Finite element approximation of a sixth order nonlinear degenerate parabolic equation
Degenerate nonlinear parabolic problems of the type \(u_t=\nabla(b(u)\nabla\Delta^2u)\) with \(b(u)=| u| ^\gamma\) 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.
scientific article; zbMATH DE number 2055238