FEM2D_POISSON
Finite Element Solution of Poisson's Equation
on a Triangulated Region

FEM2D_POISSON is a FORTRAN90 program which applies the finite element method to solve a form of Poisson's equation over an arbitrary triangulated region.

The computational region is unknown by the program. The user specifies it by preparing a file containing the coordinates of the nodes, and a file containing the indices of nodes that make up triangles that form a triangulation of the region.

Normally, the user does not type in this information by hand, but has a program fill in the nodes, and perhaps another program that constructs the triangulation. However, in the simplest case, the user might construct a very crude triangulation by hand, and have TRIANGULATION_REFINE refine it to something more reasonable.

For the following ridiculously small example:

        4----5
        |\   |\
        | \  | \
        |  \ |  \
        |   \|   \
        1----2----3
      

         0.0 0.0
         1.0 0.0
         2.0 0.0
         0.0 1.0
         1.0 1.0
      

        1 2 4
        5 4 2
        2 3 5
      

The program is set up to handle the linear Poisson equation with a right hand side function, and nonhomogeneous Dirichlet boundary conditions. The state variable U(X,Y) is then constrained by:

        - Del H(x,y) Del U(x,y) + K(x,y) * U(x,y) = F(x,y)  inside the region;
                                           U(x,y) = G(x,y)  on the boundary.
      

An improved version of the program would handle a more interesting nonlinear PDE, and include optional Neumann boundary conditions.

To specify the boundary condition function G(x,y), the coefficients H(x,y) and K(x,y) and the right hand side function F(x,y), the user has to modify a file containing three subroutines,

• SUBROUTINE DIRICHLET_CONDITION ( NODE_NUM, NODE_XY, NODE_BC ) evaluates G(X,Y).
• SUBROUTINE H_COEF ( NODE_NUM, NODE_XY, NODE_H ) evaluates H(x,y).
• SUBROUTINE K_COEF ( NODE_NUM, NODE_XY, NODE_K ) evaluates K(x,y).
• SUBROUTINE RHS ( NODE_NUM, NODE_XY, NODE_F ) evaluates F(x,y).

To run the program, the user compiles the user routines, links them with FEM2D_POISSON, and runs the executable.

The program writes out a file containing an Encapsulated PostScript image of the nodes and elements, with numbers. If there are too many nodes, the plot may be 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.

Usage:

The user must create an executable by compiling the user routines and linking them with the main program, perhaps by commands like:

        gfortran -c fem2d_poisson.f90
        gfortran -c user.f90
        gfortran fem2d_poisson.o user.o
        mv a.out fem2d_poisson
      

Assuming the executable program is called "fem2d_poisson", then the program is executed by

fem2d_poisson prefix

• prefix_nodes.txt, is a file containing the node coordinates;
• prefix_elements.txt, is a file listing the 3 nodes that make up each element;

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

Related Programs:

FEM2D_POISSON_CG, a FORTRAN90 program which solves Poisson's equation on a triangulated region, using the finite element method, sparse storage, and a conjugate gradient solver.

FEM2D_POISSON_ELL, a FORTRAN90 library which defines the geometry of an L-shaped region, as well as boundary conditions for a given Poisson problem, and is called by FEM2D_POISSON as part of a solution procedure.

FEM2D_POISSON_LAKE, a FORTRAN90 library which defines the geometry of a lake-shaped region, as well as boundary conditions for a given Poisson problem, and is called by FEM2D_POISSON as part of a solution procedure.

FEM2D_POISSON_SPARSE, a FORTRAN90 program which solves the steady (time independent) Poisson equation on an arbitrary 2D triangulated region using a version of GMRES for a sparse solver.

Reference:

1. Hans Rudolf Schwarz,
   Methode der Finiten Elemente,
   Teubner Studienbuecher, 1980,
   ISBN: 3-519-02349-0.
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.

Source Code:

• fem2d_poisson.f90, the source code;

List of Routines:

• MAIN is the main program of FEM2D_POISSON.
• ASSEMBLE_POISSON assembles the system for the Poisson equation.
• BANDWIDTH determines the bandwidth of the coefficient matrix.
• BASIS_ONE_T3 evaluates a linear basis function.
• CH_CAP capitalizes a single character.
• CH_EQI is a case insensitive comparison of two characters for equality.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• DGB_FA performs a LINPACK-style PLU factorization of an DGB matrix.
• DGB_MXV multiplies a DGB matrix times a vector.
• DGB_PRINT_SOME prints some of a DGB matrix.
• DGB_SL solves a system factored by DGB_FA.
• DIRICHLET_APPLY accounts for Dirichlet boundary conditions.
• FILE_COLUMN_COUNT counts the number of columns in the first line of a file.
• FILE_ROW_COUNT counts the number of row records in a file.
• GET_UNIT returns a free FORTRAN unit number.
• I4_MODP returns the nonnegative remainder of integer division.
• I4_WRAP forces an integer to lie between given limits by wrapping.
• I4COL_COMPARE compares columns I and J of an I4COL.
• I4COL_SORT_A ascending sorts an I4COL.
• I4COL_SWAP swaps columns I and J of an I4COL.
• I4MAT_TRANSPOSE_PRINT_SOME prints some of the transpose of an I4mat.
• I4MAT_DATA_READ reads data from an I4MAT file.
• I4MAT_HEADER_READ reads the header from an I4MAT.
• LVEC_PRINT prints a logical vector.
• POINTS_PLOT plots a pointset.
• QUAD_RULE sets the quadrature rule for assembly.
• R8MAT_DATA_READ reads data from an R8MAT file.
• R8MAT_HEADER_READ reads the header from an R8MAT file.
• R8MAT_TRANSPOSE_PRINT_SOME prints some of an R8MAT, transposed.
• R8MAT_WRITE writes an R8MAT file.
• R8VEC_PRINT_SOME prints "some" of an R8VEC.
• REFERENCE_TO_PHYSICAL_T3 maps reference points to physical points.
• RESIDUAL_POISSON evaluates the residual for the Poisson equation.
• S_TO_I4 reads an I4 from a string.
• S_TO_I4VEC reads an I4VEC from a string.
• S_TO_R8 reads an R8 from a string.
• S_TO_R8VEC reads an R8VEC from a string.
• S_WORD_COUNT counts the number of "words" in a string.
• SOLUTION_EVALUATE evaluates the solution at a point in an element.
• SORT_HEAP_EXTERNAL externally sorts a list of items into ascending order.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TRIANGLE_AREA_2D computes the area of a triangle in 2D.
• TRIANGULATION_ORDER3_BOUNDARY_NODE indicates which nodes are on the boundary.
• TRIANGULATION_ORDER3_PLOT plots a 3-node triangulation of a pointset.

You can go up one level to the FORTRAN90 source codes.