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Error estimates for multidimensional singular parabolic problems. (English) Zbl 0657.65132
The numerical approximation of the following singular parabolic equation \[ \gamma (u)_ t-\nabla_ x(\nabla_ xu+b(u))+f(u)=0,\quad x\in \Omega \subset R^ d,\quad d\geq 1 \] is the subject of this paper, when the conditions on the fixed boundary are non-homogeneous Dirichlet or Neumann conditions. The problem is from the general class of mathematical models of the two-phase Stefan problem or the porous medium equation.
The regularization of this problem given by the following formula:\(\gamma_ 2(s):=\min \{s/\epsilon,\quad \gamma (x,t,s)\}\) if \(\epsilon >0\), \(\gamma_ 2(s):=0\) if \(\epsilon =0\) and \(\gamma_ 2(s):=\max \{s/\epsilon,\quad \gamma (x,s,t)\}\) if \(\epsilon <0\) and the discretization of the regularized problem is given by means of \(C^ 0\)- piecewise linear finite elements in space and backward-differences in time (implicit scheme). By using an auxiliary nonlinear parabolic operator technique estimates in \(L^ p\) norms for the errors \(u- u_{\epsilon,h,\tau}\) and \(\gamma (u)- \gamma_{\epsilon}(u_{\epsilon,h,\tau})\) are given, where \(u_{\epsilon,h,\tau}\) is the approximate solution. Moreover, for two- phase Stefan problems in some special cases an \(L^ 2\)-error estimate for enthalpy is given.
Reviewer: Gy.Molnárka

65Z05 Applications to the sciences
76S05 Flows in porous media; filtration; seepage
65N15 Error bounds for boundary value problems involving PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
80A17 Thermodynamics of continua
35R35 Free boundary problems for PDEs
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