FEM2D_POISSON_RECTANGLE
Finite Element Solution of the 2D Poisson Equation

FEM2D_POISSON_RECTANGLE is a FORTRAN77 program which solves the 2D Poisson equation using the finite element method.

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

        - ( Uxx + Uyy ) = F(x,y)  in the box
                 U(x,y) = G(x,y)  on the box 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". Because quadratic basis functions are to be used, each triangle will be associated not only with the three corner nodes that defined it, but with three extra midside nodes. If we include these additional nodes, there are now a total of (2*NX-1)*(2*NY-1) nodes in the region.

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. The value of U at the boundary nodes is obvious, so we concentrate on the NUNK interior nodes where U(x,y) is unknown. 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)
      

The program allows the user to supply two routines:

• FUNCTION RHS ( X, Y ) returns the right hand side F(x,y) of the Poisson equation.
• SUBROUTINE EXACT ( X, Y, U, DUDX, DUDY ) returns the exact solution of the Poisson equation (assuming this is known.) This routine is necessary so that the boundary conditions may be set, and so that error analysis can be performed, reporting the L2 and H1 seminorm errors between the true and computed solutions.

There are a few variables that are easy to manipulate. In particular, the user can change the variables NX and NY in the main program, to change the number of nodes and elements. The variables (XL,YB) and (XR,YT) define the location of the lower left and upper right corners of the rectangular region, and these can also be changed in a single place in the main program.

The program writes out a file containing an Encapsulated PostScript image of the nodes and elements, with numbers. Unfortunately, for values of NX and NY over 10, the plot is too cluttered to read. For lower values, however, it is a valuable map of what is going on in the geometry.

The program is also able to write out a file containing the solution value at every node. This file may be used to create contour plots of the solution

The original version of this code comes from Professor Janet Peterson.

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 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_LINEAR, a FORTRAN77 program which solves the 2D Poisson equation on a rectangle, using the finite element method, and piecewise linear 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.f, the source code;
• fem2d_poisson_rectangle.sh, commands to compile and run the program;

Examples and Tests:

• rectangle_output.txt, the output file.
• rectangle_elements.png, a PNG image of the elements, for NX = NY = 7 (the picture is hard to read for larger values of NX and NY);
• rectangle_solution.txt, a text file containing a list, for each node, of its coordinates, and the value of the solution there;

List of Routines:

• MAIN is the main program for FEM2D_POISSON_RECTANGLE.
• ASSEM assembles the matrix and right-hand side using piecewise quadratics.
• DGB_FA performs a LINPACK-style PLU factorization of an DGB matrix.
• DGB_PRINT_SOME prints some of a DGB matrix.
• DGB_SL solves a system factored by DGB_FA.
• EROR calculates the L2 and H1-seminorm errors.
• EXACT calculates the exact solution and its first derivatives.
• GEOM sets up geometric information for a rectangular domain.
• GET_UNIT returns a free FORTRAN unit number.
• I4_HUGE returns a "huge" I4.
• I4VEC_PRINT_SOME prints "some" of an integer vector.
• MESH_EPS creates an EPS file containing an image of the mesh.
• QBF evaluates the quadratic basis functions.
• QUAD13 sets up quadrature information for a 13-point rule in a given element.
• R8_IS_INT determines if an R8 represents an integer value.
• R8_SWAP switches two R8's.
• R8VEC_PRINT_SOME prints "some" of a double precision vector.
• RHS gives the right-hand side of the differential equation.
• S_LEN_TRIM returns the length of a string to the last nonblank.
• SOLUTION_WRITE writes the solution to a file.
• TIMESTAMP prints out the current YMDHMS date as a timestamp.

You can go up one level to the FORTRAN77 source codes.