Today we will be solving several boundary value
problems for second order differential equations
using the Galerkin finite element method.
Here is an advance of what I plan to do.
Local FE basis functions for the inside of each element
will be created by Lagrange interpolation
from the values of u at the nodes of the element,
i.e. if the nodes of the element are labeled 1 and 2
the linear
Lagrange interpolation is
u(x) =[ (x-x2)/(x1-x2)] u1 +
[ (x-x1)/(x2-x1] u2
where u1 and u2 are the values of u
at x1 and x2, respectively and the coefficients
(x-x1)/(x1-x2) = p1 and
(x-x1)/(x2-x1) = p2 are
the local FE basis functions for nodes 1 and 2
of the element in question.
The global FE basis functions will be obtained
from the local function given above such that
for node i, connecting any two contiguous elements,
the global FE basis function p_i
inside the region occupied by the element
to the LEFT of the node will be equal to p2 for that
element while
inside the region occupied by the element
to the RIGHT of the node will be equal to p1 for that
element.
Moreover, the global FE basis function
will be taken as equal to zero everywhere else
(i.e. it will be non zero ONLY inside the two elements
connected by the node). This detail is KEY
for the whole FE methodology.
Using the thus obtained global FE basis function as
our test function, we will proceed with
Galerkin's method, i.e. approximating the
solution as
uG = u1 p_1 + u2 p_2 + u3 p_3 + ...+ uN p_N
(where N is the number of NODES)
multiplying left and right hand side of the
differential equation
by the test functions p1, p2, p3, .... pi,...pN, one by
one,
performing integration by parts on the term on the left
and carry out the integrations to obtain
the final system of linear algebraic equations
that must be solved to obtain the values of
u1, u2, u3 ... and hence uG.