FEM1D_BVP_QUADRATIC is a MATLAB program which applies the finite element method, with piecewise quadratic elements, to a two point boundary value problem in one spatial dimension, and compares the computed and exact solutions with the L2 and seminorm errors.
The boundary value problem (BVP) that is to be solved has the form:
- d/dx ( a(x) * du/dx ) + c(x) * u(x) = f(x)in the interval 0 < x < 1. The functions a(x), c(x), and f(x) are given.
Boundary conditions are applied at the endpoints, and in this case, these are assumed to have the form:
u(0.0) = 0.0; u(1.0) = 0.0.
To compute a finite element approximation, a set of n equally spaced nodes is defined from 0.0 to 1.0, a set of piecewise quadratoc basis functions is set up, with one basis function associated with each node, and then an integral form of the BVP is used, in which the differential equation is multiplied by each basis function, and integration by parts is used to simplify the integrand.
A simple three point Gauss quadrature formula is used to estimate the resulting integrals over each interval.
u = fem1d_bvp_quadratic ( n, @a, @c, @f, x )where
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
FEM1D_BVP_QUADRATIC is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version and a Python version..
BVP4C, MATLAB programs which illustrate how to use the MATLAB command bvp4c(), which can solve boundary value problems (BVP's) in one spatial dimension.
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FEM2D_BVP_QUADRATIC, a MATLAB program which applies the finite element method (FEM), with piecewise quadratic elements, to a 2D boundary value problem (BVP) in a rectangle, and compares the computed and exact solutions with the L2 and seminorm errors.
FEM1D_BVP_TEST runs example problems described by Dianne O'Leary, and several others.
One of the tests makes convergence plots in the H1, L2 and Max norms.
You can go up one level to the MATLAB source codes.