fem1d_nonlinear, a MATLAB code which applies the finite element method to a simple nonlinear boundary value problem (BVP) in one spatial dimension.
The nonlinear boundary value problem involves an unknown function u(x) on an interval [a,b]. Typically, a single boundary condition, either u(a) or u'(a), is given at the left endpoint, and u(b) or u'(b) at the right endpoint. The associated differential equation, which must hold in the interior of [a,b], happens to have the form:
-d/dx ( p(x) du/dx ) + q(x) * u + u * du/dx = f(x)
where p(x), q(x) and f(x) are given functions.
The nonlinearity arises because of the term u*du/dx; if this term were not present, standard finite element techniques would allow the system to be set up and solved almost as easily as a linear system of algebraic equations is solved.
Newton's method, which works well for scalar nonlinear equations, can also be applied to this problem, as long as we are willing to extend our notions of a function and derivative. We need to imagine our differential equation as a function F(u):
F(u) = -d/dx ( p(x) du/dx ) + q(x) * u + u * du/dx - f(x)
(with the boundary conditions wrapped in here somewhere as well!)
If we differentiate this function, we get a Jacobian operator, which is
evaluated at u, and applied to any small increment v. This equation
implicitly describes the tangent plane of solutions near to a given
solution u.
J(u,v) = -d/dx ( p(x) dv/dx ) + q(x) * v + u * dv/dx + v * du/dx
Now if we apply the finite element formulation to represent u and
v in terms of sums of basis functions, we can set up a linear
system, to be evaluated at u and solved for the Newton increment
delta_u:
J(u,delta_u) = - F(u)
By using the Newton increment to update u and repeating the process
as needed, we can expect to get a good finite element solution of our original
nonlinear boundary value problem.
The computer code and data files described and made available on this web page are distributed under the MIT license
FEM1D_NONLINEAR is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.
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