The global dynamics of discrete semilinear parabolic equations
SIAM Journal on Numerical Analysis
On the Cahn-Hilliard equation with degenerate mobility
SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Finite Element Approximation of the Cahn--Hilliard Equation with Degenerate Mobility
SIAM Journal on Numerical Analysis
Positivity-Preserving Numerical Schemes for Lubrication-Type Equations
SIAM Journal on Numerical Analysis
Finite Element Method for Elliptic Problems
Finite Element Method for Elliptic Problems
Finite Element Approximation of a Three Dimensional Phase Field Model for Void Electromigration
Journal of Scientific Computing
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We present nonnegativity-preserving finite element schemes for a general class of thin film equations in multiple space dimensions. The equations are fourth order degenerate parabolic, and may contain singular terms of second order which are to model van der Waals interactions. A subtle discretization of the arising nonlinearities allows us to prove discrete counterparts of the essential estimates found in the continuous setting. By use of the entropy estimate, strong convergence results for discrete solutions are obtained. In particular, the limit of discrete fluxes Mh(Uh)∇Ph will be identified with the flux M(u)∇(W'(u) - Δu) in the continuous setting. As a by-product, first results on existence and positivity almost everywhere of solutions to equations with singular lower order terms can be established in the continuous setting.