Hi, FEM for Time integration has not well studied and hence considered "Decoupled", where U=ui(t)*N(x). The equation does not seem to be self-adjoint, hence popular Galerkin's Method may not be applicable.

Could you write the whole equation? Generally, the time integration is explicitly discretized after a FEM discretization is introduced in space. To use a full time-space FEM formulation you should introduce some special shape function depending on time and space. You can find some studies about such approach but is somehow less common.

I think one sensible approach would be to treat that term in an implicit-explicit manner. That is, (1) use FEM to discretize in space, then (2) use time integration method to disctretize the time derivatives - in this case, the nonlinear term, udw/dT, can be discretized as u^n*(w^{n+n}-w^n)/k where k is your time step size. This will lead a linear scheme. Alternatively, use ( u^{n+n} + u^n)*(w^{n+n}-w^n)/(2k), this will lead to nonlinear scheme but can retain 2nd order accuracy in time if every other term in the equation is also discretized with 2nd order time accuracy.

There are plent of FEM books in literature. Please check the following paper. It precisely deals with what you are asking

Akin, J. E., A Least Squares Finite Element Solution of Nonlinear Equations, in The Mathematics of Finite Elements and Applications, J. R. Whiteman (Ed.),Academic Press, London, 153—62, 1973.

Dear Amir,

Integration method can be use to discretize the u(x,t)*(dw(x,t)/dt) , the standard nodal basis approach. unknown coefficient vector can be obtained by ordinary differential equation. Book given below, may be helpful for you.

Refer the site:

Numerical Methods for Two-phase Incompressible Flows

By Sven Gross, Arnold Reusken

Ashish