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Multigrid or multilevel methods are among the most efficient methods for solving the large linear systems arising from approximation of elliptic partial differential equations

Local Multilevel Methods for Adaptive Finite Element Methods for Nonsymmetric and Indefinite Elliptic Boundary Value Problems

SIAM J. Numerical Analysis, no. 6 (2010): 4492-4516

Cited: 4|Views7
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Abstract

In this paper, we propose some local multilevel algorithms for solving linear systems arising from adaptive finite element approximations of nonsymmetric and indefinite elliptic boundary value problems. Two types of local smoothers are constructed. One is based on the original nonsymmetric problems, and the other is defined in terms of th...More

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Introduction
  • Multigrid or multilevel methods are among the most efficient methods for solving the large linear systems arising from approximation of elliptic partial differential equations (PDEs).
  • The authors adopt the abstract theory developed in [30] to analyze the convergence of local multilevel methods for the nonsymmetric and indefinite elliptic boundary value problems.
  • Under the minimal elliptic regularity assumption, the following estimates hold true: Given > 0, there exist positive constants CI and 0( ) such that for h0 ∈ (0, 0], (2.9)
Highlights
  • Multigrid or multilevel methods are among the most efficient methods for solving the large linear systems arising from approximation of elliptic partial differential equations (PDEs)
  • In [4], Bramble, Kwak, and Pasciak applied perturbation analysis to get the uniform convergence estimate of V-cycle iteration with smoothers based on the original nonsymmetric problems or the associated symmetric problems
  • We show that the local multilevel methods converge uniformly provided that the coarsest grid is sufficiently fine
  • In [36], we have proved the following stability result for local multilevel methods for symmetric positive definite (SPD) problems: There exists a positive constant C1 such that a(v, v) ≤ C1a(Tv, v), v ∈ VJ, where T =
  • We should mention here that the local multilevel methods and the corresponding convergence estimates can be extended to the nonhomogeneous Dirichlet boundary condition
Results
  • The following estimate can be shown for local multilevel methods with local Jacobi and local Gauss–Seidel smoothers, respectively, which cannot be directly deduced by the Cauchy–Schwarz inequality for Ti. a(Tiv, u) ≤ K2 i=0
  • The authors first provide convergence analysis for the algorithms LMG and LMAA with local Jacobi and local Gauss–Seidel smoothers defined in terms of the original nonsymmetric problems.
  • In [36], the authors have proved the following stability result for local multilevel methods for SPD problems: There exists a positive constant C1 such that a(v, v) ≤ C1a(Tv, v), v ∈ VJ , where T =
  • For sufficiently small h0, there exists a constant K1, which is dependent only on the shape regularity of the meshes, and the data D, γ, , and h0, such that assumption A3 is satisfied.
  • The authors shall apply the abstract theory to local Gauss–Seidel smoother Ri, which is defined as follows: (4.25)
  • For sufficiently small h0, there exists a constant K1, which is dependent only on the shape regularity of the meshes, and the data D, , and h0, such that assumption A3 holds true for {Ti, i = 0, 1, .
  • The authors consider local smoothers based on the symmetric problems, which shall be denoted by Ri. The abstract theory can be applied to the multilevel methods with the local smoothers.
  • The local Jacobi smoother based on the associated symmetric problems is defined by
  • Based on the above perturbation estimate, the remainder is fully the same as the proof of (5.3), and the authors have a(v, v) ≤ C1 a(Tiv, v) + C12( B1/μa + γh0B0/μa)a(v, v).
Conclusion
  • The authors should mention here that the local multilevel methods and the corresponding convergence estimates can be extended to the nonhomogeneous Dirichlet boundary condition.
  • The authors denote by LMGGS∗ and LMG-Jacobi∗ algorithm LMG with local smoothers based on the symmetric problems.
  • The authors find that the convergence properties of algorithm LMG with different types of smoothers are almost the same as the case of the initial mesh h0 = from the SPD problem.
Tables
  • Table1: Example 6.1: The error between the exact solution and the discrete solution in the L2-norm and the energy norm
  • Table2: Example 6.1: Average reduction factor and the number of iterations on each level
  • Table3: Example 6.1: Iteration steps on each level for the GMRES and preconditioned GMRES methods
  • Table4: Example 6.2: Average reduction factor and the number of iterations on each level
  • Table5: Example 6.2: Iteration steps on each level for the GMRES and preconditioned GMRES methods
Download tables as Excel
Funding
  • This work was supported by the special funds for major state basic research projects (973) under 2005CB321701 and the National Science Foundation (NSF) of China (10731060)
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