Modified Adaptive Two-Grid FEMs for Nonsymmetric or Indefinite Elliptic Problems

doi:10.1007/s42967-024-00475-x

Communications on Applied Mathematics and Computation ›› 2026, Vol. 8 ›› Issue (3): 831-850.doi: 10.1007/s42967-024-00475-x

• • 下一篇

Modified Adaptive Two-Grid FEMs for Nonsymmetric or Indefinite Elliptic Problems

Fei Li¹, Sha Li¹, Liuqiang Zhong¹, Wanwan Zhu²

1. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, Guangdong, China;
2. School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou, 510520, Guangdong, China

收稿日期:2024-07-04 修回日期:2024-11-19 出版日期:2026-06-20 发布日期:2026-05-29
通讯作者: Wanwan Zhu, Email: zhuww@gdut.edu.cn E-mail:zhuww@gdut.edu.cn
作者简介:Fei Li, Email: feilimath@outlook.com;Sha Li, Email: lisha@m.scnu.edu.cn;Liuqiang Zhong, Email: zhong@scnu.edu.cn
基金资助:
The second and third authors are supported by the National Natural Science Foundation of China (Grant No. 12071160).

Modified Adaptive Two-Grid FEMs for Nonsymmetric or Indefinite Elliptic Problems

Fei Li¹, Sha Li¹, Liuqiang Zhong¹, Wanwan Zhu²

1. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, Guangdong, China;
2. School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou, 510520, Guangdong, China

Received:2024-07-04 Revised:2024-11-19 Online:2026-06-20 Published:2026-05-29
Contact: Wanwan Zhu, Email: zhuww@gdut.edu.cn E-mail:zhuww@gdut.edu.cn

摘要/Abstract

摘要： In this paper, we propose and analyze a modified adaptive two-grid (MATG) finite element method (FEM) for nonsymmetric or indefinite elliptic problems. In this method, we need to solve an extra symmetric positive definite (SPD) residual equation and correct the solution in the previous step before solving the SPD approximate problem. We construct a new residual-based a posteriori error estimator and prove its reliability. We also prove the contraction of the quasi-error and the convergence of the modified algorithm. At last, we report some numerical results to show the effectiveness and robustness of the proposed method.

关键词: Nonsymmetric or indefinite elliptic problems, A posteriori error estimates, Modified adaptive two-grid (MATG) finite element method (FEM), Convergence

Abstract: In this paper, we propose and analyze a modified adaptive two-grid (MATG) finite element method (FEM) for nonsymmetric or indefinite elliptic problems. In this method, we need to solve an extra symmetric positive definite (SPD) residual equation and correct the solution in the previous step before solving the SPD approximate problem. We construct a new residual-based a posteriori error estimator and prove its reliability. We also prove the contraction of the quasi-error and the convergence of the modified algorithm. At last, we report some numerical results to show the effectiveness and robustness of the proposed method.

Key words: Nonsymmetric or indefinite elliptic problems, A posteriori error estimates, Modified adaptive two-grid (MATG) finite element method (FEM), Convergence

中图分类号:

Fei Li, Sha Li, Liuqiang Zhong, Wanwan Zhu. Modified Adaptive Two-Grid FEMs for Nonsymmetric or Indefinite Elliptic Problems[J]. Communications on Applied Mathematics and Computation, 2026, 8(3): 831-850.

参考文献

[1] Adams, R.A.: Sobolev Space. Academic Press, New York (1975)
[2] Babuška, I., Rheinboldt, W.C.: Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15(4), 736-754 (1978)
[3] Bespalov, A., Haberl, A., Praetorius, D.: Adaptive FEM with coarse initial mesh guarantees optimal convergence rates for compactly perturbed elliptic problems. Comput. Methods Appl. Mech. Eng. 317, 318-340 (2017)
[4] Bi, C.J., Wang, C., Lin, Y.P.: A posteriori error estimates of two-grid finite element methods for nonlinear elliptic problems. J. Sci. Comput. 74(1), 23-48 (2018)
[5] Burman, E., Hansbo, P.: Edge stabilization for Galerkin approximations of convection-diffusion-reaction problems. Comput. Methods Appl. Mech. Eng. 193(15/16), 1437-1453 (2004)
[6] Carstensen, C., Ma, R.: Adaptive mixed finite element methods for non-self-adjoint indefinite second-order elliptic PDEs with optimal rates. SIAM J. Numer. Anal. 59(2), 955-982 (2021)
[7] Cascón, J.M., Nochetto, R.H.: Quasioptimal cardinality of AFEM driven by nonresidual estimators. IMA J. Numer. Anal. 32(1), 1-29 (2012)
[8] Chen, H.X., Xu, X.J., Hoppe, R.H.W.: Convergence and quasi-optimality of adaptive nonconforming finite element methods for some nonsymmetric and indefinite problems. Numer. Math. 116, 383-419 (2010)
[9] Dörfler, W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33(3), 1106-1124 (1996)
[10] Feischl, M., Führer, T., Praetorius, D.: Adaptive FEM with optimal convergence rates for a certain class of non-symmetric and possibly non-linear problems. SIAM J. Math. Anal. 52(2), 601-625 (2014)
[11] Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001)
[12] Grisvard, P.: Singularities in Boundary Value Problems. Springer, Berlin (1992)
[13] Hackbusch, W.: Elliptic Differential Equations Theory and Numerical Treatment, 2nd edn. Springer, Berlin (2017)
[14] Lazarov, R., Tomov, S.: A posteriori error estimates for finite volume element approximations of convection-diffusion-reaction equations. Comput. Geosci. 6(3), 483-503 (2002)
[15] Li, Y.K., Zhang, Y.: Analysis of adaptive two-grid finite element algorithms for linear and nonlinear problems. SIAM J. Sci. Comput. 43(2), A908-A928 (2021)
[16] Mekchay, K., Nochetto, R.H.: Convergence of adaptive finite element methods for general second order linear elliptic PDEs. SIAM J. Numer. Anal. 43(5), 1803-1827 (2005)
[17] Mitchell, W.F.: A collection of 2D elliptic problems for testing adaptive grid refinement algorithms. Appl. Math. Comput. 220, 350-364 (2013)
[18] Schatz, A., Wang, J.P.: Some new error estimates for Ritz-Galerkin methods with minimal regularity assumptions. Math. Comput. 65(213), 19-27 (1996)
[19] Scott, L.R., Zhang, S.Y.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54(190), 483-493 (1990)
[20] Stevenson, R.: The completion of locally refined simplicial partitions created by bisection. Math. Comput. 77(261), 227-241 (2008)
[21] Tang, M., Xing, X.Q., Yang, Y., Zhong, L.Q.: Iterative two-level algorithm for nonsymmetric or indefinite elliptic problems. Appl. Math. Lett. 140, 108594 (2023)
[22] Verfürth, R.: A Review of a Posteriori Error Estimation and Adaptive Mesh Refinement Techniques. Wiley, Hoboken (1996)
[23] Verfürth, R.: A Posteriori Error Estimation Techniques for Finite Element Methods. Oxford University Press, Oxford (2013)
[24] Xu, J.C.: Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal. 33(5), 1759-1777 (1996)

Modified Adaptive Two-Grid FEMs for Nonsymmetric or Indefinite Elliptic Problems

Modified Adaptive Two-Grid FEMs for Nonsymmetric or Indefinite Elliptic Problems

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

编辑推荐

Metrics

本文评价

[1]	Hongying Man, Shangyou Zhang. On the Superconvergence of a Conforming Mixed Finite Element for Linear Elasticity on Uniform n-Square Grids[J]. Communications on Applied Mathematics and Computation, 2026, 8(3): 851-867.
[2]	Reetika Chawla, Devendra Kumar, Haitao Qi. A Numerical Technique to Solve Time-Fractional Delay Diffusion Wave Equation via Trigonometric Collocation Approach[J]. Communications on Applied Mathematics and Computation, 2026, 8(3): 909-929.
[3]	B. Yu. Irgashev. Robin Type Problem for a Degenerate Equation of Even Order with Caputo Fractional Derivative[J]. Communications on Applied Mathematics and Computation, 2026, 8(3): 930-952.
[4]	Zhengge Huang. New Gradient-Based Iterative-Like Algorithms for Solving a Class of Sylvester Tensor Equations[J]. Communications on Applied Mathematics and Computation, 2026, 8(3): 1005-1049.
[5]	Jinhua Feng, Yanjun Li, Hai Bi, Yidu Yang. An Adaptive Conforming Mixed Method for the Modified Transmission Eigenvalue Problem for Absorbing Medium[J]. Communications on Applied Mathematics and Computation, 2026, 8(3): 1095-1114.
[6]	Bei-Bei Li, Jing-Jing Cui, Zheng-Ge Huang, Xiao-Feng Xie. Two Quasi-combining Real and Imaginary Parts Iteration Methods for Solving Complex Symmetric System of Linear Equations[J]. Communications on Applied Mathematics and Computation, 2026, 8(2): 427-455.
[7]	Lisha Chen, Zhibo Wang. A Second-Order Scheme with Nonuniform Time Grids for the Two-Dimensional Time-Fractional Zakharov-Kuznetsov Equation[J]. Communications on Applied Mathematics and Computation, 2026, 8(2): 456-471.
[8]	Yongjun Chen, Liping Zhang. Linear Convergence of the Collatz Method for Computing the Perron Eigenpair of a Primitive Dual Number Matrix[J]. Communications on Applied Mathematics and Computation, 2026, 8(1): 232-247.
[9]	Chunxiu Liu, Junying Cao, Tong Lyu, Xingyang Ye. Numerical Analysis of a High-Order Scheme with Nonuniform Time Grids for Caputo-Hadamard Fractional Reaction Sub-diffusion Equations[J]. Communications on Applied Mathematics and Computation, 2026, 8(1): 338-365.
[10]	Qin-Qin Shen, Geng-Chen Yang, Chen-Can Zhou. Convergence Analysis of the Projected SOR Iteration Method for Horizontal Linear Complementarity Problems[J]. Communications on Applied Mathematics and Computation, 2025, 7(5): 1617-1638.
[11]	Xu Li, Jian-Sheng Feng. Parameterized QHSS Iteration Method and Its Variants for Non-Hermitian Positive Definite Linear Systems of Strong Skew-Hermitian Parts[J]. Communications on Applied Mathematics and Computation, 2025, 7(5): 1665-1683.
[12]	Jin-Feng Mao, Fang Chen. On Multi-step Partially Randomized Extended Kaczmarz Method for Solving Large Sparse Inconsistent Linear Systems[J]. Communications on Applied Mathematics and Computation, 2025, 7(5): 1724-1743.
[13]	Lu-Xin Wang, Yang Cao, Qin-Qin Shen. Two Variants of Robust Two-Step Modulus-Based Matrix Splitting Iteration Methods for Mixed-Cell-Height Circuit Legalization Problem[J]. Communications on Applied Mathematics and Computation, 2025, 7(5): 1769-1790.
[14]	Yan-Xia Dai, Ren-Yi Yan, Ai-Li Yang. Minimum Residual BAS Iteration Method for Solving the System of Absolute Value Equations[J]. Communications on Applied Mathematics and Computation, 2025, 7(5): 1815-1825.
[15]	Rong Ma, Yu-Jiang Wu, Lun-Ji Song. Modified Newton-PAGSOR Method for Solving Nonlinear Systems with Complex Symmetric Jacobian Matrices[J]. Communications on Applied Mathematics and Computation, 2025, 7(5): 1880-1906.