fem1d, a Python code which applies the finite element method (FEM) to a 1D linear two point boundary value problem (BVP), using piecewise linear basis functions.
The BVP to be solved is:
-u'' = x * ( x + 3 ) * exp ( x ) over the interval 0 < x < 1 u(0) = 0.0 u(1) = 0.0
The exact solution is:
u(x) = x * ( 1 - x ) * exp ( x )
A version of the finite element method is used. Six equally spaced nodes are defined, from 0.0 to 1.0, dividing the interval into 5 elements. At node I, we associate a "hat" function, or piecewise linear basis function, PSI(I)(X), which has the value 1 at that node, is 0 at all other nodes.
We look for an approximate solution to our problem of the form
UH(X) = sum ( 1 <= I <= 6 ) C(I) * PSI(I,X)so that now the problem becomes the determination of the unknown coefficients C.
We take the original BVP, multiply by test function PSI(J,X), integrate over the region, and apply integration by parts, to obtain a linear system of the form
A * C = FWe modify the first and last rows of the linear system to enforce the boundary conditions, then solve to determine the values of C.
The computer code and data files described and made available on this web page are distributed under the MIT license
fem1d is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.
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