Adjoint Based A Posteriori Analysis of Multiscale Mortar Discretizations with Multinumerics

Abstract

In this paper we derive a posteriori error estimates for linear functionals of the solution to an elliptic problem discretized using a multiscale nonoverlapping domain decomposition method. The error estimates are based on the solution of an appropriately defined adjoint problem. We present a general framework that allows us to consider both primal and mixed formulations of the forward and adjoint problems within each subdomain. The primal subdomains are discretized using either an interior penalty discontinuous Galerkin method or a continuous Galerkin method with weakly imposed Dirichlet conditions. The mixed subdomains are discretized using RaviartThomas mixed finite elements. The a posteriori error estimate also accounts for the errors due to adjoint-inconsistent subdomain discretizations. The coupling between the subdomain discretizations is achieved via a mortar space. We show that the numerical discretization error can be broken down into subdomain and mortar components which may be used to drive adaptive refinement.

An adaptive multiscale hybrid-mixed method for the Oseen equations

“…Other members of this family are the Heterogeneous Multiscale method (HMM) [27], the Variational Multiscale method (VMS) [3], the Generalized Multiscale finite element method [28], the Localized Orthogonal Decomposition method (LOD) [40], the Petrov-Galerkin Enriched method (PGEM) [7,16,36], the Residual Local Projection method (RELP) [5,17,34], to mention a few. A posteriori error estimator for some of these schemes can be reviewed in [1,10,14,21,41,44,47,49,53], and the references therein.…”

The Prager–Synge theorem in reconstruction based a posteriori error estimation

“…The Stokes problem and two phase fluid-flow have been considered in [HSV12,DPVY15]. An intense activity is related to multiscale and mortar elements [PVWW13,TW13] as well as to porous media and porous elasticity [MN17, RDPE + 17, VY18]. Obstacle and contact problems have been studied in [BHS08,WW10,HW12].…”

On a multiscalea posteriorierror estimator for the stokes and Brinkman equations

“…Since, several variants of it have been proposed for fluid flow problems as the Heterogeneous Multiscale method (HMM) [38], the Variational Multiscale method (VMS) [29], the Generalized Multiscale finite element method [18], and the Localized Orthogonal Decomposition method (LOD) [32], to mention a few. For some of those methods a posteriori error analysis is provided, see for instance [9,30,1,33,14,27,36,34] and the references therein.…”