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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

##### MSC:
 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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