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Approximation of the phase-field transition system via fractional steps method. (English) Zbl 0909.35008

The author studies the phase-field transition system in a bounded domain \(\Omega\subset \mathbb{R}^2\), i.e. \[ u_t+{l\over 2} \varphi_t- k\Delta u= f,\quad \tau\varphi_t- \xi^2\Delta\varphi= {1\over 2a} (\varphi- \varphi^3)+ 2u+ g\quad\text{in }\Omega\times (0,T), \]
\[ {\partial u\over\partial\nu}+ hu= 0,\;\varphi= 0\quad\text{on }\partial\Omega\times (0,T),\quad u(.,0)= u_0,\;\varphi(.,0)= \varphi_0\quad\text{in }\Omega. \] Here, \(u\) denotes the reduced temperature, and \(\varphi\) is the order parameter. The above problem is approximated by a sequence of linear systems on the subintervals \((k\varepsilon,(k+ 1)\varepsilon)\subset (0,T)\), \(k= 0,\dots, N_\varepsilon\) as follows: \[ u^\varepsilon_t+{l\over 2} \varphi^\varepsilon_t- k\Delta u^\varepsilon= f,\quad\tau\varphi^\varepsilon_t- \xi^2\Delta\varphi^\varepsilon= {1\over 2a} \varphi^\varepsilon+ 2u^\varepsilon+ g\quad\text{in }\Omega\times (k\varepsilon,(k+ 1)\varepsilon), \]
\[ {\partial u^\varepsilon\over\partial\nu}+ hu^\varepsilon= 0,\;\varphi^\varepsilon= 0\quad\text{on }\partial\Omega\times (k\varepsilon,(k+ 1)\varepsilon), \]
\[ \varphi^\varepsilon_+(.,k\varepsilon)= z(\varepsilon, \varphi^\varepsilon_-(., k\varepsilon)),\;u^\varepsilon_+(., k\varepsilon)= u^\varepsilon_-(., k\varepsilon)\quad\text{in }\Omega, \] and \(z(t,x)\) is the solution of \(z'+ (1/2a)z^3= 0\), \(z(0,x)= x\). Here \(f_+\) and \(f_-\) denote upper and lower limits of \(f\). The main result of the paper says that \(\lim_{\varepsilon\to 0}(u^\varepsilon(t), \varphi^\varepsilon(t))= (u(t),\varphi(t))\) in \(L^2(\Omega)\times L^2(\Omega)\) for all \(t\in [0,T]\). Based on this convergence result, the author introduces a finite element method and presents numerical results.

MSC:

35A35 Theoretical approximation in context of PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65C20 Probabilistic models, generic numerical methods in probability and statistics
65H10 Numerical computation of solutions to systems of equations
65J15 Numerical solutions to equations with nonlinear operators
65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
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