Kamel Slimani

A posteriori estimate and asymptotic partial domain decomposition

The method of asymptotic partial decomposition of a domain aims at replacing a 3D or 2D problem by a hybrid problem 3D − 1D; or 2D − 1D, where the dimension of the problem decreases in part of the domain. The location of the junction between the heterogeneous problems is asymptotically estimated in certain circumstances, but for numerical simulations it is important to be able to determine the location of the junction accurately. In this article, by reformulating the problem in a mixed formulation context and by using an a posteriori error estimate, we propose an indicator of the error due to a wrong position of the junction. Minimizing this indicator allows us to determine accurately the location of the junction. Some numerical results are presented for a toy problem.

Asymptotic domain decomposition and a posteriori estimates for a semi linear problem.

Keywords: Method of asymptotic partial domain decomposition; a posteriori error estimates; Indicator of the error; semi linear elliptic equations. 2010 Mathematics Subject Classification: 35F40; 65 1 Introduction The method of asymptotic partial decomposition of a domain (MAPDD) originates with the works of G.Panasenko Panasenko (2005). The idea is to replace an original 3D or 2D problem by an hybrid one 3D − 1D; or 2D − 1D where the dimension of the problem decreases in part of the domain. The location of the junction between the heterogenous problems is asymptotically estimated in the works of G.Panasenko mainly for linear problems. Nevertheless for numerical simulations it is essential to detect with accuracy the location of the junction. Let us also mention the interest of locating with accuracy the position of the junction in blood flows simulations when different nonlinear mathematical models are used Quarteroni and Veneziani (2003), or in fluid/solid problems for which subproblems *Corresponding author: Tel:+91-9741148002, E-mail: jerome.pousin@insa-lyon.fr Research Article Coupling heterogeneous mathematical models is today commonly used, and effective solution methods for the resulting hybrid problem have recently become available for several systems. Even if in certain circumstances, asymptotic evaluations of the location of the interfaces are available, no strategy are proposed for locating the interfaces in numerical simulations. In this article, a semi- linear elliptic problem is considered. By reformulating the problem in a mixed formulation context and by using an a posteriori error estimate, we propose an indicator of the error due to a wrong position of the junction. Minimizing this indicator allows us to determine accurately the location of the junction. By comparing this indicator with a mesh error indicator, this allows to decide if it is better to refine the mesh or to move the interface. Some numerical results are presented showing the efficiency of the proposed indicator.