Galerkin method

In this study, we showed Banach space operator in the implementation of equations Galerkin approximate solution method. In here we gave some notions and definitions. Especially, we showed the convergence of Galerkin's approximations implementation of the Galerkin series. The Galerkin approximations stability condition and Galerkin's convergence of scheme was written and examined. We considered the solution of certain variation problems by the Galerkin method.We examined the finite elements method with implementation of the Galerkin's schema. We showed that the Galerkin series coincide with the Fourier series when we considered the problem in the Hilbert space setting and when the operator is linear self-adjoint operator. We showed how to use the Galerkin method to solve first order differential equations as an example we applied this to the Cauchy Problem.