ON THE CONVERGENCE OF ENTROPY CONSISTENT SCHEMES FOR LUBRICATION TYPE EQUATIONS IN MULTIPLE SPACE DIMENSIONS
ON THE CONVERGENCE OF ENTROPY CONSISTENT SCHEMES FOR LUBRICATION TYPE EQUATIONS IN MULTIPLE SPACE DIMENSIONS
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多维空间润滑型方程熵一致格式的收敛性
DOI：
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发表时间：
2000
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通讯作者：
Unther Gr¨un
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作者：
Unther Gr¨un
我们给出了一类一般的多维薄膜方程的保非负性有限元格式。这些方程是四阶退化抛物型的，可能包含用来模拟范德华相互作用的二阶奇异项。对产生的非线性进行微妙的离散化，使我们能够证明在连续环境中发现的基本估计的离散对应。利用熵估计，得到了离散解的强收敛结果。特别地，离散通量Mh(Uh)∇Ph的极限将用连续设置中的通量M(U)∇(W‘(U)−∆u)来识别。作为副产品，在连续条件下，首次得到了具有奇异低阶项的方程的解的存在性和几乎处处正性的第一个结果。
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 fluxM(u)∇(W ′(u)−∆u) 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.