FEM1D_SPECTRAL_SYMBOLIC, a MATLAB program which applies the spectral finite element method to solve the problem u'' = - pi^2 sin(x) over [-1,+1] with zero boundary conditions, using as basis elements the functions x^n*(x-1)*(x+1), and carrying out the integration using MATLAB's symbolic toolbox, by Miro Stoyanov.
This function solves the problem u'' = - pi^2 * sin( pi * x ) for x in (-1,1), with boundary conditions u(-1) = u(1) = 0.
This problem has exact solution u(x) = sin(pi*x).
We use the spectral finite element method. We assume that the solution can be written in terms of polynomials:
phi_1 = x^0 * (x-1)*(x+1),
phi_2 = x^1 * (x-1)*(x+1)
...
phi_i = x^(i-1) * (x-1)*(x+1) ...
For this particular problem, the solution is an odd function, so only every
other basis function will actually be useful.
MATLAB's symbolic toolbox is necessary for this program to run.
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
FEM1D_SPECTRAL_SYMBOLIC is available in a MATLAB version.
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fem1d_spectral_symbolic, a MATLAB program which applies the spectral finite element method (FEM) to solve the two point boundary value problem (BVP) u'' = - pi^2 sin(x) over [-1,+1] with zero boundary conditions, using as basis elements the functions x^n*(x-1)*(x+1), and carrying out the integration using MATLAB's symbolic toolbox, by Miro Stoyanov.
Miroslav Stoyanov, Oak Ridge National Laboratory, mkstoyanov@gmail.com.