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

**FEM2D_BVP_LINEAR**,
a C++ 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.

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

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

FEM1D_BVP_LINEAR, a C++ 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 C++ 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 C++ 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.

- fem2d_bvp_linear.cpp, the source code.
- fem2d_bvp_linear.sh, compiles the source code.
- fem2d_bvp_linear.hpp, the include file.