Error estimates for two-phase stefan problems in several space variables, I: Linear boundary conditions
Error estimates for two-phase stefan problems in several space variables, I: Linear boundary conditions
复制标题
多个空间变量中两相 Stefan 问题的误差估计，I：线性边界条件
作者：
R. Nochetto
采用线性边界条件，在空间上用C 0-分片线性有限元，在时间上用向后差分，并结合正则化方法，对两相Stefan问题的焓公式进行了近似.得到了L2型误差估计。对于一般的正则化问题，证明了其阶为ε1/2，而对于非退化问题，则证明了其阶为ε。对于离散问题，可以得到h2ε−1+h+τε−1/2+τ2/3阶。这些阶对一般情况施加关系ε <$τ <$h 4/3，对非退化问题施加关系ε <$h <$τ2/3，以便分别获得收敛速度h 2/3或h。此外，一个命令h|坯料h|对于具有一定逼近性质的有限元网格，给出了+τ1/2。也证明了离散解对数据的连续依赖性。
The enthalpy formulation of two-phase Stefan problems, with linear boundary conditions, is approximated by C0-piecewise linear finite elements in space and backward-differences in time combined with a regularization procedure. Error estimates of L2-type are obtained. For general regularized problems an order ε1/2 is proved, while the order is shown to be ε for non-degenerate cases. For discrete problems an order h2ε−1+h+τε−1/2+τ2/3 is obtained. These orders impose the relations ε∼τ∼h4/3 for the general case and ε∼h∼τ2/3 for non-degenerate problems, in order to obtain rates of convergence h2/3 or h respectively. Besides, an order h|log h|+τ1/2 is shown for finite element meshes with certain approximation property. Also continuous dependence of discrete solutions upon the data is proved.