|u-v_h\|_{H^1(0,1)},$$ where $u\in H^1(0,1)$ is the exact solution, $u_h\in\mathbb{P}^1$ is the finite element solution. Since in your case the exact solution $u$ is actually in $\mathbb{P}^1$, the minimum is attained for $v_h = u$, and so the error is zero.

Basically, you are solving the equation by projection onto a finite-dimensional subspace. If your solution actually lies in this subspace, you will find it.

EDIT: The key idea is the following. You are looking for a weak solution, i.e., $u\in V$ such that $$a(u,v) = f(v) \quad\text{for all} \quad v\in V.\tag{1}$$ (In your case, $a(u,v) = \int_0^1 \mu u'v' + \int_0^1 u' v$ and $f(v) = \int_0^1 v + v(1)$.) The finite element approximation consists in choosing $V_h\subset V$ (in your case, $V_h = \mathbb{P}^1$) and finding $u_h\in V_h$ such that $$a(u_h,v_h) = f(v_h) \quad\text{for all} \quad v_h\in V_h.\tag{2}$$ Setting $v=v_h\in V$ in $(1)$ and subtracting $(2)$ gives $$a(u-u_h,v_h) = 0 \quad \text{for all}\quad v_h\in V_h.$$ This tells you that your error $u-u_h$ is orthogonal to $V_h$ with respect to the "inner product" defined by $a$, or that $u_h$ is the orthogonal projection of $u$ on $V_h$ (again, with respect to $a$). And orthogonal projections on a subspace are characterized by best approximation properties with respect to appropriate norms (in this case, the $H^1$ norm since $a$ happens to be coercive in this norm).

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This should also answer your previous question, by the way. –  Christian Clason Dec 21 '12 at 11:10
    
Yes it is closely related. I tried so many different things but I think this is what I needed from the beginning. Thank you. –  BRabbit27 Dec 21 '12 at 14:25
    
I've tried to understand the Céa lemma, but I couldn't I don't know how to interpret the whole thing. Could you develop more what you said with the formulas of norms and all that stuff, or at least how to think all this stuff. –  BRabbit27 Dec 21 '12 at 16:49
    
I've added some details. You should really get a textbook on finite element methods and work through that. If you are more interested in applications than in the mathematical framework, maybe you could give the book An Analysis of the Finite Element Method by Gilbert Strang and George Fix a try. Strang also has an excellent set of video lectures at MIT Open Courseware (FEM starts at Lecture 17). –  Christian Clason Dec 21 '12 at 17:33
3  
The theory of finite element methods (including error estimates) is quite mathematical, and requires knowledge of real analysis, functional analysis and partial differential equations (as in Rudin's Real and Complex Analysis, Brezis' Functional Analysis and Evans' Partial Differential Equations, respectively - but not all of it, of course). If you have no mathematical background, that can be quite challenging. But I find the first three chapters of Endre Süli's lecture notes quite accessible. –  Christian Clason Dec 21 '12 at 19:15

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