# fem2d_bvp_linear

fem2d_bvp_linear, a MATLAB code 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 FORTRAN90 version and a MATLAB version and a Python version.

### Related Data and Programs:

fem1d_bvp_linear, a MATLAB code 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 MATLAB code 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 MATLAB code 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.

### Source Code:

• fem2d_bvp_linear.m, sets up and solves the finite element problem.
• fem2d_h1s_error_linear.m, estimates the seminorm of the error, given the piecewise linear solution of the finite element problem, and a function that evaluates the derivative of the exact solution.
• fem2d_l1_error.m, estimates the little l1 norm of the error, given the solution of the finite element problem, and a function that evaluates the exact solution.
• fem2d_l2_error_linear.m, estimates the L2 norm of the error, given the piecewise linear solution of the finite element problem, and a function that evaluates the exact solution.

Last revised on 25 March 2019.