# FEM2D_POISSON_RECTANGLE Finite Element Solution of the 2D Poisson Equation

FEM2D_POISSON_RECTANGLE is a FORTRAN90 program which solves the 2D Poisson equation using the finite element method, over a rectangular domain with a uniform mesh of triangular elements.

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

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.

### 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, a data directory which contains examples of 2D FEM files, text files that describe a 2D finite element geometry and associated nodal values;

FEM2D_HEAT_RECTANGLE, a FORTRAN90 program which solves the 2D time dependent heat equation on the unit square.

FEM2D_POISSON, a FORTRAN90 program which solves the 2D Poisson equation on an arbitrary triangulated region, using the finite element method, and piecewise linear triangular elements.

FEM2D_POISSON_RECTANGLE_LINEAR, a FORTRAN90 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,
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

### List of Routines:

• MAIN is the main for FEM2D_POISSON_RECTANGLE.
• AREA_SET sets the area of each element.
• ASSEMBLE assembles the coefficient matrix A and right hand side F.
• BANDWIDTH determines the bandwidth of the coefficient matrix.
• BOUNDARY modifies the linear system for boundary conditions.
• COMPARE compares the exact and computed solution at the nodes.
• 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.
• ELEMENT_WRITE writes the element information to a text file.
• ERRORS calculates the error in the L2 norm and H1 seminorm.
• EXACT calculates the exact solution and its first derivatives.
• GET_UNIT returns a free FORTRAN unit number.
• GRID_T6 produces a grid of pairs of 6 node triangles.
• INDX_SET assigns an unknown value index at each node.
• NODES_PLOT plots a pointset.
• NODES_WRITE writes the nodes to a file.
• QBF evaluates the quadratic basis functions.