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Finite element approximation of a sixth order nonlinear degenerate parabolic equation. (English) Zbl 1041.65076
Degenerate nonlinear parabolic problems of the type $u_t=\nabla(b(u)\nabla\Delta^2u)$ with $b(u)=\vert u\vert ^\gamma$ where $\gamma\in(0,+\infty)$ are considered. Its finite element treatment is based upon the equivalent mixed formulation $$ u_t=\nabla(b(u)\nabla w),\quad w=-\Delta v,\quad v=-\Delta u\,. $$ Spatial discretizations by continuous linear finite elements over simplices that partition the polyhedral domain are studied. For the time integration an semi-implicit Euler scheme with frozen coefficients is applied. Moreover, a physically required nonnegativity is included via variational inequalities. In detail, analytical properties are investigated and convergence of the proposed method including an iterative scheme for the discrete problem is shown.

MSC:
65M60Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE)
35K55Nonlinear parabolic equations
35K65Parabolic equations of degenerate type
65M12Stability and convergence of numerical methods (IVP of PDE)
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