FEM1D - A 1D Finite Element Solver

FEM1D applies the finite element method to the one dimensional boundary value problem

-U''= F

0 < X < 1.

U(0) = U(1) = 0.

The code is broken up into a number of functions, some of which can be call directly or used for other purposes. In particular, BF_LINEAR and FE_LINEAR can easily be called directly to produce data suitable for plotting a basis function or a finite element function:

xc = linspace ( 0.0, 1.0, 11 );
x = linspace ( 0.0, 1.0, 100 );
[ b, dbdx ] = bf_linear ( 7, x, xc );
plot ( x, b )

FEM1D is a teaching code, and therefore is not optimized. For example, the stiffness matrix is stored as a full N by N matrix, and no effort is made to use a special solver for the resulting linear system. The loops of the assembly routine behave as though every basis function interacts with all others. The stiffness matrix is assembled with a one-point quadrature rule. There are many cases where a FOR loop is used but where a vectorized loop would be much faster. The boundary conditions, the basis functions, the exact solution, the differential equation are all quite simple. However, if the student can understand this program, it can easily be modified to be more efficient and flexible.

FEM1D was supplied by Professor Janet Peterson.

Files you may copy include:

• fem1d.m, the main routine.
• assembly.m, assembles the coefficient matrix.
• bf_linear.m, evaluates a linear basis function.
• bracket.m, finds the left end of the interval containing a point.
• exact.m, evaluates the exact solution.
• fe_error.m, computes L2 and H1 error.
• fe_linear.m, evaluates the finite element linear basis function.
• geometry.m, sets up the geometric data.
• quad1.m, sets up a 1 point quadrature rule.
• quad2.m, sets up a 2 point quadrature rule.
• rhsfun.m, evaluates the right hand side of the boundary value equation.

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