Finite Element 2D,
Poisson Equation,
Conjugate Gradient Solver

FEM2D_POISSON_CG is a C++ program which applies the finite element method to solve a form of Poisson's equation over an arbitrary triangulated region, using sparse matrix storage and a conjugate gradient solver.

The storage format chosen is known as DSP or "sparse triplet" format, which essentially simply saves in three vectors A, IA, JA, which record the value, row and column of every nonzero entry.

To solve the linear system, the CG_RC routine is used, which uses reverse communication to carry out a conjugate gradient procedure.

The 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:

        |\   |\
        | \  | \
        |  \ |  \
        |   \|   \
the node file would be:
         0.0 0.0
         1.0 0.0
         2.0 0.0
         0.0 1.0
         1.0 1.0
and the triangle file would be
        1 2 4
        5 4 2
        2 3 5

The Poisson Equation:

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.

User Interface:

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

To run the program, the user compiles the user routines, links them with FEM2D_POISSON_CG, 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.


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

my_problem prefix
where prefix is the common filename prefix, so that:


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


FEM2D_POISSON_CG is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

CG_RC, a C++ library which implements the conjugate gradient method for solving a positive definite sparse linear system A*x=b, using reverse communication.

FEM2D_POISSON, a C++ program which solves Poisson's equation on a triangulated region, using the finite element method and a banded solver.

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

FEM2D_POISSON_CG_BAFFLE, a C++ library which defines the geometry of a channel with 13 hexagonal baffles, as well as boundary conditions for a given Poisson problem, and is called by fem2d_poisson_cg as part of a solution procedure.

FEM2D_POISSON_CG_ELL, a C++ 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_cg as part of a solution procedure.

FEM2D_POISSON_CG_LAKE, a C++ 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_cg as part of a solution procedure.


  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:

List of Routines:

You can go up one level to the C++ source codes.

Last revised on 25 January 2013.