When using the Galerkin Method, the condition enforced between the residual and the approximation function is that the residual should be orthogonal to the space spanned by the approximation functions.
To explain further, let's consider a partial differential equation (PDE) problem being solved using the Galerkin Method. The goal is to find an approximate solution by selecting a set of basis functions to represent the unknown solution.
The residual is the difference between the actual PDE equation and the approximate solution. The Galerkin Method enforces the condition that the residual should be orthogonal to the space spanned by the approximation functions.
Orthogonality means that the inner product (or dot product) between the residual and each of the approximation functions is zero. This condition ensures that the approximation functions capture the important features of the solution and minimize the error between the exact solution and the approximation.
By enforcing this orthogonality condition, the Galerkin Method allows us to determine the coefficients of the approximation functions and obtain a more accurate solution to the PDE problem.
In summary, the condition enforced between the residual and the approximation function when using the Galerkin Method is that the residual should be orthogonal to the space spanned by the approximation functions.