The Galerkin finite element method is a numerical method with the idea of trying to find an approximate solution in the test function space (spatial space only).  For conservation laws, the test function space is usually chosen to be a collection of polynomials with degree up to some finite number, say k. Hence we can find a finite orthogonal basis to the space and all functions in the space can be written as a linear combination of the basis functions.  In such a design, we will eventually be solving a system of ODEs with respect to time t.