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.
Is there one general approach to build a projection methods for different problems?
linear algebra;finite element;discretization
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: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 $U$.)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 areBrenner, Scott: The Mathematical Theory of Finite Element MethodsErn, Guermond: Theory and Practice of Finite Elements(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.