 of the 1dimensional nonlinear parabolic (Stefan) model problem ...... with not regular initial value data
We provide an ‘optimal’ finite element approximation error estimate for a onedimensional nonlinear parabolic model problem with nonregular initial value data. The solution concept is based on corresponding results of J. A. Nitsche for finite element approximation error estimates for the onedimension 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 3D nonstationary, nonlinear NavierStokes equations to improve the ‘notoptimal’ finite element approximation error estimates in [HeJ].
The basic idea is in line with the proposal of the "Sapcescale 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' RitzGalerkin 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 NSEcounterpart of the "nonregular" Stefan problem challenge where the solution is not necessarily an element of H(1), as the corresponding H(1)norm (momentum/energy norm) is not necessarily bounded for t > 0 is the Heywood example:
The following pdf contains a collection of related areas to the above

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 initialboundary value problems with a fixed domain.