FEM2D_POISSON_RECTANGLE_LINEAR
Finite Element Solution of the 2D Poisson Equation

FEM2D_POISSON_RECTANGLE_LINEAR is a MATLAB program which solves the 2D Poisson equation using the finite element method with piecewise linear triangular elements.

The computational region is a rectangle, with Dirichlet boundary conditions applied along the boundary. The state variable U(X,Y) is then constrained by:

        - ( Uxx + Uyy ) = F(x,y)  in the region
                 U(x,y) = G(x,y)  on the region boundary
      

The computational region is first covered with an NX by NY rectangular array of points, creating (NX-1)*(NY-1) subrectangles. Each subrectangle is divided into two triangles, creating a total of 2*(NX-1)*(NY-1) geometric "elements".

We now assume that the unknown function U(x,y) can be represented as a linear combination of the basis functions associated with each node. For each node I, we determine a basis function PHI(I)(x,y), and evaluate the following finite element integral:

        Integral ( Ux(x,y) * PHIx(I)(x,y) + Uy(x,y) * PHIy(I)(x,y) ) =
        Integral ( F(x,y) * PHI(I)(x,y)
      

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

FEM2D_POISSON_RECTANGLE_LINEAR is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

FEM2D_POISSON_RECTANGLE, a MATLAB program which solves the 2D Poisson equation on a rectangle, using the finite element method, and piecewise quadratic triangular elements.

Reference:

1. Hans Rudolf Schwarz,
   Finite Element Methods,
   Academic Press, 1988,
   ISBN: 0126330107,
   LC: TA347.F5.S3313.
2. Gilbert Strang, George Fix,
   An Analysis of the Finite Element Method,
   Cambridge, 1973,
   ISBN: 096140888X,
   LC: TA335.S77.
3. Olgierd Zienkiewicz,
   The Finite Element Method,
   Sixth Edition,
   Butterworth-Heinemann, 2005,
   ISBN: 0750663200,
   LC: TA640.2.Z54

Source Code:

• fem2d_poisson_rectangle_linear.m, the source code;

Examples and Tests:

• rectangle_nodes.txt, the node file;
• rectangle_elements.txt, the element file;
• rectangle_solution.txt, the value of the solution at each node;
• rectangle_output.txt, the output file;

You can go up one level to the MATLAB source codes.