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

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.

Keywords

Nonsymmetric or indefinite elliptic problems / A posteriori error estimates / Modified adaptive two-grid (MATG) finite element method (FEM) / Convergence / 65N12 / 65N30 / 65N50

References

[1] Adams RASobolev Space, 1975New YorkAcademic Press

[2] Babuška I, Rheinboldt WC. Error estimates for adaptive finite element computations. SIAM J. Numer. Anal., 1978, 15(4): 736-754

[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., 2017, 317: 318-340

[4] Bi CJ, Wang C, Lin YP. A posteriori error estimates of two-grid finite element methods for nonlinear elliptic problems. J. Sci. Comput., 2018, 74(1): 23-48

[5] Burman E, Hansbo P. Edge stabilization for Galerkin approximations of convection-diffusion-reaction problems. Comput. Methods Appl. Mech. Eng., 2004, 193(15/16): 1437-1453

[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., 2021, 59(2): 955-982

[7] Cascón JM, Nochetto RH. Quasioptimal cardinality of AFEM driven by nonresidual estimators. IMA J. Numer. Anal., 2012, 32(1): 1-29

[8] Chen HX, Xu XJ, Hoppe RHW. Convergence and quasi-optimality of adaptive nonconforming finite element methods for some nonsymmetric and indefinite problems. Numer. Math., 2010, 116: 383-419

[9] Dörfler W. A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal., 1996, 33(3): 1106-1124

[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., 2014, 52(2): 601-625

[11] Gilbarg D, Trudinger NSElliptic Partial Differential Equations of Second Order, 2001BerlinSpringer

[12] Grisvard PSingularities in Boundary Value Problems, 1992BerlinSpringer

[13] Hackbusch WElliptic Differential Equations Theory and Numerical Treatment, 20172BerlinSpringer

[14] Lazarov R, Tomov S. A posteriori error estimates for finite volume element approximations of convection-diffusion-reaction equations. Comput. Geosci., 2002, 6(3): 483-503

[15] Li YK, Zhang Y. Analysis of adaptive two-grid finite element algorithms for linear and nonlinear problems. SIAM J. Sci. Comput., 2021, 43(2): A908-A928

[16] Mekchay K, Nochetto RH. Convergence of adaptive finite element methods for general second order linear elliptic PDEs. SIAM J. Numer. Anal., 2005, 43(5): 1803-1827

[17] Mitchell WF. A collection of 2D elliptic problems for testing adaptive grid refinement algorithms. Appl. Math. Comput., 2013, 220: 350-364

[18] Schatz A, Wang JP. Some new error estimates for Ritz-Galerkin methods with minimal regularity assumptions. Math. Comput., 1996, 65(213): 19-27

[19] Scott LR, Zhang SY. Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput., 1990, 54(190): 483-493

[20] Stevenson R. The completion of locally refined simplicial partitions created by bisection. Math. Comput., 2008, 77(261): 227-241

[21] Tang M, Xing XQ, Yang Y, Zhong LQ. Iterative two-level algorithm for nonsymmetric or indefinite elliptic problems. Appl. Math. Lett., 2023, 140 Article ID: 108594

[22] Verfürth RA Review of a Posteriori Error Estimation and Adaptive Mesh Refinement Techniques, 1996HobokenWiley

[23] Verfürth RA Posteriori Error Estimation Techniques for Finite Element Methods, 2013OxfordOxford University Press

[24] Xu JC. Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal., 1996, 33(5): 1759-1777

Funding

National Natural Science Foundation of China(12071160)

RIGHTS & PERMISSIONS

Shanghai University