Stopping criteria for Krylov methods and finite-element approximation of variational problems

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 beensuccessfully used in the solution of the symmetric and positive definite systems obtained from thefinite-element approximation of self-adjoint elliptic partial differential equations. Taking into accountrecent results [5,6,7] which make it possible to approximate the energy norm of the error during theconjugate gradient iterative process, in [1] we introduce a stopping criterion based on an energy normand a dual space norm linked to the continuous problem. Moreover, we show that the use of efficientpreconditioners 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-definiteproblems. We show that the residual measured in the norm induced by the symmetric part of the inverseof the system matrix is relevant to measuring convergence in a finite element context. We then providealternative 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 easilyextended 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 methodframework. Numer. Math. 97 (2004), pp 1-24. [2] M. Arioli, I. S. Duff, D. Ruiz, and M. Sadkane. Block Lanczos Techniques for Accelerating theBlock 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 Algebraand 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. Strakoš and P. Tichý. On error estimation by conjugate gradient method and why it works infinite precision computations. Electronic Transactions on Numerical Analysis, 13 (2002), pp. 56–80. [7] Z. Strakoš and P. Tichý. Error estimation in preconditioned conjugate gradients. To appear on BIT.