A Posteriori Error Estimates Including Algebraic Error and Stopping Criteria for Iterative Solvers

Abstract

Abstract. We consider the finite volume and the lowest-order mixed finite element discretizations of a second-order elliptic pure diffusion model problem. The first goal of this paper is to derive guaranteed and fully computable a posteriori error estimates which take into account an inexact solution of the associated linear algebraic system. We show that the algebraic error can be simply bounded using the algebraic residual vector. Much better results are, however, obtained using the complementary energy of an equilibrated Raviart-Thomas-Nédélec discrete vector field whose divergence is given by a proper weighting of the residual vector. The second goal of this paper is to construct efficient stopping criteria for iterative solvers such as the conjugate gradients, GMRES, or Bi-CGStab. We claim that the discretization error, implied by the given numerical method, and the algebraic one should be in balance, or, more precisely, that it is enough to solve the linear algebraic system to the accuracy which guarantees that the algebraic part of the error does not contribute significantly to the whole error. Our estimates allow a reliable and cheap comparison of the discretization and algebraic errors. One can thus use them to stop the iterative algebraic solver at the desired accuracy level, without performing an excessive number of unnecessary additional iterations. Under the assumption of the relative balance between the two errors, we also prove the efficiency of our a posteriori estimates, i.e., we show that they also represent a lower bound, up to a generic constant, for the overall energy error. A local version of this result is also stated. Several numerical experiments illustrate the theoretical results.Key words. Second-order elliptic partial differential equation, finite volume method, mixed finite element method, a posteriori error estimates, iterative methods for linear algebraic systems, stopping criteria.

Guaranteed and robust a posteriori error estimates and balancing discretization and linearization errors for monotone nonlinear problems

“…The present approach can also be combined with a linear iterative solver, and to further save computational effort, the latter can be stopped whenever the algebraic error does not contribute significantly to the overall error, following [21]. This is a two-dimensional extension of a test case from [11].…”

“…Therefore, the second objective of this work is to design a posteriori error estimates distinguishing linearization and discretization errors in the context of an adaptive procedure. This type of analysis has been started by Chaillou and Suri [11,12] for a certain class of nonlinear problems similar to the present one and in the context of iterative solution of linear algebraic systems in [21]. Chaillou and Suri only considered a fixed stage of the linearization process, while we take here the analysis one step further in the context of an iterative loop.…”

Four closely related equilibrated flux reconstructions for nonconforming finite elements

“…Moreover, such an effort is unnecessary in view of the unavoidable presence of the discretization error ∇(u − u h ) . Guaranteed a posteriori error estimates not requiring (2) or (3) and distinguishing the discretization and algebraic errors are now available, see [8,6] and the references therein, and we present them here in the Crouzeix-Raviart context.…”

“…Note that this complementary energy minimization problem locally minimizes the size of the first estimator in (4). Let RTN N,0 0 (S De ) be defined as (8), but with the normal flux condition v h ·n De | ∂De\∂Ω = 0. Finally, let P * 0 (S De ) be spanned by piecewise constants on S De with zero mean value on the dual cell D e when e ∈ E int h and by constants when e ∈ E ext h .…”

Combineda posteriorimodeling-discretization error estimate for elliptic problems with complicated interfaces

“…Also, we refer to [17], where a closely related approach has been presented. An approach, which is based on the Prager-Synge hypercircle method, is presented in [10,11].…”