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Standard Galerkin method reduces the problem

Find $u\in V$ such that $a(u,v) = f(v)$ for all $v \in V$, where $V$ is Hilbert space, $a$ is bilinear form and $f\in V^*$.

to a finite dimensional problem by introducing a $n$-dimensional subspace $V_n\subset V$, then we look for an approximation $u_n$ of the solution $u$ such that

Find $u_n \in V_n$ such that $a(u_n,v) = f(v)$ for all $v \in V_n$.

I would like to replace $V_n$ by a $n$-dimensional manifold $\mathcal{M}_n$ in $V$. So the reduced problem would be

Find $u_n\in \mathcal{M}_n$ such that $a(u_n,v)=f(v)$ for all $v\in T_{u_n}\mathcal{M}_n$, where $T_{u_n}\mathcal{M}_n$ is tangent space to the manifold $\mathcal{M}_n$ at the point $u_n$.

What do we know about this problem? What are the conditions on $\mathcal{M}_n$ for existence of $u_n$? Does $u_n$ converge to $u$ as we make $\mathcal{M}_n$ bigger and bigger ($n\rightarrow \infty$)? What is this method called?(I called it Manifold Galerkin method)

Can you please point me to the literature where they discuss this problem?

tom
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  • The answers to the following scicomp question may be helpful, though the question is more computationally oriented than the question here: http://scicomp.stackexchange.com/questions/19007/finite-elements-on-manifold – Nick Alger Aug 26 '16 at 18:15
  • @NickAlger Sorry but it is not helpful, the question is very different from mine and only the name of the question is similar. Please read my question a little bit more carefully. – tom Aug 26 '16 at 18:20
  • I did read your question carefully. My comment above is a helpful link to additional related information, not an answer – Nick Alger Aug 26 '16 at 18:28

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Although this is probably not exactly what you are looking for, I would like to point you to the Reduced Basis Method which might contain some valuable insight.
It is designed for parametrized PDEs of the form $$ a\big(u(\mu), v ; \mu \big) = f(v;\mu)$$ which allow a structured way to construct a manifold. However, the whole methods starts from a high-dimensional "classical" Galerkin approximation in a vector space. Then, for parameters $\mu \in \mathcal{D}$ the manifold $$\mathcal{M} = \{ u_\mathcal{N}(\mu_i) \}, \quad u_\mathcal{N}(\mu_i) \in V_\mathcal{N} $$ can be constructed which forms the basis of the subsequent analysis. However, this manifold is immediately turned back into a space through the Gram-Schmidt procedure. For this reduced space, a rather comprehensive theory on the properties of approximations based on a parametrized manifold is available.

To sum things up, the reason why you might no find much on the "Manifold Galerking Method" is that you lose the whole theory based on vector spaces (coercivity, continuity, error bounds, ...) but not get nothing in return. In fact, it seems to be more beneficial to construct a vector space based on your manifold (question remains how to choose this if you have not a parametrized PDE) and find links between the different vector spaces.

Dan Doe
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  • From theoretical point of view, yes you might not get much by considering manifolds. However, I was experimenting with Navier Stokes equation and my 'basis functions' were vortices with variable amplitude, position and radius. In particular, the position and radius are nonlinear parameters resulting in an approximating manifold and not a linear space. I think at some point I even had a proof that in a simple case this reproduces a vortex method. The advantage was that this approach is equation agnostic unlike vortex method that revolves around vorticity equation. – tom Jan 27 '22 at 07:19
  • One really interesting behaviour was that viscosity cause the radius to grow, that is quite expected. However, when I fixed the radius to a constant value, vortices started repulsing each another, as modifying their position was the only mechanism to "spread" velocity in space(i.e. exactly what viscosity does) – tom Jan 27 '22 at 07:22