Lecture 3c Boundary reduction of equations

When using the Finite Element Method, FEM, for solving
partial differential equations, a system of linear
equations is constructed. There may be known Dirichlet
boundary values that are unknown values in the system
of equations. The system may be reduced to a smaller
system of equations using the following reduction.


Given a system of linear equations A x = y
with known matrix A, known vector y
and needing to solve for vector x,
if one of the values of x, say x4, is known
then perform the following reduction:

 | A11  A12  A13  A14 |   |x1|    |y1|
 | A21  A22  A23  A24 | * |x2| =  |y2|
 | A31  A32  A33  A34 |   |x3|    |y3|
 | A41  A42  A43  A44 |   |x4|    |y4|
 
Because A x = y and knowing the value of x4
the first three rows become

 | A11  A12  A13 |   |x1|   |y1 - A14*x4|
 | A21  A22  A23 | * |x2| = |y2 - A24*x4|
 | A31  A32  A33 |   |x3|   |y3 - A34*x4|

The fourth row is not needed since we know x4.

Because the simultaneous linear equations may be
written in any order, interchange rows so that the
last row has the known x value. For n equations in n
unknowns, the reduction is

  for i=1:n-1
    y(i) = y(i) - A(i,n) * x(n)
  end
  n=n-1

More reductions may be performed if more
values of x are known.

x(n) is known when y(n) is known
and A(n,n)=1.0 and A(n,j)=0.0 for j/=n
thus use y(n) in place of x(n) in the above process.