I often read in finite element text books that we seek a conforming approximation (say) for $X_0^h \subset H^1_0$, where $H^1_0$ is a Sobolev space and $X_0^h$ is a finite dimensional approximation. Then they go on to introduce the finite elements which are conforming. However, I've not read anywhere which tells me:
- Why is it necessary to have a conforming approximation space
- What exactly is a confirming space (in simple terms, without heavy functional analysis)
- What implications this has on the approximation (in terms of convergence/error estimates)
This is a typical example from a text book:
For a given partition of $\Omega$, a conforming approximation of $H^1 (\Omega)$ is a space of continuous functions defined by a finite number of parameters (degrees of freedom). This is usually achieved by using a space of piecewise polynomial functions on the elements K of the partition for $\Omega$. The degrees of freedom are then a set of linear forms on the set of polynomials on K.
Then they begin to introduce the finite element basis functions. From that I can just about figure out what a conforming approximation is, nothing more.
Can someone tell me more about this? - or point me in the right direction.
Many thanks.
It is not necessary to have a conforming method, but it simplifies the numerical analysis (e.g., error estimates). A popular non-conforming method is the Discontinuous Galerkin method.
A function space $X_0^h$ is conforming iff $X_0^h \subset H_0^1$. If you know that functions in $X_0^h$ are polynomials on the cells of your mesh, you have $X_0^h \subset H_0^1$ iff all your functions are continuous, i.e., you have no jumps between different cells.
I am not really sure, what you would like to ask. With non-conforming methods, you can have the same approximation quality as with conforming methods, but the proofs differ. For conforming methods, you typically use Céa's lemma + interpolation error estimates, for non-conforming methods you need the Lemma of Strang + interpolation error estimates.