Summary
The Conjugate Gradient method has always been successfully used in solving the symmetric and positive definite systems obtained by the finite element approximation of self-adjoint elliptic partial differential equations. Taking into account recent results [13, 19, 20, 22] which make it possible to approximate the energy norm of the error during the conjugate gradient iterative process, we adapt the stopping criterion introduced in [3]. Moreover, we show that the use of efficient preconditioners does not require to change the energy norm used by the stopping criterion. Finally, we present the results of several numerical tests that experimentally validate the effectiveness of our stopping criterion.
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Arioli, M., Baldini, L.: A backward error analysis of a null space algorithm in sparse quadratic programming. SIAM J. Matrix Anal. Appl. 23, 425–442 (2001)
Arioli, M., Duff, I.S., Ruiz, D.: Stopping criteria for iterative solvers. SIAM J. Matrix Anal. Appl. 13, 138–144 (1992)
Arioli, M., Noulard, E., Russo, A.: Stopping criteria for iterative methods: Applications to PDE’s. CALCOLO 38, 97–112 (2001)
Ashby, S.F., Holst, M.J., Manteuffel, T.A., Saylor, P.E.: The role of the inner product in stopping criteria for conjugate gradient iterations. BIT 41, 26–52 (2001)
Axelsson, O., Kaporin, I.: Error norm estimation and stopping criteria in preconditioned conjugate gradient iterations. J. Numerical Linear Algebra with Applications 8, 265–286 (2001)
Babuska, I.: Error-bounds for finite element method. Numer. Math. 16, 322–333 (1971)
Babuska, I.: A finite element scheme for domains with corners. Numer. Math. 20, 1–21 (1972)
Babuska, I., Strouboulis, T., Upadhyay, C.S., Gangaraj, S.K.: Computer-based proof of the existence of superconvergence points in the finite element method; superconvergence of the derivatives in finite element solutions of Laplace’s, Poisson’s, and the elasticity equations. Numer. Methods Partial Diff Equations 21, 643–665 (2001)
Calvetti, D., Morigi, S., Reichel, L., Sgallari, F.: Computable error bounds and estimates for the conjugate gradient method. Numerical Algorithms 25, 75–88 (2000)
Calvetti, D., Morigi, S., Reichel, L., Sgallari, F.: An iterativa method with error estimators, J. Comp. Appl. Math. 127, 93–119 (2001)
Ciarlet, P.: The finite element method for elliptic problems. North-Holland, Amsterdam, The Netherlands, (1978)
Dupont, T., Scott, R.: Polynomial approximation of function in sobolev spaces. Math. Comp. 34, 441–463 (1980)
Golub, G., Meurant, G.: Matrices, moments and quadrature II; how to compute the norm of the error in iterative methods. BIT 37, 687–705 (1997)
Golub, G., Strakoš, Z.: Estimates in quadratic formulas. Numerical Algorithms 8, 241–268 (1994)
Greenbaum, A.: Iterative Methods for Solving Linear Systems, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, (1997)
Grisvard, P.: Singularity in Boundary Value problems. Masson & Springer-Verlag, Paris, France, (1992)
Hestenes, M., Stiefel, E.: Methods of conjugate gradients for solving linear systems. J. Res. Nat. Bur. Standards 49, 409–436 (1952)
Higham, N.J.: Accuracy and Stability of Numerical Algorithms. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, (1996)
Meurant, G.: The computation of bounds for the norm of the error in the conjugate gradient algorithm. Numerical Algorithms, 16 (1997), pp. 77–87.
Meurant, G.: Computer solution of large linear systems, 28 of Studies in Mathematics and its Application, Elsevier/North-Holland, Amsterdam, The Netherlands, (1999)
Meurant, G.: Numerical experiments in computing bounds for the norm of the error in the preconditioned conjugate gradient algorithm, Numerical Algorithms, 22, 353–365 (1999)
Strakoš, Z., Tichý, P.: On error estimation by conjugate gradient method and why it works in finite precision computations. Electronic Transactions on Numerical Analysis, 13, 56–80 (2002)
Wahlbin, L.B.: Superconvergence in Galerkin finite element methods. vol. 1605 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, (1995)
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Mathematics Subject Classification (2000):65F10, 65N30, 65F50
28 April 2001
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Arioli, M. A stopping criterion for the conjugate gradient algorithm in a finite element method framework. Numer. Math. 97, 1–24 (2004). https://doi.org/10.1007/s00211-003-0500-y
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DOI: https://doi.org/10.1007/s00211-003-0500-y