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when using the Finite Element Method to solve the Poisson equation, one usually arrives at an (a priori) error estimate like $$ \lVert u-u_h \rVert \leq c h |u|$$ where $\lVert \cdot \rVert$ and $| \cdot |$ are the Norm and Semi-Norm in $H^{1,2}$. $u$ is the solution of the Poisson equation, $u_h$ the solution of the FE method. I am omitting some details here, but as my question is not specific to those details, that should be ok. See this lecture (chapter 7) for the relevant results.

My question: Why does one use the Seminorm of the unknown solution for an error estimate? To me, one should use the $L^2$ norms of the source term or the boundary data. How I am able to use the solution if I can only compute an aproximation $u_h$?

Any help is appreciated.

g7349290
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  • You can only use a priori estimates to compare the asymptotic convergence rates of different methods. You cannot even know what $c$ will be so why would you want to know what $|u|$ is? – knl Nov 05 '16 at 08:06

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Why does one use the Seminorm of the unknown solution for an error estimate?

Because we already know that $u$ exists. What we want is to prove we can construct an approximation $u_h$ of $u$, and that, indeed $u_h \to u$.

Sure $|u|$ is, a priori, unknow, but it's a constant, so we're sure that $ \| u_h -u\| \to 0$, and that's what we're interested in here.

If you want an estimate of $|u|$ with $f$ and the boundary condition $g$, you can use the results of Theorem 2.7 in your link.

Tryss
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