Guaranteed and computable bounds of approximation errors for the semi-discrete Biot problem

Dr. Svetlana Matculevich

Oct. 2, 2018, 1:30 p.m. S2 054

The talk is concerned with guaranteed and fully computable a posteriori error estimates for evolutionary problems associated with the poroelastic media governed by the quasi-static linear Biot equations [1]. It addresses the question of approximation error control, which arises in the iterative and monolithic approaches used for semi-discrete approximations obtained by the implicit Euler time-discretization scheme. The derivation of the error bounds is based on a combination of the Ostrowski-type estimates [2] derived for iterative schemes and a posteriori error estimates of the functional type for elliptic problems originally (also called error majorants and minorants) introduced in [3]. The validity of the first estimates is based on the contraction property of the fixed stress splitting scheme [4] used for decoupling.

The talk is based on the report [5].

[1] M.A. Biot. Theory of elasticity and consolidation for a porous anisotropic solid. Journal of applied physics, 26:182–185, 195.
[2] A. Ostrowski. Les estimations des erreurs a posteriori dans les procédés itératifs. C.R. Acad.Sci. Paris Sér. A–B, 275:A275–A278, 1972.
[3] S. Repin. A posteriori error estimation for variational problems with uniformly convex functionals. Math. Comput., 69(230):481–500, 2000.
[4] A. Mikeli´c and M. F. Wheeler. Convergence of iterative coupling for coupled flow and geomechanics. Comput. Geosci., 17(3):455–461, 2013.
[5] Kundan Kumar, Svetlana Matculevich, Jan Nordbotten, and Sergey Repin. Guaranteed and computable bounds of approximation errors for the semi-discrete Biot problem. arXiv[math.NA]:1808.08036, 2018.