作者:马俊驰，索宇洋
Authors:MA Jun chi,SUO Yu yang摘要:提出了一种浸入式虚拟元方法 ,该方法用来求解二维空间中二阶椭圆界面问题 ,克服了构造的空间无法同时满足协调性和跳跃条件的不足 。其核心思想是将形状规则的非适配网格上的协调虚拟元空间进行离散化 , 并将其投影到界面单元上的浸入式有限元空间上 ,这些界面单元是由背景网格上的界面线切割而成的 ,使得所提出的方法继承了拟合和非拟合网格方法的优点 。进一步给出此方法的严格理论推导和最优误差估计 ,最后进行数值算例求解 。通过计算数值解与解析解误差的 L² 和 H¹ 范数 ,并通过误差的线性回归图 ,可以看出 L² 和 H¹ 范数分 别以 Ο(h² )和 Ο(h)速率收敛 ,表明理论分析结果是正确的。
Abstract:An immersed virtual element method is developed to solve the second-order interface problems in two-dimensional space , which overcomes the problem that the constructed space cannot satisfy both the coordination and jump conditions. The key idea is to use the conforming virtual element spaces on a shape regular background unfitted mesh for discretization, and project them to the immersed finite element spaces on interface elements which are cut by the interfaces from the background mesh, making the proposed method combine the advantages of both body-fitted mesh methods and unfitted mesh methods. Then the strict theoretical derivation and optimal error estimation of the method are given. Finally, the numerical example is solved. By calculating the L² and H¹ norms of the numerical solution and analytical solution, and making the linear regression diagram of errors, the method achieves second order convergence in the L² norm and first order convergence in the H¹ norm, indicating that the theoretical analysis results are correct.

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