Adaptive inexact Newton methods and their application to two-phase flows

We consider nonlinear algebraic systems resulting from numerical discretizations of nonlinear partial differential equations. To solve these sys-tems, some iterative nonlinear solver, and, on each step of this solver, some iterative linear solver are used. We derive adaptive stopping criteria for both iterative solvers. Our criteria are based on an a posteriori error estimate which distinguishes the different error components, namely the discretization error, the linearization error, and the algebraic error. We stop the iterations whenever the corresponding error does no longer affect the overall error sig-nificantly. Our estimates also yield a guaranteed upper bound on the overallerror at each step of the nonlinear and linear solvers. We prove the (local) efficiency and robustness of the estimates with respect to the size of the non-linearity owing, in particular, to the error measure involving the dual normof the residual. Our developments hinge on equilibrated flux reconstructions and yield a general framework.
We show how to apply this framework to vari-ous discretization schemes like finite elements, nonconforming finite elements,discontinuous Galerkin, finite volumes, and mixed finite elements; to different linearizations like fixed point and Newton; and to arbitrary iterative linear solvers. Numerical experiments for the p-Laplacian illustrate the tight overall error control and important computational savings achieved in our approach.
In the second part of the talk, we present extensions to the two-phase flow in porous media, where again all error components (spatial, temporal, lineariza-tion, algebraic) are distinguished and fully adaptive algorithms are designed and numerically tested. Details can be found in [1, 2, 3].
References
[1] A. Ern and M. Vohral´?k, Adaptive inexact Newton methods: a posteriori error control and speedup of calculations, SIAM News, 46 (2013), pp. 1,4.
[2] , Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs, SIAM J. Sci. Comput., 35 (2013), pp. A1761–A1791.
[3] M. Vohral´?k and M. F. Wheeler, A posteriori error estimates, stopping criteria, and adaptivity for two-phase flows, Comput. Geosci., 17 (2013), pp. 789–812.
Key words and phrases. nonlinear PDE, a posteriori error estimate, adaptive linearization,
adaptive algebraic solution, adaptive mesh refinement, stopping criterion, two-phase flow.

CONTACT: luca.formaggia@polimi.it