<- previous index next ->
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.
<- previous index next ->