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FEM2D_POISSON_RECTANGLE_LINEAR\ Finite Element Solution of the 2D Poisson Equation {#fem2d_poisson_rectangle_linear-finite-element-solution-of-the-2d-poisson-equation align=”center”} ==================================================

FEM2D_POISSON_RECTANGLE_LINEAR is a C++ 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)

The set of all such equations yields a linear system for the coefficients of the representation of U.

Licensing: {#licensing align=”center”}

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

Languages: {#languages align=”center”}

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: {#related-data-and-programs align=”center”}

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

Reference: {#reference align=”center”}

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: {#source-code align=”center”}

• fem2d_poisson_rectangle_linear.cpp, the source code;

Examples and Tests: {#examples-and-tests align=”center”}

• fem2d_poisson_rectangle_linear_output.txt, the output file;

List of Routines: {#list-of-routines align=”center”}

• MAIN is the main routine of the finite element program FEM2D_POISSON_RECTANGLE_LINEAR.
• EXACT calculates the exact solution and its first derivatives.
• R8_ABS returns the absolute value of an R8.
• R8GE_FS factors and solves a R8GE system.

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

Last revised on 28 November 2008.