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My question is probably going to be too general to answer it with a couple words. Could you please suggest a good reading in that case. Projection methods are used to reduce size of the solution space for the problems. And there are at least two very interesting applications (from my point of view). The first is the solving of continuum mechanics problems (Finite Element,Ritz methods) and the second is solving of systems of linear equations (Krylov subspace methods).

The question is the following: Is there a theory or some part of analysis that studies projection methods in all their applications? If so, can other methods, like finite volume methods, be built from this starting point?

I studied FEA at university but at the moment, all the discrete approximations are like a set of isolated "tools" that I can use in some particular case. Thanks.

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The Galerkin approach (seeking an approximation from a given subspace $U$ such that the residual is orthogonal to another given subspace $V$) is indeed very general (and not restricted to finite-dimensional spaces). In the context of the numerical solution of partial differential equations, there are essentially two conditions that $U$ and $V$ have to satisfy:

  1. The discrete problem must have a unique solution; this often requires verifying so-called inf-sup-conditions. (In the standard Ritz-Galerkin method, this is basically the Lax-Milgram theorem; for linear systems $Ax=b$, this amounts to $\dim U = \dim V$ and that no vector in $V$ is $A$-orthogonal to $U$.)

  2. The discretization error must become smaller as the dimension of $U$ and $V$ are increased. This requires certain approximation properties for the subspaces. Usually, one takes spaces of piecewise polynomials (as in the standard finite element method), but other choices are possible (e.g., spectral methods). (Similarly, projection methods for linear systems are often based on Krylov spaces, since these have nice approximation properties.)

In fact, (some) finite volume methods can be described as discontinuous Galerkin methods (where $U$ and/or $V$ consist of piecewise constant functions.

Most modern mathematical textbooks on finite element methods follow this approach. Two good examples are

(I particularly like the latter, as takes a very general approach to Galerkin methods, including mixed and hybrid finite elements and discontinuous Galerkin methods.)

For linear systems, a good general discussion of the projection approach is given in Saad's book.

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For the solution of differential equations, it may be useful to think in terms of The Method of Weighted Residuals (MWR) as coined by Crandall (1956) and described in a first review by Finnlayson and Scriven (1966) as

"The Method of Weighted Residuals unifies many approximate methods of solution of differential equations that are being used currently."

and

"The Method of Weighted Residuals is an engineer’s tool for finding approximate solutions to the equations of change of distributed systems."

In short, the MWR method unifies in a systematic way several common discretization methods.

Is this what you were thinking of?

For the solution of systems of linear equations, I see Krylov subspace methods as a general approach to build projection methods. The most problem-specific part of these methods is the choice of preconditioner for acceleration of convergence -- and this is usually problem specific how to make that choice.

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