Convergence of a crystalline approximation for an area-preserving motion
Convergence of a crystalline approximation for an area-preserving motion
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DOI：
10.1016/j.cam.2003.08.041
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发表时间：
2004-04
影响因子：
2.4
通讯作者：
T. Ushijima;S. Yazaki
中科院分区：
文献类型：
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作者：
T. Ushijima;S. Yazaki
我们考虑用广义晶体运动来近似平面上的保面积运动。保面积运动用带非定域项的抛物型偏微分方程组描述，而晶体运动用常微分方程组描述。我们证明了这两个运动之间的收敛。收敛定理分两步证明：第一步，建立广义晶体运动解的先验估计；第二步，估计所有1⩽p<；∞的误差的离散W1，p范数，并将p传递到无穷远，得到离散W1，∞误差估计。构造了一个具有保面积、缩短曲线等优良性质的隐式方案，并与一个简单的方案进行了比较。
We consider an approximation of area-preserving motion in the plane by a generalized crystalline motion. The area-preserving motion is described by a parabolic partial differential equation with a nonlocal term, while the crystalline motion is governed by a system of ordinary differential equations. We show the convergence between these two motions. The convergence theorem is proved in two steps: first, an a priori estimate is established for a solution to the generalized crystalline motion; second, a discrete W1,pnorms of the error is estimated for all 1⩽p<∞ and, passing p to infinity, a discrete W1,∞error estimate is obtained. We also construct an implicit scheme which enjoys several nice properties such as the area-preserving and curve-shortening, and compare our scheme with a simple scheme.