10

The reason is that with the exception of linear problems, if you do a Fourier (or other) decomposition in time, you end up with a significant number of problems that are coupled globally in time. In other words, you have to solve lots of problems on the entire time interval concurrently. That will typically bust your computational or memory budget. The ...


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 ...


9

If your discrete equations represent your problem properly you should have the right eigenvalues, although numerically you can get a small imaginary part when solving the eigenvalue problem. As an example, let us consider the axisymmetric case of a cylindrical well. The Schrödinger equation would be written as $$\frac{\partial^2 \psi}{\partial r^2} + \frac{...


5

If you know the Chebyshev expansion for $f(z)$, why don't you formally integrate the polynomials using the recurrence relation for Chebyshev polynomials ? The Clenshaw-Curtis method is based on this approach (combined with an intelligent use of the FFT).


4

This is probably too late to help you, but here is a compact code that will do points, weights and first derivatives for Gauss, Lobatto or either Radau. function [x,w,A] = OCnonsymGLReig(n,meth) % code for nonsymmetric orthogonal collocation applications on 0 < x < 1 % n - interior points % meth = 1,2,3,4 for Gauss, Lobatto, Radau (right), ...


4

For something with a spectral flavor in time, look at deferred correction methods, starting with this paper. I would argue that they're not spectral in the usual sense of the word, but they give you a family of arbitrary-order Runge-Kutta methods, so if you think of "refining" by increasing the order (by adding more nodes), then the convergence can be ...


3

The paper [1] gives an explicit construction of the Bernstein form of a set of orthogonal polynomials on simplices based on Legendre polynomials. [1] Farouki, R.T., Goodman, T.N.T and Sauer, T: Construction of orthogonal bases for polynomials in Bernstein form on triangular and simplex domains, Computer Aided Geometric Design 20 (2003), 209-230, DOI: 10....


2

Asking Tim Davis would be the best approach to finding out.


2

I think Rob Kirby told me once that he had written something on using Bernstein polynomials for FEM. Take a look at his web site at Texas Tech (or now at Baylor).


2

As I understand it, collocation method for partial differential equations is something akin to interpolation. First we characterize the solution space as a linear combination of some set of linearly independent functions $\phi_i(x)$. The appropriate choice of $\phi_i(x)$ depends on the problem. But assuming that the anticipated solution is sufficiently ...


2

It's true that you can't eliminate $y_1,y_2$ to obtain something where every variable you have has appropriate boundary values. However, you can take derivatives on both sides of the first two equations to get $$y_1'' = c y_3' \\ y_2'' = c y_4' \\ y_3' = -f(y_1, y_2)y_2 \\ y_4' = f(y_1, y_2)y_1 $$ and then eliminate $y_3',y_4'$: $$ y_1'' = ...


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

At line 473 of the source code you provided : contrl - is the actual driver of the package. This routine contains the strategy for nonlinear equation solving. This code is written in Fortran 77 so you can't take benefit from a module. That means that you probably need to create a main program from scratch, prepare the arguments to call the subroutines you ...


1

check this : http://people.sc.fsu.edu/~jburkardt/m_src/sandia_sparse/sandia_sparse.html this may help you. this is for just points. also this is very nice tool box http://www.ians.uni-stuttgart.de/spinterp/ you can generate the points in space and then use deterministic finite element solver to evaluate those points and then calculate the statistic of the ...


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