# FEM2D_BVP_LINEAR Finite Element Method, 2D, Boundary Value Problem, Piecewise Linear Elements

FEM2D_BVP_LINEAR is a FORTRAN77 program which applies the finite element method, with piecewise linear elements, to a 2D boundary value problem over a rectangle.

The boundary value problem (BVP) that is to be solved has the form:

```        - d/dx ( a(x,y) * du/dx ) - d/dy ( a(x,y) * du/dy ) + c(x,y) * u(x,y) = f(x,y)
```
This equation holds in the interior of some rectangle R. The functions a(x,y), c(x,y), and f(x,y) are given.

Zero boundary conditions are imposed on the boundary of R.

To compute a finite element approximation, the X and Y extents of R are gridded with NX and NY equally spaced values, respectively. This defines NX*NY nodes, and divides R into (NX-1)*(NY-1) rectangular elements. At the K-th node, (X(I),Y(J)), a piecewise linear basis function PHI(K)(X,Y) is defined. The solution will be represented as a linear combination of these basis functions. An integral form of the BVP is written, in which the differential equation is multiplied by each basis function, and integration by parts is used to simplify the integrand.

### Languages:

FEM2D_BVP_LINEAR is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version and a Python version.

### Related Data and Programs:

FEM1D_BVP_LINEAR, a FORTRAN90 program which applies the finite element method (FEM), with piecewise linear elements, to a two point boundary value problem (BVP) in one spatial dimension, and compares the computed and exact solutions with the L2 and seminorm errors.

FEM2D_BVP_QUADRATIC, a FORTRAN77 program which applies the finite element method (FEM), with piecewise quadratic elements, to a 2D boundary value problem (BVP) in a rectangle, and compares the computed and exact solutions with the L2 and seminorm errors.

FEM2D_BVP_SERENE, a FORTRAN77 program which applies the finite element method (FEM), with serendipity elements, to a 2D boundary value problem (BVP) in a rectangle, and compares the computed and exact solutions with the L2 and seminorm errors.

### List of Routines:

• FEM2D_BVP_LINEAR solves boundary value problem on a rectangle.
• FEM2D_H1S_ERROR_LINEAR: seminorm error of a finite element solution.
• FEM2D_L1_ERROR estimates the l1 error norm of a finite element solution.
• FEM2D_L2_ERROR_LINEAR: L2 error norm of a finite element solution.
• R8MAT_SOLVE2 computes the solution of an N by N linear system.
• R8VEC_EVEN returns an R8VEC of evenly spaced values.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

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

Last revised on 20 June 2014.