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

doi:10.1007/s42967-024-00475-x

Abstract

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

CLC Number:

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.

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References

[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)