of the 1-dimensional non-linear parabolic (Stefan) model problem ...
... with not regular initial value data
We provide an ‘optimal’ finite element approximation error estimate for a one-dimensional non-linear parabolic model problem with non-regular initial value data. The solution concept is based on corresponding results of J. A. Nitsche for finite element approximation error estimates for the one-dimension Stefan problem ([NiJ], [NiJ1], [NiJ2]) leveraging on the NSE solution concept of this homepage. As the approach is not depending from the space dimension it can also be applied to the 3-D non-stationary, non-linear Navier-Stokes equations to improve the ‘not-optimal’ finite element approximation error estimates in [HeJ].
The basic idea is in line with the proposal of the "Sapce-scale turbulence" section regarding:
let ux, uxx denote the first and second derivative of the function u(x) and e the viscosity term. In (MuA) it is proposed to replace e*uxx by -e*H[ux](x). For the latter term it holds the equality -e*H[ux](x) = e*A[uxx](x) whereby A denotes the Symm operator in the framework of L(2)-integrable periodic function on R with domain H(-1/2) (BrK1) (DeS) (MuA) (OkH).
In this case the same approach is applied to the auxiliary problem for a solution v in ([NiJ], [NiJ1], [NiJ2]) i.e. the auxiliary function v:=ux is replaced by v:=-H[ux].
For an analog NSE analysis the linearized (analog heat equation related) auxiliary problem (with corresponding 'optimal' parabolic shift theorem and related 'optimal' Ritz-Galerkin approximation estimates, [NiJ5]) needs to be replaced by the corresponding analog of the NSE theory which is about "an unusual shift theorem for Stokes flow" (see related J. A. Nitsche paper) and a corresponding "L(infinity)-analysis of the Galerkin (finite element) approximation of the solution of the Stokes equation" (J. A. Nitsche, lecture notes).
Here we are
The classical Stefan problem aims to describe the temperature distribution in a homogeneous medium undergoing a phase change, for example ice passing to water; Stefan problems are examples of free boundary problems; appropriate variable transformation leads to nonlinear parabolic initial-boundary value problems with a fixed domain.