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A robust multigrid method for one dimensional immersed finite element method
Author(s) -
Wang Saihua,
Wang Feng,
Xu Xuejun
Publication year - 2021
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22685
Subject(s) - multigrid method , mathematics , finite element method , jump , norm (philosophy) , convergence (economics) , rate of convergence , mathematical optimization , mathematical analysis , partial differential equation , computer science , structural engineering , engineering , computer network , channel (broadcasting) , physics , quantum mechanics , political science , law , economics , economic growth
In this paper, we propose a robust multigrid method for 1D immersed finite element method (IFEM). It is shown that the multigrid method is optimal, which means that the convergence rate of the multigrid method is not only independent of the mesh size h and mesh level L , but also independent of the jump of the discontinuous coefficients. Although we only consider 1D interface method, to the best of our knowledge, this is the first attempt to give a rigorous theoretical analysis for the multigrid method for the IFEM. On the way to this goal, we also revisit the IFEM for the 1D interface problem and prove that the error estimates with respect to the L 2 norm and weighted H 1 semi‐norm are optimal and independent of the jump of the discontinuous coefficients. Numerical results are given to verify our theoretical findings.