These videos were created to accompany a university course, Numerical Methods for Engineers, taught Spring 2013. The text used in the course was "Numerical Methods for Engineers, 6th ed." by Steven Chapra and Raymond Canale.
You could say it in the comments of this video that it represents the finite element Galerkin method in a very understandable. I've looked for this video for a long time and to be honest like that I don't need subtitles to understand your english :)
You have been used the weight functiona to be as the basisi function meanwhile you said the weight functions equal the first derivative of residual w.r.t alpha, how come? you have used the technique of least squares method
Wi is Partial Derivative of Error with respect to both Alpha0 and Alpha1. Alpha0 and Alpha1 are parameters, so Wi can be thought of as measure of how much error was observed due to our parameters. (Very Hand wavy explanation)
because you need to minimize the sum of squares of errors. Whenever you need to minimize, set the derivative to zero and since we need to minimize both parameters alpha_0 and alpha_1, we do it with a partial derivative.
Okay, let me ask the question another way, the "Let" appears to define the "weak" formulation, is that correct? If it is, I think you need to make that more clear.
Please, I need too learn Finite methods for Partial differential equations such as Helmholtz, laplace, conduction etc. I cannot lay my hands on materials with solved examples. Thanks.
Okay, when you define the Galerkin at around 12:00, you're letting W sub i, which would be derivatives with respect to your linear regression terms, subscripts 0 and 1, be equated to two linear equations, each with an alpha 0 and and alpha 1. It's doubly confusing since you now refer to subscripts 1 and 2, when your regression uses subscript 0 and 1. Maybe you can write out the basis functions a little more clearly, especially in terms of the alphas.
Watching it again, it appears the equation for the error minima is changed with the "Let". That we are no longer getting exactly the minima, but the Galerkin is getting us close enough; but also allowing us to manipulate the equations to be solvable. Is this what is happening?
No, it is an ordinary differential equation. The reason for the title is that the video is part of a unit on methods that *can* be used to solve PDEs, and the finite element method *can* be used to solve PDEs. I use an ODE because it makes the explanation simpler.
