# 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.

## Key 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 $\psi_1,..., \psi_N$ .

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Source: https://tomesphere.com/paper/1705.06880