Accurate and efficient numerical solutions for elliptic obstacle problems
Accurate and efficient numerical solutions for elliptic obstacle problems
复制标题
椭圆障碍问题准确高效的数值解
DOI：
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发表时间：
2017
影响因子：
1.6
通讯作者：
Seongjai Kim
中科院分区：
文献类型：
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作者：
Philku Lee;T. Kim;Seongjai Kim
椭圆障碍物问题的公式化是为了通过结合不等式约束来找到超调和解或位于障碍物上或上方的最小表面。为了使用有限差分（FD）方法有效地解决此类问题，本文研究了基于连续过松弛（SOR）方法的简单迭代算法。它引入了子网格FD方法来减少当网格与自由边界不匹配时在自由边界附近发生的精度恶化。针对非线性障碍问题，引入梯度加权的方法，更加方便、高效地求解问题。分析了线性和非线性障碍物问题的迭代算法的收敛性。还提出了一种有效的策略来找到最佳松弛参数。数值验证表明，采用最佳参数的障碍物 SOR 迭代的收敛速度比最先进的方法快一个数量级，并且在大多数情况下，子网格 FD 方法可将数值误差减少一个数量级。给出了各种数值例子来验证该主张。
Elliptic obstacle problems are formulated to find either superharmonic solutions or minimal surfaces that lie on or over the obstacles, by incorporating inequality constraints. In order to solve such problems effectively using finite difference (FD) methods, the article investigates simple iterative algorithms based on the successive over-relaxation (SOR) method. It introduces subgrid FD methods to reduce the accuracy deterioration occurring near the free boundary when the mesh grid does not match with the free boundary. For nonlinear obstacle problems, a method of gradient-weighting is introduced to solve the problem more conveniently and efficiently. The iterative algorithm is analyzed for convergence for both linear and nonlinear obstacle problems. An effective strategy is also suggested to find the optimal relaxation parameter. It has been numerically verified that the resulting obstacle SOR iteration with the optimal parameter converges about one order faster than state-of-the-art methods and the subgrid FD methods reduce numerical errors by one order of magnitude, for most cases. Various numerical examples are given to verify the claim.