A remark about Galerkin method

TL;DR

This paper proves that for any linear equation solved via Galerkin methods, there are at least as many unresolved right-hand sides as the dimension of certain finite-dimensional subspaces, highlighting limitations in the method.

Contribution

It establishes a fundamental lower bound on the number of unresolved right-hand sides in Galerkin methods for linear equations.

Findings

Existence of at least as many unresolved right-hand sides as five times the basis functions.

The lower bound applies to all linear versions of the Galerkin method.

Highlights limitations in the solvability of linear equations using Galerkin approaches.

Abstract

In this article was proved, that any linear equation $Lu=f$ in the case of any linear versions of the Galerkin method, has at least as many unsolved right-hand sides in the form of linear combinations $f=L\psi_1 +...+L\psi_N +L\psi_{N+1} +...+L\psi_{5N} +...+L\psi_{T}$, as there are finite-dimensional linear subspaces with dimensionality as much than five times as the number of basis functions $\psi_1,..., \psi_N$ .