Stopping criteria for Krylov methods and finite-element approximation of
variational problems
Mario Arioli∗and Daniel Loghin
Rutherford Appleton Laboratory Chilton, Didcot, Oxfordshire, OX11 0QX, UK
e-mail: m.arioli@rl.ac.uk
web page: http://www.numerical.rl.ac.uk/people/marioli/marioli.html
ABSTRACT
We combine linear algebra techniques with finite element techniques to obtain a reliable stopping cri-terion for Krylov method based algorithms. The Conjugate Gradient method has for a long time been successfully used in the solution of the symmetric and positive definite systems obtained from the finite-element approximation of self-adjoint elliptic partial differential equations. Taking into account recent results [5,6,7] which make it possible to approximate the energy norm of the error during the conjugate gradient iterative process, in [1] we introduce a stopping criterion based on an energy norm and a dual space norm linked to the continuous problem. Moreover, we show that the use of efficient preconditioners does not require us to change the energy norm used by the stopping criterion.
In [3], we extend the previous results on stopping criteria to the case of nonsymmetric positive-definite problems. We show that the residual measured in the norm induced by the symmetric part of the inverse of the system matrix is relevant to measuring convergence in a finite element context. We then provide alternative ways of calculating or estimating this quantity.
Finally, we extend the results of [1] to the Block Conjugate Gradient (BCG) algorithm [2,4]. In partic-ular, we show that the simple rule proposed in [1] for computing the stopping criterion can be easily extended to the BCG algorithm with a cheap cost proportional to the square of the block size.
REFERENCES
[1] M. Arioli. A stopping criterion for the conjugate gradient algorithm in a finite element method framework. Numer. Math. 97 (2004), pp 1-24.
[2] M. Arioli, I. S. Duff, D. Ruiz, and M. Sadkane. Block Lanczos Techniques for Accelerating the Block Cimmino Method. SIAM Journal of Scientific Computing 16 (1995), pp 1478–1511. [3] M. Arioli, D. Loghin, and A. Wathen. Stopping criteria for iterations in finite-element methods.
Numer. Math. 99 (2005), pp 381-410.
[4] D. P. O’Leary. The Black Conjugate Gradient Algorithm and Related Methods. Linear Algebra and its Appl. 29 (1980), pp 293–322.
[5] G. Meurant. Numerical experiments in computing bounds for the norm of the error in the precon-ditioned conjugate gradient algorithm. Numerical Algorithms, 22 (1999), pp. 353–365.
[6] Z. Strakoˇs and P. Tich´y. On error estimation by conjugate gradient method and why it works in finite precision computations. Electronic Transactions on Numerical Analysis, 13 (2002), pp. 56– 80.
