10

I would not insist on demanding a geometrical meaning from Galerkin methods in general. There is a connection, but it becomes less meaningful as you extend it further and further. (In a sense, it would be better to call Galerkin methods "generalized projection methods".) To really understand the connection between collocation and Galerkin methods requires ...


5

To understand the functional analysis (existence & uniqueness) part of the finite element method, it's helpful to see an analogy in linear algebra. In the world of linear algebra, when you assemble any system of linear equations, you want to make sure that a solution exists. Theory suggests that you have to make sure that the (square) matrix system is ...


5

You assume that a property that holds for the original function $f$ is also true for the $L_2$ projection $u_h=P_h f$. In your case, the property is that if $f$ is non-negative, then $u_h$ should also be non-negative. But this assumption is not in general correct. You already know this from taking a few terms of a Fourier or Taylor series: Just because a ...


4

There is a bug in your code :-) First, going down in each column, you have cases where the error becomes larger again, and this should clearly not happen because the finite element spaces are nested going from top to bottom. Secondly, moving left to right the error should also decrease because the finite element spaces are nested, but again this is not ...


4

Chebyshev are orthogonal wrt to a weight function which is singular at the end-points. When you approximate a function f(x) with Chebyshev, the convergence of the approximations is not affected by the values of f or its derivatives at the end points. The convergence rate depends only on the smoothness of the function f and not on its boundary values. This is ...


2

First, Galerkin in his article from 1915 does not discuss any weighting other the $\omega(x) = 1$. The Galerkin method is a direct generalization of the Rayleigh-Ritz method, and a variational procedure cannot be constructed with any other weight. That approach was the only one used until the FFT came about. If you want to use FFT, then you must use ...


2

I will answer you in order: 1) The problem to solve is independent of the weighing function you choose, e.g. you could solve for a PDE using standard finite element basis (Lagrange) without using special functions that has special properties (orthogonal functions). 2) The weight function $w(x)$ for certain polynomials to be orthogonal arise naturally, e.g. ...


2

The basis functions $\{p_n(x): n \in \mathbb{N}\}$ can be orthogonal in $[a, b]$ with respect to a weight function $w(x)$. For example, Hermite polynomials are orthogonal in $(-\infty, \infty)$ with respect to $e^{-x^2}$, $$\int_{-\infty}^\infty H_m(x) H_n(x)\, e^{-x^2}\, \mathrm{d}x = \sqrt{ \pi}\, 2^n n! \delta_{nm}\, .$$ The weight function is needed ...


1

To my knowledge these are the same things. However, this type of thing is common. For example, the proper orthogonal decomposition also has field-specific names. Others call it principal component analysis, the Karhunen--Loeve expansion, or empirical orthogonal functions. It is also no different than an autoencoder with linear activation function. I'm sure ...


1

How are you actually computing error norm and convergence rate ? If I use your error table, I get the expected result. E.g. P2, 256 and 1024 cells In [1]: log(5.527e-3/1.382e-3)/log(2) Out[1]: 1.9997389968259023 P4, 256 and 1024 cells In [4]: log(1.141e-6/7.135e-8)/log(2) ...


1

I believe these issues arise because you are solving the transient form of the heat equation which, being parabolic, can lead to some instability for continuous galerkin method. There are ways to circumvent that. GGLS approaches (Galerkin Gradient Least Square) introduce stabilization term that dampen these oscillations by minimizing the Error on ...


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